research article log-aesthetic curves for shape completion...
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Research ArticleLog-Aesthetic Curves for Shape Completion Problem
R U Gobithaasan1 Yip Siew Wei1 and Kenjiro T Miura2
1 School of Informatics amp Applied Mathematics Universiti Malaysia Terengganu 21030 Kuala Terengganu Terengganu Malaysia2 Graduate School of Engineering Shizuoka University Shizuoka 432-8561 Japan
Correspondence should be addressed to R U Gobithaasan grumtedumy
Received 20 April 2014 Revised 8 July 2014 Accepted 11 July 2014 Published 11 August 2014
Academic Editor Juan Manuel Pena
Copyright copy 2014 R U Gobithaasan et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
An object with complete boundary or silhouette is essential in various design and computer graphics feats Due to various reasonssome parts of the object can bemissing hence increasing the complexity in designing process It is therefore important to reconstructthe missing parts of an object while retaining its aesthetic appearance In this paper we propose Log-Aestheic Curves (LAC) forshape completion problem We propose an algorithm to construct LAC segment and subsequently fit into the gap of the missingparts with C-shape or S-shape For C-shape completion we define LAC segment by specifying two endpoints and their respectivetangent directions between the gaps while for S-shape the user defines an inflection point in between the endpoints The finalsection illustrates three examples to showcase the efficiency of the proposed algorithm The results are further compared withKimiarsquos method to prove that the algorithm produces equally good result Additionally the proposed algorithm provides an extradegree of freedom in which the user would be able to choose the type of spiral that they desire to solve the shape completionproblem
1 Introduction
In a design environment there are situations in which theoutlines of an object are missing or occludedThese scenariosare commonly known as shape completion gap completionor curve completion problem It is important to recompletethe shape of the missing object while retaining its aestheticappearance since a complete object can be used further forapplications inmodeling mesh analysis and computer-aideddesign as well as manufacturing purposes The completeboundary or silhouette provides geometrical information ofa model which is significant in engineering analysis andmanufacturing process In order to reconstruct the missingboundaries of an object the associated curve segment mustsatisfy the boundary conditions so that the curve passesthrough the endpoints with specified tangent direction Todate there are numbers of algorithms available for shapecompletion problem (see the works of [1ndash6])
A notable work which received much attention is byKimia et al [3] where they proposed an algorithm whichutilizes Euler spiral to solve shape completion problem
Euler spiral is also known as Cornursquos spiral or clothoidThe Euler spiral is widely applied in highwayrailway orrobotic trajectories design due to its linear curvature property(curvature varies linearly with respect to arc length) Thealgorithm proposed by Kimia et al [3] is the implementationof Euler spiral to solve the shape completion problem Intheir proposed algorithm two endpoints (119875
0 1198752) and their
tangential angles (1205790 1205792) are the inputs In order to construct
Euler spiral for the given problem curvature and arc lengthof the curve are required to be identified Since the proposedEuler spiral algorithm is an iterative process initial guess forthese two parameters (curvature and arc length) is neededKimia et al [3] recommended the use of biarc to obtainthe initial curve with its corresponding curvature and arclength for the proposed algorithm Note that the goal of theirproposed algorithm is to minimalize the distance betweenlast point and resulting curve in order for the gap to be filledusing gradient descent method
Log-Aesthetic Curves (LAC) are curves where the log-arithmic curvature graph (LCG) can be represented bya straight line The idea of LCG was formulated by
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 960302 10 pageshttpdxdoiorg1011552014960302
2 Journal of Applied Mathematics
05 10 15 20 25
05
10
15
20
X
Y
(a) Two segments of Euler spiral
00 05 10 15
00
05
10
15
20
minus05minus10 minus05
log[120588(t)s998400 (t)120588
998400 (t)]
log[120588(t)]
(b) Its corresponding logarithmic curvature graph
Figure 1 Euler spiral defined 119904 isin [02 1] (solid line) and 119904 isin [1 3] (dotted) and its corresponding LCG
Harada et al (1999) when they attempted to classify all thecurves used in automobile design It was first known aslogarithmic distribution diagram of curvature (LDDC) Itis a graph to measure the relationship between curvatureradius (denoted by 120588) and arc length of the curve segment(denoted by 119904) They found that all the curves employed inautomobile design depict linear gradient of LDDC Kanaya etal [7] refined the formulation of LDDC which is now knownas LCG The horizontal axis of LCG represents log 120588 and thevertical axis represents log 120588(119889119904
10158401198891205881015840) [8] Figure 1 illustrates
the two segments of Euler spiralIn 2005 Miura et al [9] derived a fundamental formula
of LAC based on a linear representation of LCG A yearlater Yoshida and Saito [10] investigated the characteristicsand overall shape of LAC They also proposed a method toconstruct LAC segment interactively by specifying endpointsand their tangent vectors In 2009 they further extendedLAC into space curve and proposed amethod to interactivelycontrol the curves which is similar to quadratic Beziercurves
LAC has become a promising curve in graphic andindustrial design due to its monotonic curvature propertyThere are two shape parameters of LAC denoted by 120572 andΛ The shape parameter 120572 is used to determine the type ofspiral within the family of LAC and Λ is used to satisfythe 1198661 constraint during the design process LAC comprises
Euler spiral (120572 = minus1) Nielsen spiral (120572 = 0) logarithmic orequiangular spiral (120572 = 1) circle involute (120572 = 2) and others
In 2011 Gobithaasan and Miura [11] expanded the familyof LAC into Generalized Log-Aesthetic Curve (GLAC) byadding generalized Cornu spiral (GCS) to the family of LACThis includes Euler spiral logarithmic spiral circle involuteNielsen spiral LAC and GCS as the members of GLAC
To note GCS is proposed by Ali et al [12] where it becomesEuler spiral when the shape parameter of GCS is zero GLAChas an extra shape parameter compared to LAC in which itcan be used to satisfy additional constraints that occur duringdesign process [13] The curvature function of GLAC is alsomonotonic similar to LAC In 2012 they further extendedGLAC into space curve
In 2012 Ziatdinov et al [14] defined LAC in the formof incomplete gamma functions They claimed that this newexpression of LAC is capable of reducing the computationtime up to 13 times faster In 2013 the problem of1198662 Hermiteinterpolation was dealt with by Miura and Gobithaasan [15]by introducing log-aesthetic splines which consists of tripleLAC segments Readers are referred to [16] for a detailedreview on aesthetic curves
In this paper we introduce the implementation of 1198661
LAC algorithm to fit the gap in between the outlines ofmissingocclusion objects We compare the performance ofthis algorithm with the results obtained by Kimia et alrsquos[3] method on the same examples in order to show theeffectiveness of LAC to solve shape completion problem
The rest of this paper is organized as follows First the for-mulation of LAC is reviewed followed by a detailed explana-tion on the LAC algorithm for solving the shape completionproblem In Section 3 we demonstrate the implementationof the proposed algorithm with various 120572 values with threeexamples to show the implementation of this method Thedata and calculation used in the examples are also depictedin this sectionThe output of shape completion using Kimiarsquosmethod is also depicted for comparison purposes We alsomake comparison of the results obtained using Kimia et alrsquos[3] method and our algorithm We further show some shapevariations which can be obtained by using S-shape LAC in
Journal of Applied Mathematics 3
Pc
Pa
Pb
120579f
120579d
(a) Original position of controlpoints
P0
P1P2
120579d998400120579f998400
(b) Position of control points after transformation
Figure 2 The configuration of triangle consists of three control points for case 120572 gt 1 The curve inside the triangle represents the generatedLAC segment
which the designer tweaks the segment in order to obtain thedesired shape
2 Log-Aesthetic Curves in Shape Completion
In this section we show the parametric expression of LACand the algorithm used to obtain the solution for shapecompletion problem with various 120572 values
21 The Analytical Formulas and Parameters There are twoshape parameters used to define LAC segment 120572 and Λ Theshape parameter 120572 determines the type of spiral andΛ is usedto control the curve so that the constraints are satisfied Let 120588and 120579 represent radius of curvature and tangent angle of thecurve respectively
Equation (1) shows the radius of curvature function interms of its turning angle To note (1) is used to define theparametric expression of LAC in terms of its turning angle asshown in (2) as follows
120588 (120579) =
119890Λ120579
if 120572 = 1
((120572 minus 1) Λ120579 + 1)1(120572minus1)
otherwise(1)
LAC (120579) = 119875119886+ int
120579
0
119890119894119906120588 (119906) 119889119906 (2)
where 119875119886represents starting point and 119894 and 119890 represent the
imaginary unit and exponent respectively LAC can also berepresented in the arc length parameter 119904 as follows
LAC (119904) =
119875119886+ int
119904
0
119890119894((1Λ)(1minus119890
minusΛ119906
))119889119906 if 120572 = 0
119875119886+ int
119904
0
119890119894(log[Λ119906+1]Λ)
119889119906 if 120572 = 1
119875119886+ int
119904
0
119890119894(((Λ120572119906+1)
(120572minus1)120572
minus1)Λ(120572minus1))119889119906 otherwise
(3)
Both (2) and (3) generate the same LAC The curve canbe drawn in complex plane by the 119909-axis representing realnumbers and the 119910-axis representing imaginary numbersReaders are referred to [10] for a detailed derivation of theparametric expression of LAC
22 The LAC Algorithm
C-Shape We customize the algorithm of constructing LACsegment to fill the gap by specifying two endpoints its tangentvector and120572 valueTheC-shape LAC curve is a direct processin which the user may interactively input the endpointsand the tangent direction of those points We can furtherautomate the identification of tangents at the endpoints byusing three-point circular arc approximation around theendpoints as explained in Steps 1 and 2 in the algorithmbelowUsing these tangents we may calculate the midpoint Thesethree points are then used to fit LAC segment in the gapfor shape completion problem The idea of this method isto find a LAC segment defined in a triangle composed of
4 Journal of Applied Mathematics
Pc
Pa
Pb
120579d
120579e
(a) Original position of controlpoints
P0 P1
P2
120579d998400120579e998400
(b) Position of control points after transformation
Figure 3 The configuration of triangle consists of three control points for case 120572 le 1 The curve inside the triangle represents the generatedLAC segment
three control points as proposed by Yoshida and Saito [10]Therefore by using thismethod it is only possible to generatea C-curve Figures 2 and 3 show the configuration of thetriangle formed by three control points for cases 120572 gt 1 and120572 le 1 respectively Step 5 in the algorithm below elaboratesthese configurations
S-Shape Basically a S-shape LAC segment consists of two C-shape curves joined with 119866
1 continuity In order to constructthe S-curve for a given gap the inflection point is essential Asimple approach used is that users are given the freedom todefine the inflection point so that the algorithm is carried outtwice consisting of two C-shape segments of LAC Thus thisapproach needs two endpoints its tangent vectors 120572 valueand an inflection point
The process involves breaking the S-curve into two C-curves so the first piece of curve is constructed betweenstarting point (black) and inflection point (gray) while thesecond piece is between inflection point (gray) and endpoint(black) Figure 4 illustrates the inflection point that we definefor one of the numerical examples in this paper
Algorithm 1 demonstrates the LAC algorithm
221 Computation of Tangent Angles (Step 1ndashStep 3) To auto-mate the process of defining the tangents at the endpointswe may use circular arc approximation (osculating circle)that best fits the endpoints and points surrounding them inorder to determine the tangent angle at both endpoints Sinceit is an osculating circle we substitute the points into theequation of circleThen we identify the center (ℎ 119896) radius 119903and turning angle 119905 of circle from three points which include
Figure 4 The input for S-shape completion problem
the endpoint Next we determine the unit tangent vector atthe endpoints and subsequently calculate the correspondingtangent angle by using specified formula as indicated in thealgorithm Alternately we may also allow the user to directlydefine tangent vectors at those points
222 Identification of Second Control Point (Step 4) Once theunit tangent vector at endpoints is identified we determinethe coordinate of the second control point 119875
119887which is
Journal of Applied Mathematics 5
REMARK Endpoints 119875119886 119875119888 inflection point 119875inflect and 120572 are user defined The tangent angle at both endpoints 120579
0 1205792 can be
obtained either from three point circular arc approximation around the endpoints or the user defines it Let (119883 119884) represent thecoordinate of a pointINPUT 119875
119886 119875119888 119875inflect 120572
OUTPUT 120579119889 Λ 119903scale LAC segment
BeginStep 1 Set 119891 (119905) larr ℎ + 119903 cos (119905) 119896 + 119903 sin (119905) where (ℎ 119896) is the center of circle 119903 is radius and 119905 represents angleStep 2 Identify unit tangent vector UTV (119905) larr 119891
1015840(119905)
10038171003817100381710038171198911015840(119905)
1003817100381710038171003817 and set (119883new 119884new) larr (119883 119884) + UTV (119905)Step 3 Calculate directional angle 120579 larr tanminus1((119884new minus 119884)(119883new minus 119883))Step 4 Solve simultaneous equations 119884 = 119898119883 + 119888 where 119898 larr (119884new minus 119884)(119883new minus 119883) to get position of second control point 119875
119887
Step 5 If 10038171003817100381710038171198751198861198751198871003817100381710038171003817 le
10038171003817100381710038171198751198871198751198881003817100381710038171003817 then 119875
119886larr 119875119886 119875119888larr 119875119888
else 119875119886larr 119875119888 119875119888larr 119875119886
If 120572 gt 1 then 119875start larr 119875119888 119875end larr 119875
119886
else 119875start larr 119875119886 119875end larr 119875
119888
Step 6 Translate 119875start = (0 0) rotate triangle such that 119875119887= (minus119883 0) for 120572 gt 1 119875
119887= (119883 0) for 120572 le 1
If 119875end = (119883 minus119884) or 119875end = (minus119883 minus119884) then reflect the triangle through 119909-axisStep 7 Compute 120579
119889larr 120587 minus cosminus1((minus 1003817100381710038171003817119875119886119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119887119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119887
1003817100381710038171003817
2
)(21003817100381710038171003817119875119887119875119888
1003817100381710038171003817
10038171003817100381710038171198751198861198751198871003817100381710038171003817))
Step 8 If 120572 gt 1 then 120579119891larr cosminus1((minus 1003817100381710038171003817119875119886119875119887
1003817100381710038171003817
2
+1003817100381710038171003817119875119887119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119888
1003817100381710038171003817
2
)(21003817100381710038171003817119875119886119875119888
1003817100381710038171003817
10038171003817100381710038171198751198871198751198881003817100381710038171003817))
else 120579119890larr cosminus1((minus 1003817100381710038171003817119875119887119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119887
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119888
1003817100381710038171003817
2
)(21003817100381710038171003817119875119886119875119887
1003817100381710038171003817
10038171003817100381710038171198751198861198751198881003817100381710038171003817))
Step 9 Identify bound of Λ If 120572 lt 1 then 119886 larr 1 times 10minus10 119887 larr 1(120579
119889(1 minus 120572))
else if 120572 gt 1 then 119886 larr 1 times 10minus10 119887 larr 1(120579
119889(120572 minus 1))
else 119886 larr 1 times 10minus10 119887 larr 1 times 10
5Step 10 Set tolerance Tol larr 10
minus10iteration number 119873 larr (1 In 2) In (| 119886 minus 119887 | Tol)
Step 11 Calculate Λ using Bisection methodStep 12 Determine scaling factor 119903scale larr
10038171003817100381710038171198751198861198751198881003817100381710038171003817
1003817100381710038171003817119875011987521003817100381710038171003817
Step 13 Scale 119903scale to the curve and transform inversely the triangle to its original positionStep 14 Construct LAC segment using (2)Step 15 OUTPUT (120579
119889 Λ 119903scale LAC segment)
End
Algorithm 1
the intersection point of the tangential line at both endpointsby solving the equations simultaneously
223 Affirmation of Initial and Last Point Position (Step 5)We assume the distance between 119875
119886and 119875
119887is less than 119875
119887
and 119875119888as proposed by Yoshida and Saito [10] The position
of 119875119886and 119875
119888is switched if the criterion is not satisfied
Otherwise the position remains unchanged Note that thereare two cases of constructing LAC segment based on the120572 value as illustrated in Figures 2 and 3 For case 120572 le 1the initial endpoint 119875
119886is the starting point during the curve
construction process 119875start On the other hand starting pointof curve construction process for case 120572 gt 1 is the lastendpoint 119875
119888
