research article log-aesthetic magnetic curves and their...

Research ArticleLog-Aesthetic Magnetic Curves and Their Applicationfor CAD Systems

Mei Seen Wo1 R U Gobithaasan1 and Kenjiro T Miura2

1 School of Informatics amp Applied Mathematics University Malaysia Terengganu 21030 Kuala Terengganu Malaysia2 Graduate School of Science amp Technology Shizuoka University Shizuoka 432-8561 Japan

Correspondence should be addressed to R U Gobithaasan grumtedumy

Received 9 April 2014 Revised 1 September 2014 Accepted 6 September 2014 Published 27 October 2014

Academic Editor Dan Simon

Copyright copy 2014 Mei Seen Wo et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Curves are the building blocks of shapes and designs in computer aided geometric design (CAGD) It is important to ensure thesecurves are both visually and geometrically aesthetic to meet the high aesthetic need for successful product marketing Recentlymagnetic curves that have been proposed for computer graphics purposes are a particle tracing technique that generates a widevariety of curves and spirals under the influence of a magnetic field The contributions of this paper are threefold where the firstpart reformulates magnetic curves in the form of log-aesthetic curve (LAC) denoting it as log-aesthetic magnetic curves (LMC)and log-aesthetic magnetic space curves (LMSC) the second part elucidates vital properties of LMCs and the final part proposesG2 LMC design for CAD applications The final section shows two examples of LMC surface generation along with its zebra mapsLMC holds great potential in overcoming the weaknesses found in current interactive LAC mechanism where matching a singlesegment with G2 Hermite data is still a cumbersome task

1 Introduction

Curves are widely used in various fields most commonly inart and designs The fairness of a curve dictates the productrsquosquality and in turn its sales The definition of a beautifulcurve in the eye of an artist is a curve which exhibits constantvariation of curvaturemonotonically and no part of the curveis a circular arc as expressed by Ruskin [1] It is the samefor aesthetic designs in CAGD However lines and circlesare considered fair or beautiful because of their simplicity[2] The definition of a fair curve still remains subjective andunclear which makes the condition required for ending afairing process seems rather ambiguous Conventional curvessuch as NURBS and Bezier have obvious oscillation in theircurvature profile making it less visually pleasing They alsohave more complex curvature formulas compared to those ofnatural spirals

Harada et al [3] noted that fair curves observed innature have linear logarithmic curvature histograms (LCHs)

It was later that curves which satisfy the stated conditionare categorized as log-aesthetic curve (LAC) [4 5] whichcomprises a huge family of fair curves including logarithmicspiral circle involute and clothoid LAC is derived fromlogarithmic curvature graph (LCG)which is the analytic formof logarithmic distribution diagram of curvature (LDDC)Yoshida and Saito [5] also proposed an algorithm for inter-active generation of LAC These curves satisfy G1 Hermitedata However due to the scaling of the curve at the end ofthe algorithm it is difficult to match G2 Hermite data withone LAC segment alone albeit it can be solved by joining twoLACs with G1-continuity then scaling one of the trianglessuch that the joint is G2-continuous

Gobithaasan andMiura [6] proposed the general form ofthe LACrsquos formula known as the generalized LAC (GLAC)It has an extra degree of freedom compared to LACwhich results in increased flexibility [7] Furthermore GLACextends the family of LAC to include generalized cornuspiral [8] Both curves mentioned have great potential for

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 504610 16 pageshttpdxdoiorg1011552014504610

2 Mathematical Problems in Engineering

(a) Car model (side view)

(b) Original curve (c) Triple LA curve

Figure 1 Application of LAC circular arc (1(b)) replaced by a LAspline to achieve G2-continuity (1(c)) [10]

CAGD purposes but are highly compute intensive and timeconsuming Recent progresses in this subject matter includethe development of log-aesthetic spline and its application inautomobile design (see Figure 1) Recently Gobithaasan et al[9] proposed Runge Kutta methods to compute LACGLACproven to achieve tremendous speedup

There are also curves whose LCGs are approximatelylinear These curves are known as quasi-aesthetic curvescoined by Yoshida and Saito [11] They proposed quasi-aesthetic curve segments in the form of rational cubic BezierThe resulting curve has monotone curvatures and has ahigher aesthetic value compared to the usual rational Beziersplines Class A Bezier [12] is another curve with monotoniccurvature known to have approximately linear LCG and canbe extended to space curves However its shape is limited tothat of a logarithmic spiral as the degree of the polynomialincreases Furthermore it is currently difficult to employClass A Bezier space curve for surface design Recent studyby Nabiyev and Ziatdinov [13] shows that Bezier curves withmonotonic curvatures are not always aesthetic in terms ofthe law of technical aesthetics A study by Levien and Sequin[14] suggested that LAC is the most promising family of faircurves Readers are referred to Miura and Gobithaasan [15]for a comprehensive review on aesthetic curves for variousdesign feats