224 Transformation Process (Step 6) The transformationprocess includes translation rotation and reflection of theoriginal triangle that consist of119875
119886119875119887119875119888 Initially the original
triangle is translated to the origin so that the starting point119875start is located at the origin Next we rotate the triangleso that the second control point is placed on the 119909-axisFinally we ensure the last point of the curve constructionprocess 119875end is located at the positive 119910-axis region If notthe reflection process is needed to reflect the last point with
negative 119910-coordinate through the 119909-axis The configurationafter transformation is illustrated in Figures 2 and 3
225 Computation of Tangent and Turning Angle (Step 7and Step 8) We apply the law of cosine of a triangle tocalculate corresponding angles as shown in Figures 2 and 3The tangent angle 120579
119889and turning angle 120579
119890for case 120572 le 1
or 120579119891for case 120572 gt 1 will be used in the subsequent step for
calculating shape parameter Λ
226 Upper Bound of Shape Parameter Λ (Step 9) For thecasewhen120572 = 1 there is no upper bound for shape parameterΛ So the value of Λ can be arbitrarily large However thereare upper bounds for Λ when 120572 lt 1 and 120572 gt 1 ThereforetheΛ value of these two cases is restricted within the range of(119886 119887) where 119886 and 119887 are assigned a specified value as indicatedin the algorithm above
227 Shape Parameter Λ Calculation (Step 10 and Step 11)We determine the Λ value that satisfies the constraintsof endpoints and angles by using bisection method Firsttolerance and bound of Λ are used in the calculation ofiteration number for bisection methodThen the calculatingprocess will run until the iteration number is met Once theΛ
6 Journal of Applied Mathematics
(a) Euler spiral algorithm (b) LAC algorithm with two tangents setting for s-shaped curve and 120572 = minus1
(c) LACalgorithmwith two tangents setting for s-shaped curveand 120572 = 2
Figure 5 The results of fitting Euler spiral and LAC segment for two missing parts of a jug
value of the curve is obtained it is an interpolating curve thatsatisfies boundary constraints as shape parameter Λ is usedto control the curve so that the constraints are satisfied TheobtainedΛ value ensures the generated LAC segment starts atthe starting point and ends at the last point while the shape ofcurve follows the specified tangent vectors at two endpoints
228 Computation of Scale Factor (Step 12) We calculate thescaling factor by using the formula as shown in the algorithmwhere 119875
119886and 119875
119888are endpoints of the original triangle while
1198750and 119875
2are endpoints of the transformed triangle
229 Construction of LACSegment (Step 13ndashStep 15) In orderto obtain the curve which is defined in the original trianglewe inversely transform the triangle to its original position
The curve segment is constructed using (2) with the obtainedvalue of Λ 120579
119889 119903scale
3 Numerical Examples and Discussion
This section shows the input data and numerical and graph-ical results of implementing the algorithm presented inSection 2 for three examples For comparison purpose wepresent the results of Kimiarsquos algorithm [3] algorithm on thesame examples as well
31 The Graphical Results Figure 5 shows the first exampleof fitting Euler spiral and LAC segment to two parts of a jugThe results of the second example (artifact) and third example(vase) are depicted in Figures 6 and 7 respectively
Journal of Applied Mathematics 7
(a) Euler spiral algorithm (b) LAC algorithm when 120572 = minus1
(c) LAC algorithm when 120572 = 0
Figure 6 The results of fitting Euler spiral and LAC segment for an artifact
32 The Numerical Results The input data and the obtainedparameter values to render LAC segment for C-shaped andS-shaped gaps are shown in Tables 1 2 and 3 respectively
33 Discussion We implemented LAC algorithm with var-ious 120572 values for each example In Figures 5 6 and 7 weshowed a subfigure of LAC with 120572 = minus1 in between the gapsfor the missing part of the objectsThese examples are indeedEuler spiral similar to the work of Kimia et al [3] Users areindeed fitting a circle involute when 120572 = 2 as shown in eachfigure Nielsen spiral is fitted when 120572 = 0 and logarithmicspiral when 120572 = 1 The shape parameter 120572 definitely dictatesthe type of spiral the user desires to fit into the gap
Based on the comparison of graphical results in Figures5 6 and 7 we found that Kimiarsquos algorithm [3] and theLAC algorithm behave similarly However the proposed LACalgorithm gives more flexibility to designers as they canmanipulate the shape parameter 120572 and the tangent directionat the inflection point in order to get various S-shaped LACsegments
4 Conclusion
A LAC segment can be constructed by specifying the end-points and their tangent directions LAC is a family of aes-thetic curves where their curvature changes monotonicallyIt comprises Euler spiral logarithmic spiral Nielsen spiraland circle involute with different 120572 values The algorithmfor constructing LAC segment to solve shape completion
problem is defined in Section 2 The results of implementingboth Kimia et alrsquos [3] algorithm and the proposed LACalgorithm on the same images are depicted in Section 3LAC has an upper hand due to its flexibility although bothalgorithms are capable of solving shape completion problemsSince LAC represents a bigger family rather than Eulerspiral alone it is clear that more shapes are available to fillthe missing gap Moreover the resulting LAC segments aresatisfying119866
1 continuity where the curve segments interpolateboth endpoints in the specified tangent directions In case theuser wants a119866
2 LAC solution one may plug in the techniqueproposed by Miura et al [15] or employ 119866
2 GLAC [17] inwhich we may use shape parameter V in GLAC to satisfy thedesired curvatures
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge Universiti Malaysia Terengganu(GGP grant) and Ministry of Higher Education Malaysia(FRGS 59265) for providing financial aid to carry out thisresearch They further acknowledge anonymous referees fortheir constructive comments which improved the readabilityof this paper
8 Journal of Applied Mathematics
(a) Euler spiral algorithm (b) Three types of tangents at the inflection points and 120572 = minus1
(c) Tangents setting as in 7(b) and 120572 = minus1 (d) Tangents setting as in 7(b) and 120572 = 1
Figure 7 The results of fitting Euler spiral and LAC segment to a vase
Table 1 The input data for LAC algorithm
Figure Partgap shape Starting point Inflection point Terminating point119875119886
119875inflect 119875119888
Figure 5Outer C (47533 29067) mdash (408 512)Inner C (444 3193) mdash (4067 4907)
S (6867 198) (115139 317874) (132 44533)
Figure 6 Left C (1504 3208) mdash (1232 5896)Right C (853 307) mdash (824 583)
Figure 7C (504 126) mdash (176 120)
Right S (57667 37933) (501722 483092) (44533 598)Left S (21533 59933) (166565 4812) (94 376)
Journal of Applied Mathematics 9
Table 2 The value of parameters obtained in LAC algorithm for C-shaped gap
Figure Part Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) Outer C 0181102 197352 109383 106443Inner C 0187822 250305 153612 577559
Figure 5(c) Outer C 0236827 197352 0879684 179903Inner C 0307163 250305 0966927 144024
Figure 6(b) Left C 0319685 155147 100107 105562Right C 0204861 235562 147579 896424
Figure 6(c) Left C 053364 155147 100107 902434Right C 0610563 235562 147579 654159
Figure 7(c) mdash 0140054 11637 0601359 271762Figure 7(d) mdash 0169061 11637 0601359 270036
Table 3 The value of parameters obtained in LAC algorithm for S-shaped gap
Figure Part Line style Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) mdashPiece 1 Thick 0646746 0739623 0451367 107501
Dashed 113587 043963 0288258 152739
Piece 2 Thick 0140372 132647 0689653 935698Dashed 0473033 102648 063682 766964
Figure 5(c) mdashPiece 1 Thick 219523 043963 0151372 568396
Dashed 107216 0739623 0288256 293721
Piece 2 Thick 0811123 102648 0389659 222587Dashed 0156681 132647 063682 116479
Figure 7(c) Right S
Piece 1Thick 00000429688 0571501 0285752 227044Dashed 0698712 0871501 0585746 39271Dotted 245758 0371528 0285746 23657
Piece 2Thick 00118778 0908437 0455056 145074Dashed 0403597 120905 0755673 651694Dotted 0700923 0707704 045338 100657
Figure 7(d) Right S
Piece 1Thick 253147 0871501 0585746 411231Dashed 00000429592 0571501 0285752 227044Dotted 107779 0371528 0285746 257032
Piece 2Thick 12606 120905 0755673 474646Dashed 00120078 0908437 0455056 145071Dotted 248785 0707704 045338 673212
Figure 7(c) Left S
Piece 1Thick 0188412 0981157 0509674 121657Dashed 0603719 0781142 0471473 10399Dotted 0361402 138114 0909666 524523
Piece 2Thick 00000462514 0594672 0297337 218104Dashed 0884532 0394671 0297335 35514Dotted 0564167 0994666 0697331 275123
Figure 7(d) Left S
Piece 1Thick 161781 0781142 0471473 833887Dashed 0234412 0981157 0509674 120616Dotted 142816 138114 0909666 315215
Piece 2Thick 926646 0394671 0297335 314849Dashed 00000462423 0594672 0297337 218104Dotted 266752 0994666 0697331 266104
10 Journal of Applied Mathematics
References
[1] M Kolomenkin I Shimshoni and A Tal ldquoDemarcating curvesfor shape illustrationrdquo ACM Transactions on Graphics vol 27no 5 article 157 2008
[2] D Decarlo A Finkelstein S Rusinkiewicz and A SantellaldquoSuggestive contours for conveying shaperdquo ACM Transactionson Graphics vol 22 no 3 pp 848ndash855 2003
[3] B B Kimia I Frankel and A Popescu ldquoEuler spiral for shapecompletionrdquo International Journal of Computer Vision vol 54no 1ndash3 pp 159ndash182 2003
[4] T Judd F Durand and E Adelson ldquoApparent ridges for linedrawingrdquo ACM Transactions on Graphics vol 26 no 3 article19 7 pages 2007
[5] S Yoshizawa A Belyaev and H P Seidel ldquoFast and robustdetection of crest lines on meshesrdquo in Proceedings of the ACMSymposium on Solid and Physical Modeling (SPM rsquo05) pp 227ndash232 June 2005
[6] G Harary and A Tal ldquo3D Euler spirals for 3D curve comple-tionrdquo Computational Geometry Theory and Applications vol45 no 3 pp 115ndash126 2012
[7] I Kanaya Y Nakano and K Sato ldquoSimulated designerrsquoseyesmdashclassification of aesthetic surfacesrdquo in Proceedings of theInternational Conference onVirtual Systems andMultimedia pp289ndash296 2003
[8] R U Gobithaasan and K T Miura ldquoLogarithmic curvaturegraph as a shape interrogation toolrdquo Applied MathematicalSciences vol 8 no 16 pp 755ndash765 2014
[9] K T Miura J Sone A Yamashita and T Kaneko ldquoDerivationof a general formula of aesthetic curvesrdquo in Proceedings of 8thInternational Conference on Humans and Computers pp 166ndash171 Tokyo Japan 2005
[10] N Yoshida and T Saito ldquoInteractive aesthetic curve segmentsrdquoThe Visual Computer vol 22 no 9-11 pp 896ndash905 2006
[11] R U Gobithaasan and K T Miura ldquoAesthetic spiral for designrdquoSains Malaysiana vol 40 no 11 pp 1301ndash1305 2011
[12] JMAli RM Tookey J V Ball andAA Ball ldquoThe generalisedCornu spiral and its application to span generationrdquo Journal ofComputational and Applied Mathematics vol 102 no 1 pp 37ndash47 1999
[13] R U Gobithaasan R Karpagavalli and K T Miura ldquoShapeanalysis of generalized log-aesthetic curvesrdquo International Jour-nal of Mathematical Analysis vol 7 no 33 pp 1751ndash1759 2013
[14] R Ziatdinov N Yoshida and T Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012
[15] K T Miura D Shibuya R U Gobithaasan and S UsukildquoDesigning log-aesthetic splines with119866
2 continuityrdquoComputer-Aided Design and Applications vol 10 no 6 pp 1021ndash1032 2013
[16] K T Miura and R U Gobithaasan ldquoAesthetic curves andsurfaces in computer aided geometric designrdquo InternationalJournal of Automation Technology vol 8 no 3 pp 304ndash3162014
[17] R U Gobithaasan R Karpagavalli and K T Miura ldquoG2continuous generalized log aesthetic curvesrdquo Computer AidedDesign amp Application Accepted for publication
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
05 10 15 20 25
05
10
15
20
X
Y
(a) Two segments of Euler spiral
00 05 10 15
00
05
10
15
20
minus05minus10 minus05
log[120588(t)s998400 (t)120588
998400 (t)]
log[120588(t)]
(b) Its corresponding logarithmic curvature graph
Figure 1 Euler spiral defined 119904 isin [02 1] (solid line) and 119904 isin [1 3] (dotted) and its corresponding LCG
Harada et al (1999) when they attempted to classify all thecurves used in automobile design It was first known aslogarithmic distribution diagram of curvature (LDDC) Itis a graph to measure the relationship between curvatureradius (denoted by 120588) and arc length of the curve segment(denoted by 119904) They found that all the curves employed inautomobile design depict linear gradient of LDDC Kanaya etal [7] refined the formulation of LDDC which is now knownas LCG The horizontal axis of LCG represents log 120588 and thevertical axis represents log 120588(119889119904
10158401198891205881015840) [8] Figure 1 illustrates
the two segments of Euler spiralIn 2005 Miura et al [9] derived a fundamental formula
of LAC based on a linear representation of LCG A yearlater Yoshida and Saito [10] investigated the characteristicsand overall shape of LAC They also proposed a method toconstruct LAC segment interactively by specifying endpointsand their tangent vectors In 2009 they further extendedLAC into space curve and proposed amethod to interactivelycontrol the curves which is similar to quadratic Beziercurves
LAC has become a promising curve in graphic andindustrial design due to its monotonic curvature propertyThere are two shape parameters of LAC denoted by 120572 andΛ The shape parameter 120572 is used to determine the type ofspiral within the family of LAC and Λ is used to satisfythe 1198661 constraint during the design process LAC comprises
Euler spiral (120572 = minus1) Nielsen spiral (120572 = 0) logarithmic orequiangular spiral (120572 = 1) circle involute (120572 = 2) and others
In 2011 Gobithaasan and Miura [11] expanded the familyof LAC into Generalized Log-Aesthetic Curve (GLAC) byadding generalized Cornu spiral (GCS) to the family of LACThis includes Euler spiral logarithmic spiral circle involuteNielsen spiral LAC and GCS as the members of GLAC
To note GCS is proposed by Ali et al [12] where it becomesEuler spiral when the shape parameter of GCS is zero GLAChas an extra shape parameter compared to LAC in which itcan be used to satisfy additional constraints that occur duringdesign process [13] The curvature function of GLAC is alsomonotonic similar to LAC In 2012 they further extendedGLAC into space curve
In 2012 Ziatdinov et al [14] defined LAC in the formof incomplete gamma functions They claimed that this newexpression of LAC is capable of reducing the computationtime up to 13 times faster In 2013 the problem of1198662 Hermiteinterpolation was dealt with by Miura and Gobithaasan [15]by introducing log-aesthetic splines which consists of tripleLAC segments Readers are referred to [16] for a detailedreview on aesthetic curves
In this paper we introduce the implementation of 1198661
LAC algorithm to fit the gap in between the outlines ofmissingocclusion objects We compare the performance ofthis algorithm with the results obtained by Kimia et alrsquos[3] method on the same examples in order to show theeffectiveness of LAC to solve shape completion problem
The rest of this paper is organized as follows First the for-mulation of LAC is reviewed followed by a detailed explana-tion on the LAC algorithm for solving the shape completionproblem In Section 3 we demonstrate the implementationof the proposed algorithm with various 120572 values with threeexamples to show the implementation of this method Thedata and calculation used in the examples are also depictedin this sectionThe output of shape completion using Kimiarsquosmethod is also depicted for comparison purposes We alsomake comparison of the results obtained using Kimia et alrsquos[3] method and our algorithm We further show some shapevariations which can be obtained by using S-shape LAC in
Journal of Applied Mathematics 3
Pc
Pa
Pb
120579f
120579d
(a) Original position of controlpoints
P0
P1P2
120579d998400120579f998400
(b) Position of control points after transformation
Figure 2 The configuration of triangle consists of three control points for case 120572 gt 1 The curve inside the triangle represents the generatedLAC segment
which the designer tweaks the segment in order to obtain thedesired shape
2 Log-Aesthetic Curves in Shape Completion
In this section we show the parametric expression of LACand the algorithm used to obtain the solution for shapecompletion problem with various 120572 values
21 The Analytical Formulas and Parameters There are twoshape parameters used to define LAC segment 120572 and Λ Theshape parameter 120572 determines the type of spiral andΛ is usedto control the curve so that the constraints are satisfied Let 120588and 120579 represent radius of curvature and tangent angle of thecurve respectively
Equation (1) shows the radius of curvature function interms of its turning angle To note (1) is used to define theparametric expression of LAC in terms of its turning angle asshown in (2) as follows
120588 (120579) =
119890Λ120579
if 120572 = 1
((120572 minus 1) Λ120579 + 1)1(120572minus1)
otherwise(1)
LAC (120579) = 119875119886+ int
120579
0
119890119894119906120588 (119906) 119889119906 (2)
where 119875119886represents starting point and 119894 and 119890 represent the
imaginary unit and exponent respectively LAC can also berepresented in the arc length parameter 119904 as follows
LAC (119904) =
119875119886+ int
119904
0
119890119894((1Λ)(1minus119890
minusΛ119906
))119889119906 if 120572 = 0
119875119886+ int
119904
0
119890119894(log[Λ119906+1]Λ)
119889119906 if 120572 = 1
119875119886+ int
119904
0
119890119894(((Λ120572119906+1)
(120572minus1)120572
minus1)Λ(120572minus1))119889119906 otherwise
(3)
Both (2) and (3) generate the same LAC The curve canbe drawn in complex plane by the 119909-axis representing realnumbers and the 119910-axis representing imaginary numbersReaders are referred to [10] for a detailed derivation of theparametric expression of LAC
22 The LAC Algorithm
C-Shape We customize the algorithm of constructing LACsegment to fill the gap by specifying two endpoints its tangentvector and120572 valueTheC-shape LAC curve is a direct processin which the user may interactively input the endpointsand the tangent direction of those points We can furtherautomate the identification of tangents at the endpoints byusing three-point circular arc approximation around theendpoints as explained in Steps 1 and 2 in the algorithmbelowUsing these tangents we may calculate the midpoint Thesethree points are then used to fit LAC segment in the gapfor shape completion problem The idea of this method isto find a LAC segment defined in a triangle composed of
4 Journal of Applied Mathematics
Pc
Pa
Pb
120579d
120579e
(a) Original position of controlpoints
P0 P1
P2
120579d998400120579e998400
(b) Position of control points after transformation
Figure 3 The configuration of triangle consists of three control points for case 120572 le 1 The curve inside the triangle represents the generatedLAC segment
three control points as proposed by Yoshida and Saito [10]Therefore by using thismethod it is only possible to generatea C-curve Figures 2 and 3 show the configuration of thetriangle formed by three control points for cases 120572 gt 1 and120572 le 1 respectively Step 5 in the algorithm below elaboratesthese configurations
S-Shape Basically a S-shape LAC segment consists of two C-shape curves joined with 119866
1 continuity In order to constructthe S-curve for a given gap the inflection point is essential Asimple approach used is that users are given the freedom todefine the inflection point so that the algorithm is carried outtwice consisting of two C-shape segments of LAC Thus thisapproach needs two endpoints its tangent vectors 120572 valueand an inflection point
The process involves breaking the S-curve into two C-curves so the first piece of curve is constructed betweenstarting point (black) and inflection point (gray) while thesecond piece is between inflection point (gray) and endpoint(black) Figure 4 illustrates the inflection point that we definefor one of the numerical examples in this paper
Algorithm 1 demonstrates the LAC algorithm
221 Computation of Tangent Angles (Step 1ndashStep 3) To auto-mate the process of defining the tangents at the endpointswe may use circular arc approximation (osculating circle)that best fits the endpoints and points surrounding them inorder to determine the tangent angle at both endpoints Sinceit is an osculating circle we substitute the points into theequation of circleThen we identify the center (ℎ 119896) radius 119903and turning angle 119905 of circle from three points which include
Figure 4 The input for S-shape completion problem
the endpoint Next we determine the unit tangent vector atthe endpoints and subsequently calculate the correspondingtangent angle by using specified formula as indicated in thealgorithm Alternately we may also allow the user to directlydefine tangent vectors at those points
222 Identification of Second Control Point (Step 4) Once theunit tangent vector at endpoints is identified we determinethe coordinate of the second control point 119875
119887which is
Journal of Applied Mathematics 5
REMARK Endpoints 119875119886 119875119888 inflection point 119875inflect and 120572 are user defined The tangent angle at both endpoints 120579
0 1205792 can be
obtained either from three point circular arc approximation around the endpoints or the user defines it Let (119883 119884) represent thecoordinate of a pointINPUT 119875
119886 119875119888 119875inflect 120572
OUTPUT 120579119889 Λ 119903scale LAC segment
BeginStep 1 Set 119891 (119905) larr ℎ + 119903 cos (119905) 119896 + 119903 sin (119905) where (ℎ 119896) is the center of circle 119903 is radius and 119905 represents angleStep 2 Identify unit tangent vector UTV (119905) larr 119891
1015840(119905)
10038171003817100381710038171198911015840(119905)
1003817100381710038171003817 and set (119883new 119884new) larr (119883 119884) + UTV (119905)Step 3 Calculate directional angle 120579 larr tanminus1((119884new minus 119884)(119883new minus 119883))Step 4 Solve simultaneous equations 119884 = 119898119883 + 119888 where 119898 larr (119884new minus 119884)(119883new minus 119883) to get position of second control point 119875
119887
Step 5 If 10038171003817100381710038171198751198861198751198871003817100381710038171003817 le
10038171003817100381710038171198751198871198751198881003817100381710038171003817 then 119875
119886larr 119875119886 119875119888larr 119875119888
else 119875119886larr 119875119888 119875119888larr 119875119886
If 120572 gt 1 then 119875start larr 119875119888 119875end larr 119875
119886
else 119875start larr 119875119886 119875end larr 119875
119888
Step 6 Translate 119875start = (0 0) rotate triangle such that 119875119887= (minus119883 0) for 120572 gt 1 119875
119887= (119883 0) for 120572 le 1
If 119875end = (119883 minus119884) or 119875end = (minus119883 minus119884) then reflect the triangle through 119909-axisStep 7 Compute 120579
119889larr 120587 minus cosminus1((minus 1003817100381710038171003817119875119886119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119887119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119887
1003817100381710038171003817
2
)(21003817100381710038171003817119875119887119875119888
1003817100381710038171003817
10038171003817100381710038171198751198861198751198871003817100381710038171003817))
Step 8 If 120572 gt 1 then 120579119891larr cosminus1((minus 1003817100381710038171003817119875119886119875119887
1003817100381710038171003817
2
+1003817100381710038171003817119875119887119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119888
1003817100381710038171003817
2
)(21003817100381710038171003817119875119886119875119888
1003817100381710038171003817
10038171003817100381710038171198751198871198751198881003817100381710038171003817))
else 120579119890larr cosminus1((minus 1003817100381710038171003817119875119887119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119887
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119888
1003817100381710038171003817
2
)(21003817100381710038171003817119875119886119875119887
1003817100381710038171003817
10038171003817100381710038171198751198861198751198881003817100381710038171003817))
Step 9 Identify bound of Λ If 120572 lt 1 then 119886 larr 1 times 10minus10 119887 larr 1(120579
119889(1 minus 120572))
else if 120572 gt 1 then 119886 larr 1 times 10minus10 119887 larr 1(120579
119889(120572 minus 1))
else 119886 larr 1 times 10minus10 119887 larr 1 times 10
5Step 10 Set tolerance Tol larr 10
minus10iteration number 119873 larr (1 In 2) In (| 119886 minus 119887 | Tol)
Step 11 Calculate Λ using Bisection methodStep 12 Determine scaling factor 119903scale larr
10038171003817100381710038171198751198861198751198881003817100381710038171003817
1003817100381710038171003817119875011987521003817100381710038171003817
Step 13 Scale 119903scale to the curve and transform inversely the triangle to its original positionStep 14 Construct LAC segment using (2)Step 15 OUTPUT (120579
119889 Λ 119903scale LAC segment)
End
Algorithm 1
the intersection point of the tangential line at both endpointsby solving the equations simultaneously
223 Affirmation of Initial and Last Point Position (Step 5)We assume the distance between 119875
119886and 119875
119887is less than 119875
119887
and 119875119888as proposed by Yoshida and Saito [10] The position
of 119875119886and 119875
119888is switched if the criterion is not satisfied
Otherwise the position remains unchanged Note that thereare two cases of constructing LAC segment based on the120572 value as illustrated in Figures 2 and 3 For case 120572 le 1the initial endpoint 119875
119886is the starting point during the curve
construction process 119875start On the other hand starting pointof curve construction process for case 120572 gt 1 is the lastendpoint 119875
119888
224 Transformation Process (Step 6) The transformationprocess includes translation rotation and reflection of theoriginal triangle that consist of119875
119886119875119887119875119888 Initially the original
triangle is translated to the origin so that the starting point119875start is located at the origin Next we rotate the triangleso that the second control point is placed on the 119909-axisFinally we ensure the last point of the curve constructionprocess 119875end is located at the positive 119910-axis region If notthe reflection process is needed to reflect the last point with
negative 119910-coordinate through the 119909-axis The configurationafter transformation is illustrated in Figures 2 and 3
225 Computation of Tangent and Turning Angle (Step 7and Step 8) We apply the law of cosine of a triangle tocalculate corresponding angles as shown in Figures 2 and 3The tangent angle 120579
119889and turning angle 120579
119890for case 120572 le 1
or 120579119891for case 120572 gt 1 will be used in the subsequent step for
calculating shape parameter Λ
226 Upper Bound of Shape Parameter Λ (Step 9) For thecasewhen120572 = 1 there is no upper bound for shape parameterΛ So the value of Λ can be arbitrarily large However thereare upper bounds for Λ when 120572 lt 1 and 120572 gt 1 ThereforetheΛ value of these two cases is restricted within the range of(119886 119887) where 119886 and 119887 are assigned a specified value as indicatedin the algorithm above
227 Shape Parameter Λ Calculation (Step 10 and Step 11)We determine the Λ value that satisfies the constraintsof endpoints and angles by using bisection method Firsttolerance and bound of Λ are used in the calculation ofiteration number for bisection methodThen the calculatingprocess will run until the iteration number is met Once theΛ
6 Journal of Applied Mathematics
(a) Euler spiral algorithm (b) LAC algorithm with two tangents setting for s-shaped curve and 120572 = minus1
(c) LACalgorithmwith two tangents setting for s-shaped curveand 120572 = 2
Figure 5 The results of fitting Euler spiral and LAC segment for two missing parts of a jug
value of the curve is obtained it is an interpolating curve thatsatisfies boundary constraints as shape parameter Λ is usedto control the curve so that the constraints are satisfied TheobtainedΛ value ensures the generated LAC segment starts atthe starting point and ends at the last point while the shape ofcurve follows the specified tangent vectors at two endpoints
228 Computation of Scale Factor (Step 12) We calculate thescaling factor by using the formula as shown in the algorithmwhere 119875
119886and 119875
119888are endpoints of the original triangle while
1198750and 119875
2are endpoints of the transformed triangle
229 Construction of LACSegment (Step 13ndashStep 15) In orderto obtain the curve which is defined in the original trianglewe inversely transform the triangle to its original position
The curve segment is constructed using (2) with the obtainedvalue of Λ 120579
119889 119903scale
3 Numerical Examples and Discussion
This section shows the input data and numerical and graph-ical results of implementing the algorithm presented inSection 2 for three examples For comparison purpose wepresent the results of Kimiarsquos algorithm [3] algorithm on thesame examples as well
31 The Graphical Results Figure 5 shows the first exampleof fitting Euler spiral and LAC segment to two parts of a jugThe results of the second example (artifact) and third example(vase) are depicted in Figures 6 and 7 respectively
Journal of Applied Mathematics 7
(a) Euler spiral algorithm (b) LAC algorithm when 120572 = minus1
(c) LAC algorithm when 120572 = 0
Figure 6 The results of fitting Euler spiral and LAC segment for an artifact
32 The Numerical Results The input data and the obtainedparameter values to render LAC segment for C-shaped andS-shaped gaps are shown in Tables 1 2 and 3 respectively
33 Discussion We implemented LAC algorithm with var-ious 120572 values for each example In Figures 5 6 and 7 weshowed a subfigure of LAC with 120572 = minus1 in between the gapsfor the missing part of the objectsThese examples are indeedEuler spiral similar to the work of Kimia et al [3] Users areindeed fitting a circle involute when 120572 = 2 as shown in eachfigure Nielsen spiral is fitted when 120572 = 0 and logarithmicspiral when 120572 = 1 The shape parameter 120572 definitely dictatesthe type of spiral the user desires to fit into the gap
Based on the comparison of graphical results in Figures5 6 and 7 we found that Kimiarsquos algorithm [3] and theLAC algorithm behave similarly However the proposed LACalgorithm gives more flexibility to designers as they canmanipulate the shape parameter 120572 and the tangent directionat the inflection point in order to get various S-shaped LACsegments
4 Conclusion
A LAC segment can be constructed by specifying the end-points and their tangent directions LAC is a family of aes-thetic curves where their curvature changes monotonicallyIt comprises Euler spiral logarithmic spiral Nielsen spiraland circle involute with different 120572 values The algorithmfor constructing LAC segment to solve shape completion
problem is defined in Section 2 The results of implementingboth Kimia et alrsquos [3] algorithm and the proposed LACalgorithm on the same images are depicted in Section 3LAC has an upper hand due to its flexibility although bothalgorithms are capable of solving shape completion problemsSince LAC represents a bigger family rather than Eulerspiral alone it is clear that more shapes are available to fillthe missing gap Moreover the resulting LAC segments aresatisfying119866
1 continuity where the curve segments interpolateboth endpoints in the specified tangent directions In case theuser wants a119866
2 LAC solution one may plug in the techniqueproposed by Miura et al [15] or employ 119866
2 GLAC [17] inwhich we may use shape parameter V in GLAC to satisfy thedesired curvatures
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge Universiti Malaysia Terengganu(GGP grant) and Ministry of Higher Education Malaysia(FRGS 59265) for providing financial aid to carry out thisresearch They further acknowledge anonymous referees fortheir constructive comments which improved the readabilityof this paper
8 Journal of Applied Mathematics
(a) Euler spiral algorithm (b) Three types of tangents at the inflection points and 120572 = minus1
(c) Tangents setting as in 7(b) and 120572 = minus1 (d) Tangents setting as in 7(b) and 120572 = 1
Figure 7 The results of fitting Euler spiral and LAC segment to a vase
Table 1 The input data for LAC algorithm
Figure Partgap shape Starting point Inflection point Terminating point119875119886
119875inflect 119875119888
Figure 5Outer C (47533 29067) mdash (408 512)Inner C (444 3193) mdash (4067 4907)
S (6867 198) (115139 317874) (132 44533)
Figure 6 Left C (1504 3208) mdash (1232 5896)Right C (853 307) mdash (824 583)
Figure 7C (504 126) mdash (176 120)
Right S (57667 37933) (501722 483092) (44533 598)Left S (21533 59933) (166565 4812) (94 376)
Journal of Applied Mathematics 9
Table 2 The value of parameters obtained in LAC algorithm for C-shaped gap
Figure Part Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) Outer C 0181102 197352 109383 106443Inner C 0187822 250305 153612 577559
Figure 5(c) Outer C 0236827 197352 0879684 179903Inner C 0307163 250305 0966927 144024
Figure 6(b) Left C 0319685 155147 100107 105562Right C 0204861 235562 147579 896424
Figure 6(c) Left C 053364 155147 100107 902434Right C 0610563 235562 147579 654159
Figure 7(c) mdash 0140054 11637 0601359 271762Figure 7(d) mdash 0169061 11637 0601359 270036
Table 3 The value of parameters obtained in LAC algorithm for S-shaped gap
Figure Part Line style Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) mdashPiece 1 Thick 0646746 0739623 0451367 107501
Dashed 113587 043963 0288258 152739
Piece 2 Thick 0140372 132647 0689653 935698Dashed 0473033 102648 063682 766964
Figure 5(c) mdashPiece 1 Thick 219523 043963 0151372 568396
Dashed 107216 0739623 0288256 293721
Piece 2 Thick 0811123 102648 0389659 222587Dashed 0156681 132647 063682 116479
Figure 7(c) Right S
Piece 1Thick 00000429688 0571501 0285752 227044Dashed 0698712 0871501 0585746 39271Dotted 245758 0371528 0285746 23657
Piece 2Thick 00118778 0908437 0455056 145074Dashed 0403597 120905 0755673 651694Dotted 0700923 0707704 045338 100657
Figure 7(d) Right S
Piece 1Thick 253147 0871501 0585746 411231Dashed 00000429592 0571501 0285752 227044Dotted 107779 0371528 0285746 257032
Piece 2Thick 12606 120905 0755673 474646Dashed 00120078 0908437 0455056 145071Dotted 248785 0707704 045338 673212
Figure 7(c) Left S
Piece 1Thick 0188412 0981157 0509674 121657Dashed 0603719 0781142 0471473 10399Dotted 0361402 138114 0909666 524523
Piece 2Thick 00000462514 0594672 0297337 218104Dashed 0884532 0394671 0297335 35514Dotted 0564167 0994666 0697331 275123
Figure 7(d) Left S
Piece 1Thick 161781 0781142 0471473 833887Dashed 0234412 0981157 0509674 120616Dotted 142816 138114 0909666 315215
Piece 2Thick 926646 0394671 0297335 314849Dashed 00000462423 0594672 0297337 218104Dotted 266752 0994666 0697331 266104
10 Journal of Applied Mathematics
References
[1] M Kolomenkin I Shimshoni and A Tal ldquoDemarcating curvesfor shape illustrationrdquo ACM Transactions on Graphics vol 27no 5 article 157 2008
[2] D Decarlo A Finkelstein S Rusinkiewicz and A SantellaldquoSuggestive contours for conveying shaperdquo ACM Transactionson Graphics vol 22 no 3 pp 848ndash855 2003
[3] B B Kimia I Frankel and A Popescu ldquoEuler spiral for shapecompletionrdquo International Journal of Computer Vision vol 54no 1ndash3 pp 159ndash182 2003
[4] T Judd F Durand and E Adelson ldquoApparent ridges for linedrawingrdquo ACM Transactions on Graphics vol 26 no 3 article19 7 pages 2007
[5] S Yoshizawa A Belyaev and H P Seidel ldquoFast and robustdetection of crest lines on meshesrdquo in Proceedings of the ACMSymposium on Solid and Physical Modeling (SPM rsquo05) pp 227ndash232 June 2005
[6] G Harary and A Tal ldquo3D Euler spirals for 3D curve comple-tionrdquo Computational Geometry Theory and Applications vol45 no 3 pp 115ndash126 2012
[7] I Kanaya Y Nakano and K Sato ldquoSimulated designerrsquoseyesmdashclassification of aesthetic surfacesrdquo in Proceedings of theInternational Conference onVirtual Systems andMultimedia pp289ndash296 2003
[8] R U Gobithaasan and K T Miura ldquoLogarithmic curvaturegraph as a shape interrogation toolrdquo Applied MathematicalSciences vol 8 no 16 pp 755ndash765 2014
[9] K T Miura J Sone A Yamashita and T Kaneko ldquoDerivationof a general formula of aesthetic curvesrdquo in Proceedings of 8thInternational Conference on Humans and Computers pp 166ndash171 Tokyo Japan 2005