In 2009 Xu and Mould proposed a particle tracingmethod to produce magnetic curve which is categorized asfair curve due to the monotonicity of its curvature functionIt was inspired by artistic motives and they further showed itspracticalities in computer graphics by rendering trees hairswater and fire Magnetic curves are defined as a particletracing method that creates curves with constantly varyingcurvature by utilizing the effect of amagnetic fieldThis paperaims to reformulate magnetic curves for CAD practicalitiesand further reformulates to construct splines which satisfyG2Hermite data

Section 1 elaborates in detail the physics behindmagneticcurves The contribution of this paper is threefold the firstpart elucidates the connection between magnetic curves andLACs and the second part reformulates magnetic curves inthe form of 2D and 3D log-aesthetic curve and denotes itas log-aesthetic magnetic curves (LMC) and log-aesthetic

magnetic space curves (LMSC) for CAD applications finallythe third part proposes an algorithm for developing G2continuous LMC

2 Formulation of Magnetic Curves

In a uniformmagnetic field themotion of a particle of charge119902 and mass 119898 travelling with velocity V under magneticinduction is the result of Lorentz Force [16]

= 119902 (V times ) (1)

which can be rewritten as

119898119889V119889119905

= 119902 (V times ) (2)

where times represents the cross product operation It describesthe motion of charged particles experiencing Lorentz forceFor convenience V is separated into two components withthe first V

||parallel to and the second V

perpperpendicular to

The steps on detailed derivation are omitted for brevityalbeit readers are referred to [17] To vary the curvature of thecurve 119902(119905) is set to be an arbitrary real function of 119905 instead ofa constant 119902 Note that the magnitude of the magnetic field119861 should not be confused with its vector and 119861 ge 0 Weobtain the components of magnetic curves as follows

119889V119909

119889119905=119902 (119905) 119861

119898V119910

119889V119910

119889119905= minus

119902 (119905) 119861

119898V119909

119889V119911

119889119905= 0

(3)

Note that V119911(119905) = V

119911(119905119900) = V|| and 119905

119900is the initial time 119905 of the

trajectory To simplify the expression we take (119902(119905)119861)119898 =

120596(119905) By using separation of variables method we arrive atthe following equations

V119909(119905) = 119860 cosΩ (119905) + 119863 sinΩ (119905) (4)

V119910(119905) = 119863 cosΩ (119905) minus 119860 sinΩ (119905) (5)

V119911(119905) = V

|| (6)

where Ω1015840(119905) = 120596(119905) and 119860 and 119863 are arbitrary constants Inorder to set the point at the origin the initial conditions areset to be V

119909(119905119900) = Vperpge 0 and V

119910(119905119900) = 0 which are the 119909 and

119910 components of Vperp These values are then substituted to (4)

These initial values dictate the direction andmagnitude of thevelocity vector at the origin of the Cartesian plane which can

Mathematical Problems in Engineering 3

120596(+)

rr

120596(minus)

B

Figure 2 Circular particle (orbs) motion in uniformmagnetic fieldwith radius of gyration 119903

1 2minus02minus04minus06minus08minus10minus12

minus1

rarr t0

rarr t1

Figure 3 Velocity vectors (arrows) of particle along its trajectory onthe Cartesian plane

be represented as a vector of magnitude V2perp= V2119909+ V2119910and the

direction is parallel to the 119909-axis Thus we have

V119909(119905119900) = 119860 cosΩ(119905

119900) + 119863 sinΩ(119905

119900)

V119910(119905119900) = 119863 cosΩ(119905

119900) minus 119860 sinΩ(119905

119900)

(7)

Expressing 119860 and 119863 in (7) in terms of 119905119900and substituting

back into (4) we obtain the position of the particle on theCartesian plane at time 119905 as

119909 (119905) = 1199090+ Vperpint

119905

119905119900

cos (Ω (119905119900) minus Ω (119905)) 119889119905 (8)

119910 (119905) = 1199100+ Vperpint

119905

119905119900

sin (Ω (119905119900) minus Ω (119905)) 119889119905 (9)

119911 (119905) = 1199110+ int

119905

119905119900

V||119889119905 (10)

where (1199090 1199100 1199110) is the initial position of the particle

Thus the magnetic curve is two-dimensional when V||= 0

and three-dimensional otherwise Note that although in thispaper the velocity vector at 119905

119900is set to be ⟨V

perp 0 0⟩ to fix

the tangential angle to 0 at 119905119900at the origin we can alter the

initial values V119909(119905119900) and V

119910(119905119900) such that the tangential angle

is tanminus1(V119910(119905119900)V119909(119905119900)) Figures 2 and 3 depict the concept of

particle motion under the influence of a constant magneticfield

The main idea of magnetic curve is to vary the radiusof gyration constantly to obtain a curve with monotoniccurvature The radius of gyration and gyro-frequency ofmagnetic curves is given by the equations

120588 (119905) =V2perp+ V2||

Vperp |120596 (119905)|

=119898 (V2perp+ V2||)

1003816100381610038161003816119902 (119905)1003816100381610038161003816 119861Vperp

120596 (119905) =119902 (119905) 119861

119898

(11)

Note that the radius of gyration is actually the same asthe radius of curvature coined in CAGD while the gyro-frequency of the particle is the angular velocity referred to inphysical theories It can also be noted that 120579(119905) = Ω(119905

119900)minusΩ(119905)

is the tangential angle of the trajectory or curveThus if 119902(119905) isa constant we will have 120588 as a constant resulting in a circulartrajectory By setting 119902 as a function of 119905 value of120588 varies withrespect to 119905 creating a spiral trajectory The radius of torsionis governed by

120583 (119905) =V2perp+ V2||

V|| |120596 (119905)|

=119898 (V2perp+ V2||)

1003816100381610038161003816119902 (119905)1003816100381610038161003816 119861V||

(12)

Magnetic curves with constant initial velocities andmagneticfield have total arc lengths of

119904 (119905) = radicV2perp+ V2||(119905 minus 119905119900) (13)

A summary of the properties of magnetic curves [18] are asfollows

(i) Radius of gyration decreases while curvatureincreases when 119902(119905) increases

(ii) The direction of particle acceleration changes when119902(119905) changes sign

(iii) Arc length of magnetic curve grows in a constant rateas 119905 increases when V

perpis a constant

Xu and Mould [18] proposed to vary particle charge 119902(119905) forcreating various spirals

119902 (119905) = 119905minus120573 120573 isin R (14)


1 2 3 4 5 6

5

10

15

20

25

30

t

120581(t)

(a)

1 2 3 4 5 6

1

2

3

4

5

t

120581998400 (t)

(b)

Figure 4 Influence of 119861 on the curvature function (a) and rate of change of curvature (b) when 120573 = minus1 From bottom 119861 =

1 15 2 25 3 35 4 45 5 119902(119905) lt 0

00 05 10 15 20 2500

05

10

15

20

25 120573 = minus1

120573 = minus2

120573 = minus3

120573 = minus4

(a)

minus10 minus05

minus05

05

05

120573 = 09

120573 = 08

120573 = 07120573 = 06

(b)

000005 00001

000002

000004

000006

120573 = 09

120573 = 08

120573 = 07

120573 = 06

minus000004

minus000005minus000002

(c)

Figure 5 (a) and (b) Influence of 120573 on the curvesrsquo shapes with 119902(119905) lt 0 Figure (c) is the magnified figure of (b) at the origin

Assuming 119898 = 1 such that the curve is on the 119909-119910 Cartesianplane the equation of the trajectory is

119909 (119905) = 1199090+

Vperpint

119905

119905119900

cos(119861 (1199051minus120573

119900minus 1199051minus120573

)

1 minus 120573)119889119905 120573 = 1

Vperpint

119905

119905119900

cos (119861 (ln (119905119900) minus ln (119905))) 119889119905 120573 = 1

119910 (119905) = 1199100+

Vperpint

119905

119905119900

sin(119861 (1199051minus120573

119900minus 1199051minus120573

)

1 minus 120573)119889119905 120573 = 1

Vperpint

119905

119905119900

sin (119861 (ln (119905119900) minus ln (119905))) 119889119905 120573 = 1

(15)

A space curve is obtained by introducing 119911(119905) as stated in (10)For planar curve it can also be written in a complex plane asfollows

119862119883119872

(119905) = 1198750+

Vperpint

119905

119905119900

119890119894(119861(1199051minus120573

119900minus1199051minus120573

)(1minus120573))119889119905 120573 = 1

Vperpint

119905

119905119900

119890119894119861(ln(119905

119900)minusln(119905))