[10] N Yoshida and T Saito ldquoInteractive aesthetic curve segmentsrdquoThe Visual Computer vol 22 no 9-11 pp 896ndash905 2006
[11] R U Gobithaasan and K T Miura ldquoAesthetic spiral for designrdquoSains Malaysiana vol 40 no 11 pp 1301ndash1305 2011
[12] JMAli RM Tookey J V Ball andAA Ball ldquoThe generalisedCornu spiral and its application to span generationrdquo Journal ofComputational and Applied Mathematics vol 102 no 1 pp 37ndash47 1999
[13] R U Gobithaasan R Karpagavalli and K T Miura ldquoShapeanalysis of generalized log-aesthetic curvesrdquo International Jour-nal of Mathematical Analysis vol 7 no 33 pp 1751ndash1759 2013
[14] R Ziatdinov N Yoshida and T Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012
[15] K T Miura D Shibuya R U Gobithaasan and S UsukildquoDesigning log-aesthetic splines with119866
2 continuityrdquoComputer-Aided Design and Applications vol 10 no 6 pp 1021ndash1032 2013
[16] K T Miura and R U Gobithaasan ldquoAesthetic curves andsurfaces in computer aided geometric designrdquo InternationalJournal of Automation Technology vol 8 no 3 pp 304ndash3162014
[17] R U Gobithaasan R Karpagavalli and K T Miura ldquoG2continuous generalized log aesthetic curvesrdquo Computer AidedDesign amp Application Accepted for publication
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
Pc
Pa
Pb
120579f
120579d
(a) Original position of controlpoints
P0
P1P2
120579d998400120579f998400
(b) Position of control points after transformation
Figure 2 The configuration of triangle consists of three control points for case 120572 gt 1 The curve inside the triangle represents the generatedLAC segment
which the designer tweaks the segment in order to obtain thedesired shape
2 Log-Aesthetic Curves in Shape Completion
In this section we show the parametric expression of LACand the algorithm used to obtain the solution for shapecompletion problem with various 120572 values
21 The Analytical Formulas and Parameters There are twoshape parameters used to define LAC segment 120572 and Λ Theshape parameter 120572 determines the type of spiral andΛ is usedto control the curve so that the constraints are satisfied Let 120588and 120579 represent radius of curvature and tangent angle of thecurve respectively
Equation (1) shows the radius of curvature function interms of its turning angle To note (1) is used to define theparametric expression of LAC in terms of its turning angle asshown in (2) as follows
120588 (120579) =
119890Λ120579
if 120572 = 1
((120572 minus 1) Λ120579 + 1)1(120572minus1)
otherwise(1)
LAC (120579) = 119875119886+ int
120579
0
119890119894119906120588 (119906) 119889119906 (2)
where 119875119886represents starting point and 119894 and 119890 represent the
imaginary unit and exponent respectively LAC can also berepresented in the arc length parameter 119904 as follows
LAC (119904) =
119875119886+ int
119904
0
119890119894((1Λ)(1minus119890
minusΛ119906
))119889119906 if 120572 = 0
119875119886+ int
119904
0
119890119894(log[Λ119906+1]Λ)
119889119906 if 120572 = 1
119875119886+ int
119904
0
119890119894(((Λ120572119906+1)
(120572minus1)120572
minus1)Λ(120572minus1))119889119906 otherwise
(3)
Both (2) and (3) generate the same LAC The curve canbe drawn in complex plane by the 119909-axis representing realnumbers and the 119910-axis representing imaginary numbersReaders are referred to [10] for a detailed derivation of theparametric expression of LAC
22 The LAC Algorithm
C-Shape We customize the algorithm of constructing LACsegment to fill the gap by specifying two endpoints its tangentvector and120572 valueTheC-shape LAC curve is a direct processin which the user may interactively input the endpointsand the tangent direction of those points We can furtherautomate the identification of tangents at the endpoints byusing three-point circular arc approximation around theendpoints as explained in Steps 1 and 2 in the algorithmbelowUsing these tangents we may calculate the midpoint Thesethree points are then used to fit LAC segment in the gapfor shape completion problem The idea of this method isto find a LAC segment defined in a triangle composed of
4 Journal of Applied Mathematics
Pc
Pa
Pb
120579d
120579e
(a) Original position of controlpoints
P0 P1
P2
120579d998400120579e998400
(b) Position of control points after transformation
Figure 3 The configuration of triangle consists of three control points for case 120572 le 1 The curve inside the triangle represents the generatedLAC segment
three control points as proposed by Yoshida and Saito [10]Therefore by using thismethod it is only possible to generatea C-curve Figures 2 and 3 show the configuration of thetriangle formed by three control points for cases 120572 gt 1 and120572 le 1 respectively Step 5 in the algorithm below elaboratesthese configurations
S-Shape Basically a S-shape LAC segment consists of two C-shape curves joined with 119866
1 continuity In order to constructthe S-curve for a given gap the inflection point is essential Asimple approach used is that users are given the freedom todefine the inflection point so that the algorithm is carried outtwice consisting of two C-shape segments of LAC Thus thisapproach needs two endpoints its tangent vectors 120572 valueand an inflection point
The process involves breaking the S-curve into two C-curves so the first piece of curve is constructed betweenstarting point (black) and inflection point (gray) while thesecond piece is between inflection point (gray) and endpoint(black) Figure 4 illustrates the inflection point that we definefor one of the numerical examples in this paper
Algorithm 1 demonstrates the LAC algorithm
221 Computation of Tangent Angles (Step 1ndashStep 3) To auto-mate the process of defining the tangents at the endpointswe may use circular arc approximation (osculating circle)that best fits the endpoints and points surrounding them inorder to determine the tangent angle at both endpoints Sinceit is an osculating circle we substitute the points into theequation of circleThen we identify the center (ℎ 119896) radius 119903and turning angle 119905 of circle from three points which include
Figure 4 The input for S-shape completion problem
the endpoint Next we determine the unit tangent vector atthe endpoints and subsequently calculate the correspondingtangent angle by using specified formula as indicated in thealgorithm Alternately we may also allow the user to directlydefine tangent vectors at those points
222 Identification of Second Control Point (Step 4) Once theunit tangent vector at endpoints is identified we determinethe coordinate of the second control point 119875
119887which is
Journal of Applied Mathematics 5
REMARK Endpoints 119875119886 119875119888 inflection point 119875inflect and 120572 are user defined The tangent angle at both endpoints 120579
0 1205792 can be
obtained either from three point circular arc approximation around the endpoints or the user defines it Let (119883 119884) represent thecoordinate of a pointINPUT 119875
119886 119875119888 119875inflect 120572
OUTPUT 120579119889 Λ 119903scale LAC segment
BeginStep 1 Set 119891 (119905) larr ℎ + 119903 cos (119905) 119896 + 119903 sin (119905) where (ℎ 119896) is the center of circle 119903 is radius and 119905 represents angleStep 2 Identify unit tangent vector UTV (119905) larr 119891
1015840(119905)
10038171003817100381710038171198911015840(119905)
1003817100381710038171003817 and set (119883new 119884new) larr (119883 119884) + UTV (119905)Step 3 Calculate directional angle 120579 larr tanminus1((119884new minus 119884)(119883new minus 119883))Step 4 Solve simultaneous equations 119884 = 119898119883 + 119888 where 119898 larr (119884new minus 119884)(119883new minus 119883) to get position of second control point 119875
119887
Step 5 If 10038171003817100381710038171198751198861198751198871003817100381710038171003817 le
10038171003817100381710038171198751198871198751198881003817100381710038171003817 then 119875
119886larr 119875119886 119875119888larr 119875119888
else 119875119886larr 119875119888 119875119888larr 119875119886
If 120572 gt 1 then 119875start larr 119875119888 119875end larr 119875
119886
else 119875start larr 119875119886 119875end larr 119875
119888
Step 6 Translate 119875start = (0 0) rotate triangle such that 119875119887= (minus119883 0) for 120572 gt 1 119875
119887= (119883 0) for 120572 le 1
If 119875end = (119883 minus119884) or 119875end = (minus119883 minus119884) then reflect the triangle through 119909-axisStep 7 Compute 120579
119889larr 120587 minus cosminus1((minus 1003817100381710038171003817119875119886119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119887119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119887
1003817100381710038171003817
2
)(21003817100381710038171003817119875119887119875119888
1003817100381710038171003817
10038171003817100381710038171198751198861198751198871003817100381710038171003817))
Step 8 If 120572 gt 1 then 120579119891larr cosminus1((minus 1003817100381710038171003817119875119886119875119887
1003817100381710038171003817
2
+1003817100381710038171003817119875119887119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119888
1003817100381710038171003817
2
)(21003817100381710038171003817119875119886119875119888
1003817100381710038171003817
10038171003817100381710038171198751198871198751198881003817100381710038171003817))
else 120579119890larr cosminus1((minus 1003817100381710038171003817119875119887119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119887
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119888
1003817100381710038171003817
2
)(21003817100381710038171003817119875119886119875119887
1003817100381710038171003817
10038171003817100381710038171198751198861198751198881003817100381710038171003817))
Step 9 Identify bound of Λ If 120572 lt 1 then 119886 larr 1 times 10minus10 119887 larr 1(120579
119889(1 minus 120572))
else if 120572 gt 1 then 119886 larr 1 times 10minus10 119887 larr 1(120579
119889(120572 minus 1))
else 119886 larr 1 times 10minus10 119887 larr 1 times 10
5Step 10 Set tolerance Tol larr 10
minus10iteration number 119873 larr (1 In 2) In (| 119886 minus 119887 | Tol)
Step 11 Calculate Λ using Bisection methodStep 12 Determine scaling factor 119903scale larr
10038171003817100381710038171198751198861198751198881003817100381710038171003817
1003817100381710038171003817119875011987521003817100381710038171003817
Step 13 Scale 119903scale to the curve and transform inversely the triangle to its original positionStep 14 Construct LAC segment using (2)Step 15 OUTPUT (120579
119889 Λ 119903scale LAC segment)
End
Algorithm 1
the intersection point of the tangential line at both endpointsby solving the equations simultaneously
223 Affirmation of Initial and Last Point Position (Step 5)We assume the distance between 119875
119886and 119875
119887is less than 119875
119887
and 119875119888as proposed by Yoshida and Saito [10] The position
of 119875119886and 119875
119888is switched if the criterion is not satisfied
Otherwise the position remains unchanged Note that thereare two cases of constructing LAC segment based on the120572 value as illustrated in Figures 2 and 3 For case 120572 le 1the initial endpoint 119875
119886is the starting point during the curve
construction process 119875start On the other hand starting pointof curve construction process for case 120572 gt 1 is the lastendpoint 119875
119888
224 Transformation Process (Step 6) The transformationprocess includes translation rotation and reflection of theoriginal triangle that consist of119875
119886119875119887119875119888 Initially the original
triangle is translated to the origin so that the starting point119875start is located at the origin Next we rotate the triangleso that the second control point is placed on the 119909-axisFinally we ensure the last point of the curve constructionprocess 119875end is located at the positive 119910-axis region If notthe reflection process is needed to reflect the last point with
negative 119910-coordinate through the 119909-axis The configurationafter transformation is illustrated in Figures 2 and 3
225 Computation of Tangent and Turning Angle (Step 7and Step 8) We apply the law of cosine of a triangle tocalculate corresponding angles as shown in Figures 2 and 3The tangent angle 120579
119889and turning angle 120579
119890for case 120572 le 1
or 120579119891for case 120572 gt 1 will be used in the subsequent step for
calculating shape parameter Λ
226 Upper Bound of Shape Parameter Λ (Step 9) For thecasewhen120572 = 1 there is no upper bound for shape parameterΛ So the value of Λ can be arbitrarily large However thereare upper bounds for Λ when 120572 lt 1 and 120572 gt 1 ThereforetheΛ value of these two cases is restricted within the range of(119886 119887) where 119886 and 119887 are assigned a specified value as indicatedin the algorithm above
227 Shape Parameter Λ Calculation (Step 10 and Step 11)We determine the Λ value that satisfies the constraintsof endpoints and angles by using bisection method Firsttolerance and bound of Λ are used in the calculation ofiteration number for bisection methodThen the calculatingprocess will run until the iteration number is met Once theΛ
6 Journal of Applied Mathematics
(a) Euler spiral algorithm (b) LAC algorithm with two tangents setting for s-shaped curve and 120572 = minus1
(c) LACalgorithmwith two tangents setting for s-shaped curveand 120572 = 2
Figure 5 The results of fitting Euler spiral and LAC segment for two missing parts of a jug
value of the curve is obtained it is an interpolating curve thatsatisfies boundary constraints as shape parameter Λ is usedto control the curve so that the constraints are satisfied TheobtainedΛ value ensures the generated LAC segment starts atthe starting point and ends at the last point while the shape ofcurve follows the specified tangent vectors at two endpoints
228 Computation of Scale Factor (Step 12) We calculate thescaling factor by using the formula as shown in the algorithmwhere 119875
119886and 119875
119888are endpoints of the original triangle while
1198750and 119875
2are endpoints of the transformed triangle
229 Construction of LACSegment (Step 13ndashStep 15) In orderto obtain the curve which is defined in the original trianglewe inversely transform the triangle to its original position
The curve segment is constructed using (2) with the obtainedvalue of Λ 120579
119889 119903scale
3 Numerical Examples and Discussion
This section shows the input data and numerical and graph-ical results of implementing the algorithm presented inSection 2 for three examples For comparison purpose wepresent the results of Kimiarsquos algorithm [3] algorithm on thesame examples as well
31 The Graphical Results Figure 5 shows the first exampleof fitting Euler spiral and LAC segment to two parts of a jugThe results of the second example (artifact) and third example(vase) are depicted in Figures 6 and 7 respectively
Journal of Applied Mathematics 7
(a) Euler spiral algorithm (b) LAC algorithm when 120572 = minus1
(c) LAC algorithm when 120572 = 0
Figure 6 The results of fitting Euler spiral and LAC segment for an artifact
32 The Numerical Results The input data and the obtainedparameter values to render LAC segment for C-shaped andS-shaped gaps are shown in Tables 1 2 and 3 respectively
33 Discussion We implemented LAC algorithm with var-ious 120572 values for each example In Figures 5 6 and 7 weshowed a subfigure of LAC with 120572 = minus1 in between the gapsfor the missing part of the objectsThese examples are indeedEuler spiral similar to the work of Kimia et al [3] Users areindeed fitting a circle involute when 120572 = 2 as shown in eachfigure Nielsen spiral is fitted when 120572 = 0 and logarithmicspiral when 120572 = 1 The shape parameter 120572 definitely dictatesthe type of spiral the user desires to fit into the gap
Based on the comparison of graphical results in Figures5 6 and 7 we found that Kimiarsquos algorithm [3] and theLAC algorithm behave similarly However the proposed LACalgorithm gives more flexibility to designers as they canmanipulate the shape parameter 120572 and the tangent directionat the inflection point in order to get various S-shaped LACsegments
4 Conclusion
A LAC segment can be constructed by specifying the end-points and their tangent directions LAC is a family of aes-thetic curves where their curvature changes monotonicallyIt comprises Euler spiral logarithmic spiral Nielsen spiraland circle involute with different 120572 values The algorithmfor constructing LAC segment to solve shape completion
problem is defined in Section 2 The results of implementingboth Kimia et alrsquos [3] algorithm and the proposed LACalgorithm on the same images are depicted in Section 3LAC has an upper hand due to its flexibility although bothalgorithms are capable of solving shape completion problemsSince LAC represents a bigger family rather than Eulerspiral alone it is clear that more shapes are available to fillthe missing gap Moreover the resulting LAC segments aresatisfying119866
1 continuity where the curve segments interpolateboth endpoints in the specified tangent directions In case theuser wants a119866
2 LAC solution one may plug in the techniqueproposed by Miura et al [15] or employ 119866
2 GLAC [17] inwhich we may use shape parameter V in GLAC to satisfy thedesired curvatures
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge Universiti Malaysia Terengganu(GGP grant) and Ministry of Higher Education Malaysia(FRGS 59265) for providing financial aid to carry out thisresearch They further acknowledge anonymous referees fortheir constructive comments which improved the readabilityof this paper
8 Journal of Applied Mathematics
(a) Euler spiral algorithm (b) Three types of tangents at the inflection points and 120572 = minus1
(c) Tangents setting as in 7(b) and 120572 = minus1 (d) Tangents setting as in 7(b) and 120572 = 1
Figure 7 The results of fitting Euler spiral and LAC segment to a vase
Table 1 The input data for LAC algorithm
Figure Partgap shape Starting point Inflection point Terminating point119875119886
119875inflect 119875119888
Figure 5Outer C (47533 29067) mdash (408 512)Inner C (444 3193) mdash (4067 4907)
S (6867 198) (115139 317874) (132 44533)
Figure 6 Left C (1504 3208) mdash (1232 5896)Right C (853 307) mdash (824 583)
Figure 7C (504 126) mdash (176 120)
Right S (57667 37933) (501722 483092) (44533 598)Left S (21533 59933) (166565 4812) (94 376)
Journal of Applied Mathematics 9
Table 2 The value of parameters obtained in LAC algorithm for C-shaped gap
Figure Part Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) Outer C 0181102 197352 109383 106443Inner C 0187822 250305 153612 577559
Figure 5(c) Outer C 0236827 197352 0879684 179903Inner C 0307163 250305 0966927 144024
Figure 6(b) Left C 0319685 155147 100107 105562Right C 0204861 235562 147579 896424
Figure 6(c) Left C 053364 155147 100107 902434Right C 0610563 235562 147579 654159
Figure 7(c) mdash 0140054 11637 0601359 271762Figure 7(d) mdash 0169061 11637 0601359 270036
Table 3 The value of parameters obtained in LAC algorithm for S-shaped gap
Figure Part Line style Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) mdashPiece 1 Thick 0646746 0739623 0451367 107501
Dashed 113587 043963 0288258 152739
Piece 2 Thick 0140372 132647 0689653 935698Dashed 0473033 102648 063682 766964
Figure 5(c) mdashPiece 1 Thick 219523 043963 0151372 568396
Dashed 107216 0739623 0288256 293721
Piece 2 Thick 0811123 102648 0389659 222587Dashed 0156681 132647 063682 116479
Figure 7(c) Right S
Piece 1Thick 00000429688 0571501 0285752 227044Dashed 0698712 0871501 0585746 39271Dotted 245758 0371528 0285746 23657
Piece 2Thick 00118778 0908437 0455056 145074Dashed 0403597 120905 0755673 651694Dotted 0700923 0707704 045338 100657
Figure 7(d) Right S
Piece 1Thick 253147 0871501 0585746 411231Dashed 00000429592 0571501 0285752 227044Dotted 107779 0371528 0285746 257032
Piece 2Thick 12606 120905 0755673 474646Dashed 00120078 0908437 0455056 145071Dotted 248785 0707704 045338 673212
Figure 7(c) Left S
Piece 1Thick 0188412 0981157 0509674 121657Dashed 0603719 0781142 0471473 10399Dotted 0361402 138114 0909666 524523
Piece 2Thick 00000462514 0594672 0297337 218104Dashed 0884532 0394671 0297335 35514Dotted 0564167 0994666 0697331 275123
Figure 7(d) Left S
Piece 1Thick 161781 0781142 0471473 833887Dashed 0234412 0981157 0509674 120616Dotted 142816 138114 0909666 315215
Piece 2Thick 926646 0394671 0297335 314849Dashed 00000462423 0594672 0297337 218104Dotted 266752 0994666 0697331 266104
10 Journal of Applied Mathematics
References
[1] M Kolomenkin I Shimshoni and A Tal ldquoDemarcating curvesfor shape illustrationrdquo ACM Transactions on Graphics vol 27no 5 article 157 2008
[2] D Decarlo A Finkelstein S Rusinkiewicz and A SantellaldquoSuggestive contours for conveying shaperdquo ACM Transactionson Graphics vol 22 no 3 pp 848ndash855 2003
[3] B B Kimia I Frankel and A Popescu ldquoEuler spiral for shapecompletionrdquo International Journal of Computer Vision vol 54no 1ndash3 pp 159ndash182 2003
[4] T Judd F Durand and E Adelson ldquoApparent ridges for linedrawingrdquo ACM Transactions on Graphics vol 26 no 3 article19 7 pages 2007
[5] S Yoshizawa A Belyaev and H P Seidel ldquoFast and robustdetection of crest lines on meshesrdquo in Proceedings of the ACMSymposium on Solid and Physical Modeling (SPM rsquo05) pp 227ndash232 June 2005
[6] G Harary and A Tal ldquo3D Euler spirals for 3D curve comple-tionrdquo Computational Geometry Theory and Applications vol45 no 3 pp 115ndash126 2012
[7] I Kanaya Y Nakano and K Sato ldquoSimulated designerrsquoseyesmdashclassification of aesthetic surfacesrdquo in Proceedings of theInternational Conference onVirtual Systems andMultimedia pp289ndash296 2003
[8] R U Gobithaasan and K T Miura ldquoLogarithmic curvaturegraph as a shape interrogation toolrdquo Applied MathematicalSciences vol 8 no 16 pp 755ndash765 2014
[9] K T Miura J Sone A Yamashita and T Kaneko ldquoDerivationof a general formula of aesthetic curvesrdquo in Proceedings of 8thInternational Conference on Humans and Computers pp 166ndash171 Tokyo Japan 2005
[10] N Yoshida and T Saito ldquoInteractive aesthetic curve segmentsrdquoThe Visual Computer vol 22 no 9-11 pp 896ndash905 2006
[11] R U Gobithaasan and K T Miura ldquoAesthetic spiral for designrdquoSains Malaysiana vol 40 no 11 pp 1301ndash1305 2011
[12] JMAli RM Tookey J V Ball andAA Ball ldquoThe generalisedCornu spiral and its application to span generationrdquo Journal ofComputational and Applied Mathematics vol 102 no 1 pp 37ndash47 1999
[13] R U Gobithaasan R Karpagavalli and K T Miura ldquoShapeanalysis of generalized log-aesthetic curvesrdquo International Jour-nal of Mathematical Analysis vol 7 no 33 pp 1751ndash1759 2013
[14] R Ziatdinov N Yoshida and T Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012
[15] K T Miura D Shibuya R U Gobithaasan and S UsukildquoDesigning log-aesthetic splines with119866
2 continuityrdquoComputer-Aided Design and Applications vol 10 no 6 pp 1021ndash1032 2013
[16] K T Miura and R U Gobithaasan ldquoAesthetic curves andsurfaces in computer aided geometric designrdquo InternationalJournal of Automation Technology vol 8 no 3 pp 304ndash3162014
[17] R U Gobithaasan R Karpagavalli and K T Miura ldquoG2continuous generalized log aesthetic curvesrdquo Computer AidedDesign amp Application Accepted for publication
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
Pc
Pa
Pb
120579d
120579e
(a) Original position of controlpoints
P0 P1
P2
120579d998400120579e998400
(b) Position of control points after transformation
Figure 3 The configuration of triangle consists of three control points for case 120572 le 1 The curve inside the triangle represents the generatedLAC segment
three control points as proposed by Yoshida and Saito [10]Therefore by using thismethod it is only possible to generatea C-curve Figures 2 and 3 show the configuration of thetriangle formed by three control points for cases 120572 gt 1 and120572 le 1 respectively Step 5 in the algorithm below elaboratesthese configurations
S-Shape Basically a S-shape LAC segment consists of two C-shape curves joined with 119866
1 continuity In order to constructthe S-curve for a given gap the inflection point is essential Asimple approach used is that users are given the freedom todefine the inflection point so that the algorithm is carried outtwice consisting of two C-shape segments of LAC Thus thisapproach needs two endpoints its tangent vectors 120572 valueand an inflection point
The process involves breaking the S-curve into two C-curves so the first piece of curve is constructed betweenstarting point (black) and inflection point (gray) while thesecond piece is between inflection point (gray) and endpoint(black) Figure 4 illustrates the inflection point that we definefor one of the numerical examples in this paper
Algorithm 1 demonstrates the LAC algorithm
221 Computation of Tangent Angles (Step 1ndashStep 3) To auto-mate the process of defining the tangents at the endpointswe may use circular arc approximation (osculating circle)that best fits the endpoints and points surrounding them inorder to determine the tangent angle at both endpoints Sinceit is an osculating circle we substitute the points into theequation of circleThen we identify the center (ℎ 119896) radius 119903and turning angle 119905 of circle from three points which include
Figure 4 The input for S-shape completion problem
the endpoint Next we determine the unit tangent vector atthe endpoints and subsequently calculate the correspondingtangent angle by using specified formula as indicated in thealgorithm Alternately we may also allow the user to directlydefine tangent vectors at those points
222 Identification of Second Control Point (Step 4) Once theunit tangent vector at endpoints is identified we determinethe coordinate of the second control point 119875
119887which is
Journal of Applied Mathematics 5
REMARK Endpoints 119875119886 119875119888 inflection point 119875inflect and 120572 are user defined The tangent angle at both endpoints 120579
0 1205792 can be
obtained either from three point circular arc approximation around the endpoints or the user defines it Let (119883 119884) represent thecoordinate of a pointINPUT 119875
119886 119875119888 119875inflect 120572
OUTPUT 120579119889 Λ 119903scale LAC segment
BeginStep 1 Set 119891 (119905) larr ℎ + 119903 cos (119905) 119896 + 119903 sin (119905) where (ℎ 119896) is the center of circle 119903 is radius and 119905 represents angleStep 2 Identify unit tangent vector UTV (119905) larr 119891
1015840(119905)
10038171003817100381710038171198911015840(119905)
1003817100381710038171003817 and set (119883new 119884new) larr (119883 119884) + UTV (119905)Step 3 Calculate directional angle 120579 larr tanminus1((119884new minus 119884)(119883new minus 119883))Step 4 Solve simultaneous equations 119884 = 119898119883 + 119888 where 119898 larr (119884new minus 119884)(119883new minus 119883) to get position of second control point 119875
119887
Step 5 If 10038171003817100381710038171198751198861198751198871003817100381710038171003817 le
10038171003817100381710038171198751198871198751198881003817100381710038171003817 then 119875
119886larr 119875119886 119875119888larr 119875119888
else 119875119886larr 119875119888 119875119888larr 119875119886
If 120572 gt 1 then 119875start larr 119875119888 119875end larr 119875
119886
else 119875start larr 119875119886 119875end larr 119875
119888
Step 6 Translate 119875start = (0 0) rotate triangle such that 119875119887= (minus119883 0) for 120572 gt 1 119875
119887= (119883 0) for 120572 le 1
If 119875end = (119883 minus119884) or 119875end = (minus119883 minus119884) then reflect the triangle through 119909-axisStep 7 Compute 120579
119889larr 120587 minus cosminus1((minus 1003817100381710038171003817119875119886119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119887119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119887
1003817100381710038171003817
2
)(21003817100381710038171003817119875119887119875119888
1003817100381710038171003817
10038171003817100381710038171198751198861198751198871003817100381710038171003817))
Step 8 If 120572 gt 1 then 120579119891larr cosminus1((minus 1003817100381710038171003817119875119886119875119887
1003817100381710038171003817
2
+1003817100381710038171003817119875119887119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119888
1003817100381710038171003817
2
)(21003817100381710038171003817119875119886119875119888
1003817100381710038171003817
10038171003817100381710038171198751198871198751198881003817100381710038171003817))
else 120579119890larr cosminus1((minus 1003817100381710038171003817119875119887119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119887
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119888
1003817100381710038171003817
2
)(21003817100381710038171003817119875119886119875119887
1003817100381710038171003817
10038171003817100381710038171198751198861198751198881003817100381710038171003817))
Step 9 Identify bound of Λ If 120572 lt 1 then 119886 larr 1 times 10minus10 119887 larr 1(120579
119889(1 minus 120572))
else if 120572 gt 1 then 119886 larr 1 times 10minus10 119887 larr 1(120579
119889(120572 minus 1))
else 119886 larr 1 times 10minus10 119887 larr 1 times 10
5Step 10 Set tolerance Tol larr 10
minus10iteration number 119873 larr (1 In 2) In (| 119886 minus 119887 | Tol)
Step 11 Calculate Λ using Bisection methodStep 12 Determine scaling factor 119903scale larr
10038171003817100381710038171198751198861198751198881003817100381710038171003817
1003817100381710038171003817119875011987521003817100381710038171003817
Step 13 Scale 119903scale to the curve and transform inversely the triangle to its original positionStep 14 Construct LAC segment using (2)Step 15 OUTPUT (120579
119889 Λ 119903scale LAC segment)
End
Algorithm 1
the intersection point of the tangential line at both endpointsby solving the equations simultaneously
223 Affirmation of Initial and Last Point Position (Step 5)We assume the distance between 119875
119886and 119875
119887is less than 119875
119887
and 119875119888as proposed by Yoshida and Saito [10] The position
of 119875119886and 119875
119888is switched if the criterion is not satisfied
Otherwise the position remains unchanged Note that thereare two cases of constructing LAC segment based on the120572 value as illustrated in Figures 2 and 3 For case 120572 le 1the initial endpoint 119875
119886is the starting point during the curve
construction process 119875start On the other hand starting pointof curve construction process for case 120572 gt 1 is the lastendpoint 119875
119888
224 Transformation Process (Step 6) The transformationprocess includes translation rotation and reflection of theoriginal triangle that consist of119875
119886119875119887119875119888 Initially the original
triangle is translated to the origin so that the starting point119875start is located at the origin Next we rotate the triangleso that the second control point is placed on the 119909-axisFinally we ensure the last point of the curve constructionprocess 119875end is located at the positive 119910-axis region If notthe reflection process is needed to reflect the last point with
negative 119910-coordinate through the 119909-axis The configurationafter transformation is illustrated in Figures 2 and 3
225 Computation of Tangent and Turning Angle (Step 7and Step 8) We apply the law of cosine of a triangle tocalculate corresponding angles as shown in Figures 2 and 3The tangent angle 120579
119889and turning angle 120579
119890for case 120572 le 1
or 120579119891for case 120572 gt 1 will be used in the subsequent step for
calculating shape parameter Λ
226 Upper Bound of Shape Parameter Λ (Step 9) For thecasewhen120572 = 1 there is no upper bound for shape parameterΛ So the value of Λ can be arbitrarily large However thereare upper bounds for Λ when 120572 lt 1 and 120572 gt 1 ThereforetheΛ value of these two cases is restricted within the range of(119886 119887) where 119886 and 119887 are assigned a specified value as indicatedin the algorithm above
227 Shape Parameter Λ Calculation (Step 10 and Step 11)We determine the Λ value that satisfies the constraintsof endpoints and angles by using bisection method Firsttolerance and bound of Λ are used in the calculation ofiteration number for bisection methodThen the calculatingprocess will run until the iteration number is met Once theΛ
6 Journal of Applied Mathematics
(a) Euler spiral algorithm (b) LAC algorithm with two tangents setting for s-shaped curve and 120572 = minus1
(c) LACalgorithmwith two tangents setting for s-shaped curveand 120572 = 2
Figure 5 The results of fitting Euler spiral and LAC segment for two missing parts of a jug
value of the curve is obtained it is an interpolating curve thatsatisfies boundary constraints as shape parameter Λ is usedto control the curve so that the constraints are satisfied TheobtainedΛ value ensures the generated LAC segment starts atthe starting point and ends at the last point while the shape ofcurve follows the specified tangent vectors at two endpoints
228 Computation of Scale Factor (Step 12) We calculate thescaling factor by using the formula as shown in the algorithmwhere 119875
119886and 119875
119888are endpoints of the original triangle while
1198750and 119875
2are endpoints of the transformed triangle
229 Construction of LACSegment (Step 13ndashStep 15) In orderto obtain the curve which is defined in the original trianglewe inversely transform the triangle to its original position
The curve segment is constructed using (2) with the obtainedvalue of Λ 120579
119889 119903scale
3 Numerical Examples and Discussion
This section shows the input data and numerical and graph-ical results of implementing the algorithm presented inSection 2 for three examples For comparison purpose wepresent the results of Kimiarsquos algorithm [3] algorithm on thesame examples as well
31 The Graphical Results Figure 5 shows the first exampleof fitting Euler spiral and LAC segment to two parts of a jugThe results of the second example (artifact) and third example(vase) are depicted in Figures 6 and 7 respectively
Journal of Applied Mathematics 7
(a) Euler spiral algorithm (b) LAC algorithm when 120572 = minus1
(c) LAC algorithm when 120572 = 0
Figure 6 The results of fitting Euler spiral and LAC segment for an artifact
32 The Numerical Results The input data and the obtainedparameter values to render LAC segment for C-shaped andS-shaped gaps are shown in Tables 1 2 and 3 respectively
33 Discussion We implemented LAC algorithm with var-ious 120572 values for each example In Figures 5 6 and 7 weshowed a subfigure of LAC with 120572 = minus1 in between the gapsfor the missing part of the objectsThese examples are indeedEuler spiral similar to the work of Kimia et al [3] Users areindeed fitting a circle involute when 120572 = 2 as shown in eachfigure Nielsen spiral is fitted when 120572 = 0 and logarithmicspiral when 120572 = 1 The shape parameter 120572 definitely dictatesthe type of spiral the user desires to fit into the gap
Based on the comparison of graphical results in Figures5 6 and 7 we found that Kimiarsquos algorithm [3] and theLAC algorithm behave similarly However the proposed LACalgorithm gives more flexibility to designers as they canmanipulate the shape parameter 120572 and the tangent directionat the inflection point in order to get various S-shaped LACsegments
4 Conclusion
A LAC segment can be constructed by specifying the end-points and their tangent directions LAC is a family of aes-thetic curves where their curvature changes monotonicallyIt comprises Euler spiral logarithmic spiral Nielsen spiraland circle involute with different 120572 values The algorithmfor constructing LAC segment to solve shape completion
problem is defined in Section 2 The results of implementingboth Kimia et alrsquos [3] algorithm and the proposed LACalgorithm on the same images are depicted in Section 3LAC has an upper hand due to its flexibility although bothalgorithms are capable of solving shape completion problemsSince LAC represents a bigger family rather than Eulerspiral alone it is clear that more shapes are available to fillthe missing gap Moreover the resulting LAC segments aresatisfying119866
1 continuity where the curve segments interpolateboth endpoints in the specified tangent directions In case theuser wants a119866
2 LAC solution one may plug in the techniqueproposed by Miura et al [15] or employ 119866
2 GLAC [17] inwhich we may use shape parameter V in GLAC to satisfy thedesired curvatures
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge Universiti Malaysia Terengganu(GGP grant) and Ministry of Higher Education Malaysia(FRGS 59265) for providing financial aid to carry out thisresearch They further acknowledge anonymous referees fortheir constructive comments which improved the readabilityof this paper
8 Journal of Applied Mathematics
(a) Euler spiral algorithm (b) Three types of tangents at the inflection points and 120572 = minus1
(c) Tangents setting as in 7(b) and 120572 = minus1 (d) Tangents setting as in 7(b) and 120572 = 1
Figure 7 The results of fitting Euler spiral and LAC segment to a vase
Table 1 The input data for LAC algorithm
Figure Partgap shape Starting point Inflection point Terminating point119875119886
119875inflect 119875119888
Figure 5Outer C (47533 29067) mdash (408 512)Inner C (444 3193) mdash (4067 4907)
S (6867 198) (115139 317874) (132 44533)
Figure 6 Left C (1504 3208) mdash (1232 5896)Right C (853 307) mdash (824 583)
Figure 7C (504 126) mdash (176 120)
Right S (57667 37933) (501722 483092) (44533 598)Left S (21533 59933) (166565 4812) (94 376)
Journal of Applied Mathematics 9
Table 2 The value of parameters obtained in LAC algorithm for C-shaped gap
Figure Part Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) Outer C 0181102 197352 109383 106443Inner C 0187822 250305 153612 577559
Figure 5(c) Outer C 0236827 197352 0879684 179903Inner C 0307163 250305 0966927 144024
Figure 6(b) Left C 0319685 155147 100107 105562Right C 0204861 235562 147579 896424
Figure 6(c) Left C 053364 155147 100107 902434Right C 0610563 235562 147579 654159
Figure 7(c) mdash 0140054 11637 0601359 271762Figure 7(d) mdash 0169061 11637 0601359 270036
Table 3 The value of parameters obtained in LAC algorithm for S-shaped gap
Figure Part Line style Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) mdashPiece 1 Thick 0646746 0739623 0451367 107501
Dashed 113587 043963 0288258 152739
Piece 2 Thick 0140372 132647 0689653 935698Dashed 0473033 102648 063682 766964
Figure 5(c) mdashPiece 1 Thick 219523 043963 0151372 568396
Dashed 107216 0739623 0288256 293721
Piece 2 Thick 0811123 102648 0389659 222587Dashed 0156681 132647 063682 116479
Figure 7(c) Right S
Piece 1Thick 00000429688 0571501 0285752 227044Dashed 0698712 0871501 0585746 39271Dotted 245758 0371528 0285746 23657
Piece 2Thick 00118778 0908437 0455056 145074Dashed 0403597 120905 0755673 651694Dotted 0700923 0707704 045338 100657
Figure 7(d) Right S
Piece 1Thick 253147 0871501 0585746 411231Dashed 00000429592 0571501 0285752 227044Dotted 107779 0371528 0285746 257032
Piece 2Thick 12606 120905 0755673 474646Dashed 00120078 0908437 0455056 145071Dotted 248785 0707704 045338 673212
Figure 7(c) Left S
Piece 1Thick 0188412 0981157 0509674 121657Dashed 0603719 0781142 0471473 10399Dotted 0361402 138114 0909666 524523
Piece 2Thick 00000462514 0594672 0297337 218104Dashed 0884532 0394671 0297335 35514Dotted 0564167 0994666 0697331 275123
Figure 7(d) Left S
Piece 1Thick 161781 0781142 0471473 833887Dashed 0234412 0981157 0509674 120616Dotted 142816 138114 0909666 315215
Piece 2Thick 926646 0394671 0297335 314849Dashed 00000462423 0594672 0297337 218104Dotted 266752 0994666 0697331 266104
10 Journal of Applied Mathematics
References
[1] M Kolomenkin I Shimshoni and A Tal ldquoDemarcating curvesfor shape illustrationrdquo ACM Transactions on Graphics vol 27no 5 article 157 2008
[2] D Decarlo A Finkelstein S Rusinkiewicz and A SantellaldquoSuggestive contours for conveying shaperdquo ACM Transactionson Graphics vol 22 no 3 pp 848ndash855 2003
[3] B B Kimia I Frankel and A Popescu ldquoEuler spiral for shapecompletionrdquo International Journal of Computer Vision vol 54no 1ndash3 pp 159ndash182 2003
[4] T Judd F Durand and E Adelson ldquoApparent ridges for linedrawingrdquo ACM Transactions on Graphics vol 26 no 3 article19 7 pages 2007
[5] S Yoshizawa A Belyaev and H P Seidel ldquoFast and robustdetection of crest lines on meshesrdquo in Proceedings of the ACMSymposium on Solid and Physical Modeling (SPM rsquo05) pp 227ndash232 June 2005
[6] G Harary and A Tal ldquo3D Euler spirals for 3D curve comple-tionrdquo Computational Geometry Theory and Applications vol45 no 3 pp 115ndash126 2012
[7] I Kanaya Y Nakano and K Sato ldquoSimulated designerrsquoseyesmdashclassification of aesthetic surfacesrdquo in Proceedings of theInternational Conference onVirtual Systems andMultimedia pp289ndash296 2003
[8] R U Gobithaasan and K T Miura ldquoLogarithmic curvaturegraph as a shape interrogation toolrdquo Applied MathematicalSciences vol 8 no 16 pp 755ndash765 2014
[9] K T Miura J Sone A Yamashita and T Kaneko ldquoDerivationof a general formula of aesthetic curvesrdquo in Proceedings of 8thInternational Conference on Humans and Computers pp 166ndash171 Tokyo Japan 2005
[10] N Yoshida and T Saito ldquoInteractive aesthetic curve segmentsrdquoThe Visual Computer vol 22 no 9-11 pp 896ndash905 2006
[11] R U Gobithaasan and K T Miura ldquoAesthetic spiral for designrdquoSains Malaysiana vol 40 no 11 pp 1301ndash1305 2011
[12] JMAli RM Tookey J V Ball andAA Ball ldquoThe generalisedCornu spiral and its application to span generationrdquo Journal ofComputational and Applied Mathematics vol 102 no 1 pp 37ndash47 1999
[13] R U Gobithaasan R Karpagavalli and K T Miura ldquoShapeanalysis of generalized log-aesthetic curvesrdquo International Jour-nal of Mathematical Analysis vol 7 no 33 pp 1751ndash1759 2013
[14] R Ziatdinov N Yoshida and T Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012
[15] K T Miura D Shibuya R U Gobithaasan and S UsukildquoDesigning log-aesthetic splines with119866
2 continuityrdquoComputer-Aided Design and Applications vol 10 no 6 pp 1021ndash1032 2013
[16] K T Miura and R U Gobithaasan ldquoAesthetic curves andsurfaces in computer aided geometric designrdquo InternationalJournal of Automation Technology vol 8 no 3 pp 304ndash3162014
[17] R U Gobithaasan R Karpagavalli and K T Miura ldquoG2continuous generalized log aesthetic curvesrdquo Computer AidedDesign amp Application Accepted for publication
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
REMARK Endpoints 119875119886 119875119888 inflection point 119875inflect and 120572 are user defined The tangent angle at both endpoints 120579
0 1205792 can be
obtained either from three point circular arc approximation around the endpoints or the user defines it Let (119883 119884) represent thecoordinate of a pointINPUT 119875
119886 119875119888 119875inflect 120572
OUTPUT 120579119889 Λ 119903scale LAC segment
BeginStep 1 Set 119891 (119905) larr ℎ + 119903 cos (119905) 119896 + 119903 sin (119905) where (ℎ 119896) is the center of circle 119903 is radius and 119905 represents angleStep 2 Identify unit tangent vector UTV (119905) larr 119891
1015840(119905)
10038171003817100381710038171198911015840(119905)
1003817100381710038171003817 and set (119883new 119884new) larr (119883 119884) + UTV (119905)Step 3 Calculate directional angle 120579 larr tanminus1((119884new minus 119884)(119883new minus 119883))Step 4 Solve simultaneous equations 119884 = 119898119883 + 119888 where 119898 larr (119884new minus 119884)(119883new minus 119883) to get position of second control point 119875
119887
Step 5 If 10038171003817100381710038171198751198861198751198871003817100381710038171003817 le
10038171003817100381710038171198751198871198751198881003817100381710038171003817 then 119875
119886larr 119875119886 119875119888larr 119875119888
else 119875119886larr 119875119888 119875119888larr 119875119886
If 120572 gt 1 then 119875start larr 119875119888 119875end larr 119875
119886
else 119875start larr 119875119886 119875end larr 119875
119888
Step 6 Translate 119875start = (0 0) rotate triangle such that 119875119887= (minus119883 0) for 120572 gt 1 119875
119887= (119883 0) for 120572 le 1
If 119875end = (119883 minus119884) or 119875end = (minus119883 minus119884) then reflect the triangle through 119909-axisStep 7 Compute 120579
119889larr 120587 minus cosminus1((minus 1003817100381710038171003817119875119886119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119887119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119887
1003817100381710038171003817
2
)(21003817100381710038171003817119875119887119875119888
1003817100381710038171003817
10038171003817100381710038171198751198861198751198871003817100381710038171003817))
Step 8 If 120572 gt 1 then 120579119891larr cosminus1((minus 1003817100381710038171003817119875119886119875119887
1003817100381710038171003817
2
+1003817100381710038171003817119875119887119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119888
1003817100381710038171003817
2
)(21003817100381710038171003817119875119886119875119888
1003817100381710038171003817
10038171003817100381710038171198751198871198751198881003817100381710038171003817))
else 120579119890larr cosminus1((minus 1003817100381710038171003817119875119887119875119888
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119887
1003817100381710038171003817
2
+1003817100381710038171003817119875119886119875119888
1003817100381710038171003817
2
)(21003817100381710038171003817119875119886119875119887
1003817100381710038171003817
10038171003817100381710038171198751198861198751198881003817100381710038171003817))
Step 9 Identify bound of Λ If 120572 lt 1 then 119886 larr 1 times 10minus10 119887 larr 1(120579
119889(1 minus 120572))
else if 120572 gt 1 then 119886 larr 1 times 10minus10 119887 larr 1(120579
119889(120572 minus 1))
else 119886 larr 1 times 10minus10 119887 larr 1 times 10
5Step 10 Set tolerance Tol larr 10
minus10iteration number 119873 larr (1 In 2) In (| 119886 minus 119887 | Tol)
Step 11 Calculate Λ using Bisection methodStep 12 Determine scaling factor 119903scale larr
10038171003817100381710038171198751198861198751198881003817100381710038171003817
1003817100381710038171003817119875011987521003817100381710038171003817
Step 13 Scale 119903scale to the curve and transform inversely the triangle to its original positionStep 14 Construct LAC segment using (2)Step 15 OUTPUT (120579
119889 Λ 119903scale LAC segment)
End
Algorithm 1
the intersection point of the tangential line at both endpointsby solving the equations simultaneously
223 Affirmation of Initial and Last Point Position (Step 5)We assume the distance between 119875
119886and 119875
119887is less than 119875
119887
and 119875119888as proposed by Yoshida and Saito [10] The position
of 119875119886and 119875
119888is switched if the criterion is not satisfied
Otherwise the position remains unchanged Note that thereare two cases of constructing LAC segment based on the120572 value as illustrated in Figures 2 and 3 For case 120572 le 1the initial endpoint 119875
119886is the starting point during the curve
construction process 119875start On the other hand starting pointof curve construction process for case 120572 gt 1 is the lastendpoint 119875
119888
224 Transformation Process (Step 6) The transformationprocess includes translation rotation and reflection of theoriginal triangle that consist of119875
119886119875119887119875119888 Initially the original
triangle is translated to the origin so that the starting point119875start is located at the origin Next we rotate the triangleso that the second control point is placed on the 119909-axisFinally we ensure the last point of the curve constructionprocess 119875end is located at the positive 119910-axis region If notthe reflection process is needed to reflect the last point with
negative 119910-coordinate through the 119909-axis The configurationafter transformation is illustrated in Figures 2 and 3
225 Computation of Tangent and Turning Angle (Step 7and Step 8) We apply the law of cosine of a triangle tocalculate corresponding angles as shown in Figures 2 and 3The tangent angle 120579
119889and turning angle 120579
119890for case 120572 le 1
or 120579119891for case 120572 gt 1 will be used in the subsequent step for
calculating shape parameter Λ
226 Upper Bound of Shape Parameter Λ (Step 9) For thecasewhen120572 = 1 there is no upper bound for shape parameterΛ So the value of Λ can be arbitrarily large However thereare upper bounds for Λ when 120572 lt 1 and 120572 gt 1 ThereforetheΛ value of these two cases is restricted within the range of(119886 119887) where 119886 and 119887 are assigned a specified value as indicatedin the algorithm above
227 Shape Parameter Λ Calculation (Step 10 and Step 11)We determine the Λ value that satisfies the constraintsof endpoints and angles by using bisection method Firsttolerance and bound of Λ are used in the calculation ofiteration number for bisection methodThen the calculatingprocess will run until the iteration number is met Once theΛ
6 Journal of Applied Mathematics
(a) Euler spiral algorithm (b) LAC algorithm with two tangents setting for s-shaped curve and 120572 = minus1
(c) LACalgorithmwith two tangents setting for s-shaped curveand 120572 = 2
Figure 5 The results of fitting Euler spiral and LAC segment for two missing parts of a jug
value of the curve is obtained it is an interpolating curve thatsatisfies boundary constraints as shape parameter Λ is usedto control the curve so that the constraints are satisfied TheobtainedΛ value ensures the generated LAC segment starts atthe starting point and ends at the last point while the shape ofcurve follows the specified tangent vectors at two endpoints
228 Computation of Scale Factor (Step 12) We calculate thescaling factor by using the formula as shown in the algorithmwhere 119875
119886and 119875
119888are endpoints of the original triangle while
1198750and 119875
2are endpoints of the transformed triangle
229 Construction of LACSegment (Step 13ndashStep 15) In orderto obtain the curve which is defined in the original trianglewe inversely transform the triangle to its original position
The curve segment is constructed using (2) with the obtainedvalue of Λ 120579
119889 119903scale
3 Numerical Examples and Discussion
This section shows the input data and numerical and graph-ical results of implementing the algorithm presented inSection 2 for three examples For comparison purpose wepresent the results of Kimiarsquos algorithm [3] algorithm on thesame examples as well
31 The Graphical Results Figure 5 shows the first exampleof fitting Euler spiral and LAC segment to two parts of a jugThe results of the second example (artifact) and third example(vase) are depicted in Figures 6 and 7 respectively
Journal of Applied Mathematics 7
(a) Euler spiral algorithm (b) LAC algorithm when 120572 = minus1
(c) LAC algorithm when 120572 = 0
Figure 6 The results of fitting Euler spiral and LAC segment for an artifact
32 The Numerical Results The input data and the obtainedparameter values to render LAC segment for C-shaped andS-shaped gaps are shown in Tables 1 2 and 3 respectively
33 Discussion We implemented LAC algorithm with var-ious 120572 values for each example In Figures 5 6 and 7 weshowed a subfigure of LAC with 120572 = minus1 in between the gapsfor the missing part of the objectsThese examples are indeedEuler spiral similar to the work of Kimia et al [3] Users areindeed fitting a circle involute when 120572 = 2 as shown in eachfigure Nielsen spiral is fitted when 120572 = 0 and logarithmicspiral when 120572 = 1 The shape parameter 120572 definitely dictatesthe type of spiral the user desires to fit into the gap
Based on the comparison of graphical results in Figures5 6 and 7 we found that Kimiarsquos algorithm [3] and theLAC algorithm behave similarly However the proposed LACalgorithm gives more flexibility to designers as they canmanipulate the shape parameter 120572 and the tangent directionat the inflection point in order to get various S-shaped LACsegments
4 Conclusion
A LAC segment can be constructed by specifying the end-points and their tangent directions LAC is a family of aes-thetic curves where their curvature changes monotonicallyIt comprises Euler spiral logarithmic spiral Nielsen spiraland circle involute with different 120572 values The algorithmfor constructing LAC segment to solve shape completion
problem is defined in Section 2 The results of implementingboth Kimia et alrsquos [3] algorithm and the proposed LACalgorithm on the same images are depicted in Section 3LAC has an upper hand due to its flexibility although bothalgorithms are capable of solving shape completion problemsSince LAC represents a bigger family rather than Eulerspiral alone it is clear that more shapes are available to fillthe missing gap Moreover the resulting LAC segments aresatisfying119866
1 continuity where the curve segments interpolateboth endpoints in the specified tangent directions In case theuser wants a119866
2 LAC solution one may plug in the techniqueproposed by Miura et al [15] or employ 119866
2 GLAC [17] inwhich we may use shape parameter V in GLAC to satisfy thedesired curvatures
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge Universiti Malaysia Terengganu(GGP grant) and Ministry of Higher Education Malaysia(FRGS 59265) for providing financial aid to carry out thisresearch They further acknowledge anonymous referees fortheir constructive comments which improved the readabilityof this paper
8 Journal of Applied Mathematics
(a) Euler spiral algorithm (b) Three types of tangents at the inflection points and 120572 = minus1
(c) Tangents setting as in 7(b) and 120572 = minus1 (d) Tangents setting as in 7(b) and 120572 = 1
Figure 7 The results of fitting Euler spiral and LAC segment to a vase
Table 1 The input data for LAC algorithm
Figure Partgap shape Starting point Inflection point Terminating point119875119886
119875inflect 119875119888
Figure 5Outer C (47533 29067) mdash (408 512)Inner C (444 3193) mdash (4067 4907)
S (6867 198) (115139 317874) (132 44533)
Figure 6 Left C (1504 3208) mdash (1232 5896)Right C (853 307) mdash (824 583)
Figure 7C (504 126) mdash (176 120)
Right S (57667 37933) (501722 483092) (44533 598)Left S (21533 59933) (166565 4812) (94 376)
Journal of Applied Mathematics 9
Table 2 The value of parameters obtained in LAC algorithm for C-shaped gap
Figure Part Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) Outer C 0181102 197352 109383 106443Inner C 0187822 250305 153612 577559
Figure 5(c) Outer C 0236827 197352 0879684 179903Inner C 0307163 250305 0966927 144024
Figure 6(b) Left C 0319685 155147 100107 105562Right C 0204861 235562 147579 896424
Figure 6(c) Left C 053364 155147 100107 902434Right C 0610563 235562 147579 654159
Figure 7(c) mdash 0140054 11637 0601359 271762Figure 7(d) mdash 0169061 11637 0601359 270036
Table 3 The value of parameters obtained in LAC algorithm for S-shaped gap
Figure Part Line style Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) mdashPiece 1 Thick 0646746 0739623 0451367 107501
Dashed 113587 043963 0288258 152739
Piece 2 Thick 0140372 132647 0689653 935698Dashed 0473033 102648 063682 766964
Figure 5(c) mdashPiece 1 Thick 219523 043963 0151372 568396
Dashed 107216 0739623 0288256 293721
Piece 2 Thick 0811123 102648 0389659 222587Dashed 0156681 132647 063682 116479
Figure 7(c) Right S
Piece 1Thick 00000429688 0571501 0285752 227044Dashed 0698712 0871501 0585746 39271Dotted 245758 0371528 0285746 23657
Piece 2Thick 00118778 0908437 0455056 145074Dashed 0403597 120905 0755673 651694Dotted 0700923 0707704 045338 100657
Figure 7(d) Right S
Piece 1Thick 253147 0871501 0585746 411231Dashed 00000429592 0571501 0285752 227044Dotted 107779 0371528 0285746 257032
Piece 2Thick 12606 120905 0755673 474646Dashed 00120078 0908437 0455056 145071Dotted 248785 0707704 045338 673212
Figure 7(c) Left S
Piece 1Thick 0188412 0981157 0509674 121657Dashed 0603719 0781142 0471473 10399Dotted 0361402 138114 0909666 524523
Piece 2Thick 00000462514 0594672 0297337 218104Dashed 0884532 0394671 0297335 35514Dotted 0564167 0994666 0697331 275123
Figure 7(d) Left S
Piece 1Thick 161781 0781142 0471473 833887Dashed 0234412 0981157 0509674 120616Dotted 142816 138114 0909666 315215
Piece 2Thick 926646 0394671 0297335 314849Dashed 00000462423 0594672 0297337 218104Dotted 266752 0994666 0697331 266104
10 Journal of Applied Mathematics
References
[1] M Kolomenkin I Shimshoni and A Tal ldquoDemarcating curvesfor shape illustrationrdquo ACM Transactions on Graphics vol 27no 5 article 157 2008
[2] D Decarlo A Finkelstein S Rusinkiewicz and A SantellaldquoSuggestive contours for conveying shaperdquo ACM Transactionson Graphics vol 22 no 3 pp 848ndash855 2003
[3] B B Kimia I Frankel and A Popescu ldquoEuler spiral for shapecompletionrdquo International Journal of Computer Vision vol 54no 1ndash3 pp 159ndash182 2003
[4] T Judd F Durand and E Adelson ldquoApparent ridges for linedrawingrdquo ACM Transactions on Graphics vol 26 no 3 article19 7 pages 2007
[5] S Yoshizawa A Belyaev and H P Seidel ldquoFast and robustdetection of crest lines on meshesrdquo in Proceedings of the ACMSymposium on Solid and Physical Modeling (SPM rsquo05) pp 227ndash232 June 2005
[6] G Harary and A Tal ldquo3D Euler spirals for 3D curve comple-tionrdquo Computational Geometry Theory and Applications vol45 no 3 pp 115ndash126 2012
[7] I Kanaya Y Nakano and K Sato ldquoSimulated designerrsquoseyesmdashclassification of aesthetic surfacesrdquo in Proceedings of theInternational Conference onVirtual Systems andMultimedia pp289ndash296 2003
[8] R U Gobithaasan and K T Miura ldquoLogarithmic curvaturegraph as a shape interrogation toolrdquo Applied MathematicalSciences vol 8 no 16 pp 755ndash765 2014
[9] K T Miura J Sone A Yamashita and T Kaneko ldquoDerivationof a general formula of aesthetic curvesrdquo in Proceedings of 8thInternational Conference on Humans and Computers pp 166ndash171 Tokyo Japan 2005
[10] N Yoshida and T Saito ldquoInteractive aesthetic curve segmentsrdquoThe Visual Computer vol 22 no 9-11 pp 896ndash905 2006
[11] R U Gobithaasan and K T Miura ldquoAesthetic spiral for designrdquoSains Malaysiana vol 40 no 11 pp 1301ndash1305 2011
[12] JMAli RM Tookey J V Ball andAA Ball ldquoThe generalisedCornu spiral and its application to span generationrdquo Journal ofComputational and Applied Mathematics vol 102 no 1 pp 37ndash47 1999
[13] R U Gobithaasan R Karpagavalli and K T Miura ldquoShapeanalysis of generalized log-aesthetic curvesrdquo International Jour-nal of Mathematical Analysis vol 7 no 33 pp 1751ndash1759 2013
[14] R Ziatdinov N Yoshida and T Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012
[15] K T Miura D Shibuya R U Gobithaasan and S UsukildquoDesigning log-aesthetic splines with119866
2 continuityrdquoComputer-Aided Design and Applications vol 10 no 6 pp 1021ndash1032 2013
[16] K T Miura and R U Gobithaasan ldquoAesthetic curves andsurfaces in computer aided geometric designrdquo InternationalJournal of Automation Technology vol 8 no 3 pp 304ndash3162014
[17] R U Gobithaasan R Karpagavalli and K T Miura ldquoG2continuous generalized log aesthetic curvesrdquo Computer AidedDesign amp Application Accepted for publication
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
(a) Euler spiral algorithm (b) LAC algorithm with two tangents setting for s-shaped curve and 120572 = minus1
(c) LACalgorithmwith two tangents setting for s-shaped curveand 120572 = 2
Figure 5 The results of fitting Euler spiral and LAC segment for two missing parts of a jug
value of the curve is obtained it is an interpolating curve thatsatisfies boundary constraints as shape parameter Λ is usedto control the curve so that the constraints are satisfied TheobtainedΛ value ensures the generated LAC segment starts atthe starting point and ends at the last point while the shape ofcurve follows the specified tangent vectors at two endpoints
228 Computation of Scale Factor (Step 12) We calculate thescaling factor by using the formula as shown in the algorithmwhere 119875
119886and 119875
119888are endpoints of the original triangle while
1198750and 119875
2are endpoints of the transformed triangle
229 Construction of LACSegment (Step 13ndashStep 15) In orderto obtain the curve which is defined in the original trianglewe inversely transform the triangle to its original position
The curve segment is constructed using (2) with the obtainedvalue of Λ 120579
119889 119903scale
3 Numerical Examples and Discussion
This section shows the input data and numerical and graph-ical results of implementing the algorithm presented inSection 2 for three examples For comparison purpose wepresent the results of Kimiarsquos algorithm [3] algorithm on thesame examples as well
31 The Graphical Results Figure 5 shows the first exampleof fitting Euler spiral and LAC segment to two parts of a jugThe results of the second example (artifact) and third example(vase) are depicted in Figures 6 and 7 respectively
Journal of Applied Mathematics 7
(a) Euler spiral algorithm (b) LAC algorithm when 120572 = minus1
(c) LAC algorithm when 120572 = 0
Figure 6 The results of fitting Euler spiral and LAC segment for an artifact
32 The Numerical Results The input data and the obtainedparameter values to render LAC segment for C-shaped andS-shaped gaps are shown in Tables 1 2 and 3 respectively
33 Discussion We implemented LAC algorithm with var-ious 120572 values for each example In Figures 5 6 and 7 weshowed a subfigure of LAC with 120572 = minus1 in between the gapsfor the missing part of the objectsThese examples are indeedEuler spiral similar to the work of Kimia et al [3] Users areindeed fitting a circle involute when 120572 = 2 as shown in eachfigure Nielsen spiral is fitted when 120572 = 0 and logarithmicspiral when 120572 = 1 The shape parameter 120572 definitely dictatesthe type of spiral the user desires to fit into the gap
Based on the comparison of graphical results in Figures5 6 and 7 we found that Kimiarsquos algorithm [3] and theLAC algorithm behave similarly However the proposed LACalgorithm gives more flexibility to designers as they canmanipulate the shape parameter 120572 and the tangent directionat the inflection point in order to get various S-shaped LACsegments
4 Conclusion
A LAC segment can be constructed by specifying the end-points and their tangent directions LAC is a family of aes-thetic curves where their curvature changes monotonicallyIt comprises Euler spiral logarithmic spiral Nielsen spiraland circle involute with different 120572 values The algorithmfor constructing LAC segment to solve shape completion
problem is defined in Section 2 The results of implementingboth Kimia et alrsquos [3] algorithm and the proposed LACalgorithm on the same images are depicted in Section 3LAC has an upper hand due to its flexibility although bothalgorithms are capable of solving shape completion problemsSince LAC represents a bigger family rather than Eulerspiral alone it is clear that more shapes are available to fillthe missing gap Moreover the resulting LAC segments aresatisfying119866
1 continuity where the curve segments interpolateboth endpoints in the specified tangent directions In case theuser wants a119866
2 LAC solution one may plug in the techniqueproposed by Miura et al [15] or employ 119866
2 GLAC [17] inwhich we may use shape parameter V in GLAC to satisfy thedesired curvatures
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge Universiti Malaysia Terengganu(GGP grant) and Ministry of Higher Education Malaysia(FRGS 59265) for providing financial aid to carry out thisresearch They further acknowledge anonymous referees fortheir constructive comments which improved the readabilityof this paper
8 Journal of Applied Mathematics
(a) Euler spiral algorithm (b) Three types of tangents at the inflection points and 120572 = minus1
(c) Tangents setting as in 7(b) and 120572 = minus1 (d) Tangents setting as in 7(b) and 120572 = 1
Figure 7 The results of fitting Euler spiral and LAC segment to a vase
Table 1 The input data for LAC algorithm
Figure Partgap shape Starting point Inflection point Terminating point119875119886
119875inflect 119875119888
Figure 5Outer C (47533 29067) mdash (408 512)Inner C (444 3193) mdash (4067 4907)
S (6867 198) (115139 317874) (132 44533)
Figure 6 Left C (1504 3208) mdash (1232 5896)Right C (853 307) mdash (824 583)
Figure 7C (504 126) mdash (176 120)
Right S (57667 37933) (501722 483092) (44533 598)Left S (21533 59933) (166565 4812) (94 376)
Journal of Applied Mathematics 9
Table 2 The value of parameters obtained in LAC algorithm for C-shaped gap
Figure Part Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) Outer C 0181102 197352 109383 106443Inner C 0187822 250305 153612 577559
Figure 5(c) Outer C 0236827 197352 0879684 179903Inner C 0307163 250305 0966927 144024
Figure 6(b) Left C 0319685 155147 100107 105562Right C 0204861 235562 147579 896424
Figure 6(c) Left C 053364 155147 100107 902434Right C 0610563 235562 147579 654159
Figure 7(c) mdash 0140054 11637 0601359 271762Figure 7(d) mdash 0169061 11637 0601359 270036
Table 3 The value of parameters obtained in LAC algorithm for S-shaped gap
Figure Part Line style Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) mdashPiece 1 Thick 0646746 0739623 0451367 107501
Dashed 113587 043963 0288258 152739
Piece 2 Thick 0140372 132647 0689653 935698Dashed 0473033 102648 063682 766964
Figure 5(c) mdashPiece 1 Thick 219523 043963 0151372 568396
Dashed 107216 0739623 0288256 293721
Piece 2 Thick 0811123 102648 0389659 222587Dashed 0156681 132647 063682 116479
Figure 7(c) Right S
Piece 1Thick 00000429688 0571501 0285752 227044Dashed 0698712 0871501 0585746 39271Dotted 245758 0371528 0285746 23657
Piece 2Thick 00118778 0908437 0455056 145074Dashed 0403597 120905 0755673 651694Dotted 0700923 0707704 045338 100657
Figure 7(d) Right S
Piece 1Thick 253147 0871501 0585746 411231Dashed 00000429592 0571501 0285752 227044Dotted 107779 0371528 0285746 257032
Piece 2Thick 12606 120905 0755673 474646Dashed 00120078 0908437 0455056 145071Dotted 248785 0707704 045338 673212
Figure 7(c) Left S
Piece 1Thick 0188412 0981157 0509674 121657Dashed 0603719 0781142 0471473 10399Dotted 0361402 138114 0909666 524523
Piece 2Thick 00000462514 0594672 0297337 218104Dashed 0884532 0394671 0297335 35514Dotted 0564167 0994666 0697331 275123
Figure 7(d) Left S
Piece 1Thick 161781 0781142 0471473 833887Dashed 0234412 0981157 0509674 120616Dotted 142816 138114 0909666 315215
Piece 2Thick 926646 0394671 0297335 314849Dashed 00000462423 0594672 0297337 218104Dotted 266752 0994666 0697331 266104
10 Journal of Applied Mathematics
References
[1] M Kolomenkin I Shimshoni and A Tal ldquoDemarcating curvesfor shape illustrationrdquo ACM Transactions on Graphics vol 27no 5 article 157 2008
[2] D Decarlo A Finkelstein S Rusinkiewicz and A SantellaldquoSuggestive contours for conveying shaperdquo ACM Transactionson Graphics vol 22 no 3 pp 848ndash855 2003
[3] B B Kimia I Frankel and A Popescu ldquoEuler spiral for shapecompletionrdquo International Journal of Computer Vision vol 54no 1ndash3 pp 159ndash182 2003
[4] T Judd F Durand and E Adelson ldquoApparent ridges for linedrawingrdquo ACM Transactions on Graphics vol 26 no 3 article19 7 pages 2007
[5] S Yoshizawa A Belyaev and H P Seidel ldquoFast and robustdetection of crest lines on meshesrdquo in Proceedings of the ACMSymposium on Solid and Physical Modeling (SPM rsquo05) pp 227ndash232 June 2005
[6] G Harary and A Tal ldquo3D Euler spirals for 3D curve comple-tionrdquo Computational Geometry Theory and Applications vol45 no 3 pp 115ndash126 2012
[7] I Kanaya Y Nakano and K Sato ldquoSimulated designerrsquoseyesmdashclassification of aesthetic surfacesrdquo in Proceedings of theInternational Conference onVirtual Systems andMultimedia pp289ndash296 2003
[8] R U Gobithaasan and K T Miura ldquoLogarithmic curvaturegraph as a shape interrogation toolrdquo Applied MathematicalSciences vol 8 no 16 pp 755ndash765 2014
[9] K T Miura J Sone A Yamashita and T Kaneko ldquoDerivationof a general formula of aesthetic curvesrdquo in Proceedings of 8thInternational Conference on Humans and Computers pp 166ndash171 Tokyo Japan 2005
[10] N Yoshida and T Saito ldquoInteractive aesthetic curve segmentsrdquoThe Visual Computer vol 22 no 9-11 pp 896ndash905 2006
[11] R U Gobithaasan and K T Miura ldquoAesthetic spiral for designrdquoSains Malaysiana vol 40 no 11 pp 1301ndash1305 2011
[12] JMAli RM Tookey J V Ball andAA Ball ldquoThe generalisedCornu spiral and its application to span generationrdquo Journal ofComputational and Applied Mathematics vol 102 no 1 pp 37ndash47 1999
[13] R U Gobithaasan R Karpagavalli and K T Miura ldquoShapeanalysis of generalized log-aesthetic curvesrdquo International Jour-nal of Mathematical Analysis vol 7 no 33 pp 1751ndash1759 2013
[14] R Ziatdinov N Yoshida and T Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012
[15] K T Miura D Shibuya R U Gobithaasan and S UsukildquoDesigning log-aesthetic splines with119866
2 continuityrdquoComputer-Aided Design and Applications vol 10 no 6 pp 1021ndash1032 2013
[16] K T Miura and R U Gobithaasan ldquoAesthetic curves andsurfaces in computer aided geometric designrdquo InternationalJournal of Automation Technology vol 8 no 3 pp 304ndash3162014
[17] R U Gobithaasan R Karpagavalli and K T Miura ldquoG2continuous generalized log aesthetic curvesrdquo Computer AidedDesign amp Application Accepted for publication
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
(a) Euler spiral algorithm (b) LAC algorithm when 120572 = minus1
(c) LAC algorithm when 120572 = 0
Figure 6 The results of fitting Euler spiral and LAC segment for an artifact
32 The Numerical Results The input data and the obtainedparameter values to render LAC segment for C-shaped andS-shaped gaps are shown in Tables 1 2 and 3 respectively
33 Discussion We implemented LAC algorithm with var-ious 120572 values for each example In Figures 5 6 and 7 weshowed a subfigure of LAC with 120572 = minus1 in between the gapsfor the missing part of the objectsThese examples are indeedEuler spiral similar to the work of Kimia et al [3] Users areindeed fitting a circle involute when 120572 = 2 as shown in eachfigure Nielsen spiral is fitted when 120572 = 0 and logarithmicspiral when 120572 = 1 The shape parameter 120572 definitely dictatesthe type of spiral the user desires to fit into the gap
Based on the comparison of graphical results in Figures5 6 and 7 we found that Kimiarsquos algorithm [3] and theLAC algorithm behave similarly However the proposed LACalgorithm gives more flexibility to designers as they canmanipulate the shape parameter 120572 and the tangent directionat the inflection point in order to get various S-shaped LACsegments
4 Conclusion
A LAC segment can be constructed by specifying the end-points and their tangent directions LAC is a family of aes-thetic curves where their curvature changes monotonicallyIt comprises Euler spiral logarithmic spiral Nielsen spiraland circle involute with different 120572 values The algorithmfor constructing LAC segment to solve shape completion
problem is defined in Section 2 The results of implementingboth Kimia et alrsquos [3] algorithm and the proposed LACalgorithm on the same images are depicted in Section 3LAC has an upper hand due to its flexibility although bothalgorithms are capable of solving shape completion problemsSince LAC represents a bigger family rather than Eulerspiral alone it is clear that more shapes are available to fillthe missing gap Moreover the resulting LAC segments aresatisfying119866
1 continuity where the curve segments interpolateboth endpoints in the specified tangent directions In case theuser wants a119866
2 LAC solution one may plug in the techniqueproposed by Miura et al [15] or employ 119866
2 GLAC [17] inwhich we may use shape parameter V in GLAC to satisfy thedesired curvatures
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge Universiti Malaysia Terengganu(GGP grant) and Ministry of Higher Education Malaysia(FRGS 59265) for providing financial aid to carry out thisresearch They further acknowledge anonymous referees fortheir constructive comments which improved the readabilityof this paper
8 Journal of Applied Mathematics
(a) Euler spiral algorithm (b) Three types of tangents at the inflection points and 120572 = minus1
(c) Tangents setting as in 7(b) and 120572 = minus1 (d) Tangents setting as in 7(b) and 120572 = 1
Figure 7 The results of fitting Euler spiral and LAC segment to a vase
Table 1 The input data for LAC algorithm
Figure Partgap shape Starting point Inflection point Terminating point119875119886
119875inflect 119875119888
Figure 5Outer C (47533 29067) mdash (408 512)Inner C (444 3193) mdash (4067 4907)
S (6867 198) (115139 317874) (132 44533)
Figure 6 Left C (1504 3208) mdash (1232 5896)Right C (853 307) mdash (824 583)
Figure 7C (504 126) mdash (176 120)
Right S (57667 37933) (501722 483092) (44533 598)Left S (21533 59933) (166565 4812) (94 376)
Journal of Applied Mathematics 9
Table 2 The value of parameters obtained in LAC algorithm for C-shaped gap
Figure Part Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) Outer C 0181102 197352 109383 106443Inner C 0187822 250305 153612 577559
Figure 5(c) Outer C 0236827 197352 0879684 179903Inner C 0307163 250305 0966927 144024
Figure 6(b) Left C 0319685 155147 100107 105562Right C 0204861 235562 147579 896424
Figure 6(c) Left C 053364 155147 100107 902434Right C 0610563 235562 147579 654159
Figure 7(c) mdash 0140054 11637 0601359 271762Figure 7(d) mdash 0169061 11637 0601359 270036
Table 3 The value of parameters obtained in LAC algorithm for S-shaped gap
Figure Part Line style Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) mdashPiece 1 Thick 0646746 0739623 0451367 107501
Dashed 113587 043963 0288258 152739
Piece 2 Thick 0140372 132647 0689653 935698Dashed 0473033 102648 063682 766964
Figure 5(c) mdashPiece 1 Thick 219523 043963 0151372 568396
Dashed 107216 0739623 0288256 293721
Piece 2 Thick 0811123 102648 0389659 222587Dashed 0156681 132647 063682 116479
Figure 7(c) Right S
Piece 1Thick 00000429688 0571501 0285752 227044Dashed 0698712 0871501 0585746 39271Dotted 245758 0371528 0285746 23657
Piece 2Thick 00118778 0908437 0455056 145074Dashed 0403597 120905 0755673 651694Dotted 0700923 0707704 045338 100657
Figure 7(d) Right S
Piece 1Thick 253147 0871501 0585746 411231Dashed 00000429592 0571501 0285752 227044Dotted 107779 0371528 0285746 257032
Piece 2Thick 12606 120905 0755673 474646Dashed 00120078 0908437 0455056 145071Dotted 248785 0707704 045338 673212
Figure 7(c) Left S
Piece 1Thick 0188412 0981157 0509674 121657Dashed 0603719 0781142 0471473 10399Dotted 0361402 138114 0909666 524523
Piece 2Thick 00000462514 0594672 0297337 218104Dashed 0884532 0394671 0297335 35514Dotted 0564167 0994666 0697331 275123
Figure 7(d) Left S
Piece 1Thick 161781 0781142 0471473 833887Dashed 0234412 0981157 0509674 120616Dotted 142816 138114 0909666 315215
Piece 2Thick 926646 0394671 0297335 314849Dashed 00000462423 0594672 0297337 218104Dotted 266752 0994666 0697331 266104
10 Journal of Applied Mathematics
References
[1] M Kolomenkin I Shimshoni and A Tal ldquoDemarcating curvesfor shape illustrationrdquo ACM Transactions on Graphics vol 27no 5 article 157 2008
[2] D Decarlo A Finkelstein S Rusinkiewicz and A SantellaldquoSuggestive contours for conveying shaperdquo ACM Transactionson Graphics vol 22 no 3 pp 848ndash855 2003
[3] B B Kimia I Frankel and A Popescu ldquoEuler spiral for shapecompletionrdquo International Journal of Computer Vision vol 54no 1ndash3 pp 159ndash182 2003
[4] T Judd F Durand and E Adelson ldquoApparent ridges for linedrawingrdquo ACM Transactions on Graphics vol 26 no 3 article19 7 pages 2007
[5] S Yoshizawa A Belyaev and H P Seidel ldquoFast and robustdetection of crest lines on meshesrdquo in Proceedings of the ACMSymposium on Solid and Physical Modeling (SPM rsquo05) pp 227ndash232 June 2005
[6] G Harary and A Tal ldquo3D Euler spirals for 3D curve comple-tionrdquo Computational Geometry Theory and Applications vol45 no 3 pp 115ndash126 2012
[7] I Kanaya Y Nakano and K Sato ldquoSimulated designerrsquoseyesmdashclassification of aesthetic surfacesrdquo in Proceedings of theInternational Conference onVirtual Systems andMultimedia pp289ndash296 2003
[8] R U Gobithaasan and K T Miura ldquoLogarithmic curvaturegraph as a shape interrogation toolrdquo Applied MathematicalSciences vol 8 no 16 pp 755ndash765 2014
[9] K T Miura J Sone A Yamashita and T Kaneko ldquoDerivationof a general formula of aesthetic curvesrdquo in Proceedings of 8thInternational Conference on Humans and Computers pp 166ndash171 Tokyo Japan 2005
[10] N Yoshida and T Saito ldquoInteractive aesthetic curve segmentsrdquoThe Visual Computer vol 22 no 9-11 pp 896ndash905 2006
[11] R U Gobithaasan and K T Miura ldquoAesthetic spiral for designrdquoSains Malaysiana vol 40 no 11 pp 1301ndash1305 2011
[12] JMAli RM Tookey J V Ball andAA Ball ldquoThe generalisedCornu spiral and its application to span generationrdquo Journal ofComputational and Applied Mathematics vol 102 no 1 pp 37ndash47 1999
[13] R U Gobithaasan R Karpagavalli and K T Miura ldquoShapeanalysis of generalized log-aesthetic curvesrdquo International Jour-nal of Mathematical Analysis vol 7 no 33 pp 1751ndash1759 2013
[14] R Ziatdinov N Yoshida and T Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012
[15] K T Miura D Shibuya R U Gobithaasan and S UsukildquoDesigning log-aesthetic splines with119866
2 continuityrdquoComputer-Aided Design and Applications vol 10 no 6 pp 1021ndash1032 2013
[16] K T Miura and R U Gobithaasan ldquoAesthetic curves andsurfaces in computer aided geometric designrdquo InternationalJournal of Automation Technology vol 8 no 3 pp 304ndash3162014
[17] R U Gobithaasan R Karpagavalli and K T Miura ldquoG2continuous generalized log aesthetic curvesrdquo Computer AidedDesign amp Application Accepted for publication
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Applied Mathematics
(a) Euler spiral algorithm (b) Three types of tangents at the inflection points and 120572 = minus1
(c) Tangents setting as in 7(b) and 120572 = minus1 (d) Tangents setting as in 7(b) and 120572 = 1
Figure 7 The results of fitting Euler spiral and LAC segment to a vase
Table 1 The input data for LAC algorithm
Figure Partgap shape Starting point Inflection point Terminating point119875119886
119875inflect 119875119888
Figure 5Outer C (47533 29067) mdash (408 512)Inner C (444 3193) mdash (4067 4907)
S (6867 198) (115139 317874) (132 44533)
Figure 6 Left C (1504 3208) mdash (1232 5896)Right C (853 307) mdash (824 583)
Figure 7C (504 126) mdash (176 120)
Right S (57667 37933) (501722 483092) (44533 598)Left S (21533 59933) (166565 4812) (94 376)
Journal of Applied Mathematics 9
Table 2 The value of parameters obtained in LAC algorithm for C-shaped gap
Figure Part Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) Outer C 0181102 197352 109383 106443Inner C 0187822 250305 153612 577559
Figure 5(c) Outer C 0236827 197352 0879684 179903Inner C 0307163 250305 0966927 144024
Figure 6(b) Left C 0319685 155147 100107 105562Right C 0204861 235562 147579 896424
Figure 6(c) Left C 053364 155147 100107 902434Right C 0610563 235562 147579 654159
Figure 7(c) mdash 0140054 11637 0601359 271762Figure 7(d) mdash 0169061 11637 0601359 270036
Table 3 The value of parameters obtained in LAC algorithm for S-shaped gap
Figure Part Line style Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) mdashPiece 1 Thick 0646746 0739623 0451367 107501
Dashed 113587 043963 0288258 152739
Piece 2 Thick 0140372 132647 0689653 935698Dashed 0473033 102648 063682 766964
Figure 5(c) mdashPiece 1 Thick 219523 043963 0151372 568396
Dashed 107216 0739623 0288256 293721
Piece 2 Thick 0811123 102648 0389659 222587Dashed 0156681 132647 063682 116479
Figure 7(c) Right S
Piece 1Thick 00000429688 0571501 0285752 227044Dashed 0698712 0871501 0585746 39271Dotted 245758 0371528 0285746 23657
Piece 2Thick 00118778 0908437 0455056 145074Dashed 0403597 120905 0755673 651694Dotted 0700923 0707704 045338 100657
Figure 7(d) Right S
Piece 1Thick 253147 0871501 0585746 411231Dashed 00000429592 0571501 0285752 227044Dotted 107779 0371528 0285746 257032
Piece 2Thick 12606 120905 0755673 474646Dashed 00120078 0908437 0455056 145071Dotted 248785 0707704 045338 673212
Figure 7(c) Left S
Piece 1Thick 0188412 0981157 0509674 121657Dashed 0603719 0781142 0471473 10399Dotted 0361402 138114 0909666 524523
Piece 2Thick 00000462514 0594672 0297337 218104Dashed 0884532 0394671 0297335 35514Dotted 0564167 0994666 0697331 275123
Figure 7(d) Left S
Piece 1Thick 161781 0781142 0471473 833887Dashed 0234412 0981157 0509674 120616Dotted 142816 138114 0909666 315215
Piece 2Thick 926646 0394671 0297335 314849Dashed 00000462423 0594672 0297337 218104Dotted 266752 0994666 0697331 266104
10 Journal of Applied Mathematics
References
[1] M Kolomenkin I Shimshoni and A Tal ldquoDemarcating curvesfor shape illustrationrdquo ACM Transactions on Graphics vol 27no 5 article 157 2008
[2] D Decarlo A Finkelstein S Rusinkiewicz and A SantellaldquoSuggestive contours for conveying shaperdquo ACM Transactionson Graphics vol 22 no 3 pp 848ndash855 2003
[3] B B Kimia I Frankel and A Popescu ldquoEuler spiral for shapecompletionrdquo International Journal of Computer Vision vol 54no 1ndash3 pp 159ndash182 2003
[4] T Judd F Durand and E Adelson ldquoApparent ridges for linedrawingrdquo ACM Transactions on Graphics vol 26 no 3 article19 7 pages 2007
[5] S Yoshizawa A Belyaev and H P Seidel ldquoFast and robustdetection of crest lines on meshesrdquo in Proceedings of the ACMSymposium on Solid and Physical Modeling (SPM rsquo05) pp 227ndash232 June 2005
[6] G Harary and A Tal ldquo3D Euler spirals for 3D curve comple-tionrdquo Computational Geometry Theory and Applications vol45 no 3 pp 115ndash126 2012
[7] I Kanaya Y Nakano and K Sato ldquoSimulated designerrsquoseyesmdashclassification of aesthetic surfacesrdquo in Proceedings of theInternational Conference onVirtual Systems andMultimedia pp289ndash296 2003
[8] R U Gobithaasan and K T Miura ldquoLogarithmic curvaturegraph as a shape interrogation toolrdquo Applied MathematicalSciences vol 8 no 16 pp 755ndash765 2014
[9] K T Miura J Sone A Yamashita and T Kaneko ldquoDerivationof a general formula of aesthetic curvesrdquo in Proceedings of 8thInternational Conference on Humans and Computers pp 166ndash171 Tokyo Japan 2005
[10] N Yoshida and T Saito ldquoInteractive aesthetic curve segmentsrdquoThe Visual Computer vol 22 no 9-11 pp 896ndash905 2006
[11] R U Gobithaasan and K T Miura ldquoAesthetic spiral for designrdquoSains Malaysiana vol 40 no 11 pp 1301ndash1305 2011
[12] JMAli RM Tookey J V Ball andAA Ball ldquoThe generalisedCornu spiral and its application to span generationrdquo Journal ofComputational and Applied Mathematics vol 102 no 1 pp 37ndash47 1999
[13] R U Gobithaasan R Karpagavalli and K T Miura ldquoShapeanalysis of generalized log-aesthetic curvesrdquo International Jour-nal of Mathematical Analysis vol 7 no 33 pp 1751ndash1759 2013
[14] R Ziatdinov N Yoshida and T Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012
[15] K T Miura D Shibuya R U Gobithaasan and S UsukildquoDesigning log-aesthetic splines with119866
2 continuityrdquoComputer-Aided Design and Applications vol 10 no 6 pp 1021ndash1032 2013
[16] K T Miura and R U Gobithaasan ldquoAesthetic curves andsurfaces in computer aided geometric designrdquo InternationalJournal of Automation Technology vol 8 no 3 pp 304ndash3162014
[17] R U Gobithaasan R Karpagavalli and K T Miura ldquoG2continuous generalized log aesthetic curvesrdquo Computer AidedDesign amp Application Accepted for publication
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 9
Table 2 The value of parameters obtained in LAC algorithm for C-shaped gap
Figure Part Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) Outer C 0181102 197352 109383 106443Inner C 0187822 250305 153612 577559
Figure 5(c) Outer C 0236827 197352 0879684 179903Inner C 0307163 250305 0966927 144024
Figure 6(b) Left C 0319685 155147 100107 105562Right C 0204861 235562 147579 896424
Figure 6(c) Left C 053364 155147 100107 902434Right C 0610563 235562 147579 654159
Figure 7(c) mdash 0140054 11637 0601359 271762Figure 7(d) mdash 0169061 11637 0601359 270036
Table 3 The value of parameters obtained in LAC algorithm for S-shaped gap
Figure Part Line style Shape parameter Angles (radian) Scale factorΛ 120579
119889120579119890or 120579119891
119903scale
Figure 5(b) mdashPiece 1 Thick 0646746 0739623 0451367 107501
Dashed 113587 043963 0288258 152739
Piece 2 Thick 0140372 132647 0689653 935698Dashed 0473033 102648 063682 766964
Figure 5(c) mdashPiece 1 Thick 219523 043963 0151372 568396
Dashed 107216 0739623 0288256 293721
Piece 2 Thick 0811123 102648 0389659 222587Dashed 0156681 132647 063682 116479
Figure 7(c) Right S
Piece 1Thick 00000429688 0571501 0285752 227044Dashed 0698712 0871501 0585746 39271Dotted 245758 0371528 0285746 23657
Piece 2Thick 00118778 0908437 0455056 145074Dashed 0403597 120905 0755673 651694Dotted 0700923 0707704 045338 100657
Figure 7(d) Right S
Piece 1Thick 253147 0871501 0585746 411231Dashed 00000429592 0571501 0285752 227044Dotted 107779 0371528 0285746 257032
Piece 2Thick 12606 120905 0755673 474646Dashed 00120078 0908437 0455056 145071Dotted 248785 0707704 045338 673212
Figure 7(c) Left S
Piece 1Thick 0188412 0981157 0509674 121657Dashed 0603719 0781142 0471473 10399Dotted 0361402 138114 0909666 524523
Piece 2Thick 00000462514 0594672 0297337 218104Dashed 0884532 0394671 0297335 35514Dotted 0564167 0994666 0697331 275123
Figure 7(d) Left S
Piece 1Thick 161781 0781142 0471473 833887Dashed 0234412 0981157 0509674 120616Dotted 142816 138114 0909666 315215
Piece 2Thick 926646 0394671 0297335 314849Dashed 00000462423 0594672 0297337 218104Dotted 266752 0994666 0697331 266104
10 Journal of Applied Mathematics
References
[1] M Kolomenkin I Shimshoni and A Tal ldquoDemarcating curvesfor shape illustrationrdquo ACM Transactions on Graphics vol 27no 5 article 157 2008
[2] D Decarlo A Finkelstein S Rusinkiewicz and A SantellaldquoSuggestive contours for conveying shaperdquo ACM Transactionson Graphics vol 22 no 3 pp 848ndash855 2003
[3] B B Kimia I Frankel and A Popescu ldquoEuler spiral for shapecompletionrdquo International Journal of Computer Vision vol 54no 1ndash3 pp 159ndash182 2003
[4] T Judd F Durand and E Adelson ldquoApparent ridges for linedrawingrdquo ACM Transactions on Graphics vol 26 no 3 article19 7 pages 2007
[5] S Yoshizawa A Belyaev and H P Seidel ldquoFast and robustdetection of crest lines on meshesrdquo in Proceedings of the ACMSymposium on Solid and Physical Modeling (SPM rsquo05) pp 227ndash232 June 2005
[6] G Harary and A Tal ldquo3D Euler spirals for 3D curve comple-tionrdquo Computational Geometry Theory and Applications vol45 no 3 pp 115ndash126 2012
[7] I Kanaya Y Nakano and K Sato ldquoSimulated designerrsquoseyesmdashclassification of aesthetic surfacesrdquo in Proceedings of theInternational Conference onVirtual Systems andMultimedia pp289ndash296 2003
[8] R U Gobithaasan and K T Miura ldquoLogarithmic curvaturegraph as a shape interrogation toolrdquo Applied MathematicalSciences vol 8 no 16 pp 755ndash765 2014
[9] K T Miura J Sone A Yamashita and T Kaneko ldquoDerivationof a general formula of aesthetic curvesrdquo in Proceedings of 8thInternational Conference on Humans and Computers pp 166ndash171 Tokyo Japan 2005
[10] N Yoshida and T Saito ldquoInteractive aesthetic curve segmentsrdquoThe Visual Computer vol 22 no 9-11 pp 896ndash905 2006
[11] R U Gobithaasan and K T Miura ldquoAesthetic spiral for designrdquoSains Malaysiana vol 40 no 11 pp 1301ndash1305 2011
[12] JMAli RM Tookey J V Ball andAA Ball ldquoThe generalisedCornu spiral and its application to span generationrdquo Journal ofComputational and Applied Mathematics vol 102 no 1 pp 37ndash47 1999
[13] R U Gobithaasan R Karpagavalli and K T Miura ldquoShapeanalysis of generalized log-aesthetic curvesrdquo International Jour-nal of Mathematical Analysis vol 7 no 33 pp 1751ndash1759 2013
[14] R Ziatdinov N Yoshida and T Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012
[15] K T Miura D Shibuya R U Gobithaasan and S UsukildquoDesigning log-aesthetic splines with119866
2 continuityrdquoComputer-Aided Design and Applications vol 10 no 6 pp 1021ndash1032 2013
[16] K T Miura and R U Gobithaasan ldquoAesthetic curves andsurfaces in computer aided geometric designrdquo InternationalJournal of Automation Technology vol 8 no 3 pp 304ndash3162014
[17] R U Gobithaasan R Karpagavalli and K T Miura ldquoG2continuous generalized log aesthetic curvesrdquo Computer AidedDesign amp Application Accepted for publication
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Applied Mathematics
References
[1] M Kolomenkin I Shimshoni and A Tal ldquoDemarcating curvesfor shape illustrationrdquo ACM Transactions on Graphics vol 27no 5 article 157 2008
[2] D Decarlo A Finkelstein S Rusinkiewicz and A SantellaldquoSuggestive contours for conveying shaperdquo ACM Transactionson Graphics vol 22 no 3 pp 848ndash855 2003
[3] B B Kimia I Frankel and A Popescu ldquoEuler spiral for shapecompletionrdquo International Journal of Computer Vision vol 54no 1ndash3 pp 159ndash182 2003
[4] T Judd F Durand and E Adelson ldquoApparent ridges for linedrawingrdquo ACM Transactions on Graphics vol 26 no 3 article19 7 pages 2007
[5] S Yoshizawa A Belyaev and H P Seidel ldquoFast and robustdetection of crest lines on meshesrdquo in Proceedings of the ACMSymposium on Solid and Physical Modeling (SPM rsquo05) pp 227ndash232 June 2005
[6] G Harary and A Tal ldquo3D Euler spirals for 3D curve comple-tionrdquo Computational Geometry Theory and Applications vol45 no 3 pp 115ndash126 2012
[7] I Kanaya Y Nakano and K Sato ldquoSimulated designerrsquoseyesmdashclassification of aesthetic surfacesrdquo in Proceedings of theInternational Conference onVirtual Systems andMultimedia pp289ndash296 2003
[8] R U Gobithaasan and K T Miura ldquoLogarithmic curvaturegraph as a shape interrogation toolrdquo Applied MathematicalSciences vol 8 no 16 pp 755ndash765 2014
[9] K T Miura J Sone A Yamashita and T Kaneko ldquoDerivationof a general formula of aesthetic curvesrdquo in Proceedings of 8thInternational Conference on Humans and Computers pp 166ndash171 Tokyo Japan 2005
[10] N Yoshida and T Saito ldquoInteractive aesthetic curve segmentsrdquoThe Visual Computer vol 22 no 9-11 pp 896ndash905 2006
[11] R U Gobithaasan and K T Miura ldquoAesthetic spiral for designrdquoSains Malaysiana vol 40 no 11 pp 1301ndash1305 2011
[12] JMAli RM Tookey J V Ball andAA Ball ldquoThe generalisedCornu spiral and its application to span generationrdquo Journal ofComputational and Applied Mathematics vol 102 no 1 pp 37ndash47 1999
[13] R U Gobithaasan R Karpagavalli and K T Miura ldquoShapeanalysis of generalized log-aesthetic curvesrdquo International Jour-nal of Mathematical Analysis vol 7 no 33 pp 1751ndash1759 2013
[14] R Ziatdinov N Yoshida and T Kim ldquoAnalytic parametricequations of log-aesthetic curves in terms of incomplete gammafunctionsrdquo Computer Aided Geometric Design vol 29 no 2 pp129ndash140 2012
[15] K T Miura D Shibuya R U Gobithaasan and S UsukildquoDesigning log-aesthetic splines with119866
2 continuityrdquoComputer-Aided Design and Applications vol 10 no 6 pp 1021ndash1032 2013
[16] K T Miura and R U Gobithaasan ldquoAesthetic curves andsurfaces in computer aided geometric designrdquo InternationalJournal of Automation Technology vol 8 no 3 pp 304ndash3162014
[17] R U Gobithaasan R Karpagavalli and K T Miura ldquoG2continuous generalized log aesthetic curvesrdquo Computer AidedDesign amp Application Accepted for publication
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of