119889119905 120573 = 1

(16)

whose gyro-frequency and radius of curvature are

120596 (119905) = 119861119905minus120573 120573 isin R

120588 (119905) =V2perp+ V2||

1003816100381610038161003816119905minus1205731003816100381610038161003816 119861Vperp

(17)

Equation (16) generates planar trajectories or curves ofvarious shapes mainly spirals and circles It produces straightlines when 120573 = 1 Note that these curves are well-defined at119905 = 0 when 120573 le 1 otherwise 1199051minus120573 is not a real number

The particle velocity Vperpuniformly scales the entire trajec-

tory and its arc length whereas 119861 the magnitude of magneticfield uniformly scales the rate of change of curvature Theeffects of scaling with 119861 on the curvature and rate of changeof curvature can be seen in Figure 4 120573 is a shape parameterwhich determines the overall shape of the magnetic curvesand is directly related to LCG gradient its effect on planarcurves is shown in Figure 5 The shapes of magnetic curvesare further discussed in the following section

3 Magnetic Curves with Constant LCG andLTG Gradient

In this section we further investigate the fairness of magneticcurves with particle charge as a variant Various aesthetic


cases ofmagnetic curves will also be discussed in this sectionShape interrogation tools used in the aesthetic analysis ofmagnetic curve include curvature profile logarithmic curva-ture graph (LCG) and logarithmic torsion graph (LTG)

Given 120588(119905) and 119904(119905) are the radius of curvature and arclength of a curve the LCG is defined as a graph whose hor-izontal and vertical axes are the logarithm of 120588(119905)1199041015840(119905)1205881015840(119905)and 120588(119905) respectively A LAC is defined as a curvewhose LCGis linear and its gradient function 120582(119905) = 120572 where 120572 is a realconstant A LA space curve is defined as a curve whose LTG islinear The LCG LTG and their respective gradient function[19] of a curve are given as

LCG (119905) = log 120588 (119905) log(120588 (119905) 119904

1015840(119905)

1205881015840 (119905)) (18)

LTG (119905) = log120583 (119905) log(120583 (119905) 119904

1015840(119905)

1205831015840 (119905)) (19)

120582 (119905) = 1 +120588 (119905)

1205881015840(119905)2(1205881015840(119905) 11990410158401015840(119905)

1199041015840 (119905)minus 12058810158401015840(119905)) (20)

120595 (119905) = 1 +120583 (119905)

1205831015840(119905)2(1205831015840(119905) 11990410158401015840(119905)

1205831015840 (119905)minus 12058310158401015840(119905)) (21)

The LCG and its gradient of (16) are

LCG (119905) =

logV2perp+ V2||

1198611003816100381610038161003816119905minus1205731003816100381610038161003816 Vperp

log(119905radicV2perp+ V2||

120573)

120582 (119905) = 120572 =1

120573

(22)


LTG (119905) =

logV2perp+ V2||

1198611003816100381610038161003816119905minus1205731003816100381610038161003816 V||


120573)

120595 (119905) = 120572 =1

120573

(23)

Therefore (16) is guaranteed to be log-aesthetic curves orspace curves It is also notable that neither 119861 nor V

perphas

an influence on the LCG nor LTG gradient We are ableto classify some of the well-known aesthetic curves fromtheir LCG gradients Xu andMouldrsquosmagnetic spirals includeclothoid curve logarithmic spiral and circle involute whichoccur when 120573 = 120572 = minus1 120573 = 120572 = 1 and 120573 =

05 (120572 = 2) Generally these spirals are divergent when 120573 lt 0

and convergent when 120573 gt 0 The result of the shape andLCG analysis of these plane curves is depicted in Figure 6Inflection points occur at 119905 = 0 whenever 120573 lt 0 Inflectionpoints occur at 119905 = infin when 120573 gt 0

Equation (16) cannot be used to derive Nielsenrsquos spiralwhich occurs when 120582(119905) = 120572 = 0 However magnetic curvesdo comprise Nielsenrsquos spiral which is proven in (23)ndash(25)

Substituting 120579(119905) = Ω(119905119900) minus Ω(119905) into (8) we employ the

general formula of radius of curvature to find

120588 (119905) =119889119904119889119905

119889120579119889119905=

Vperp

119889120579119889119905=119889119904

119889120579 (24)

Substituting the equations and their derivatives of (14) and(24) with respect to 119905 into (20) gives

120582 (119905) =1205791015840(119905) 120579(3)

(119905)

12057910158401015840(119905)2

minus 1 = 0 (25)

which yields the solution

120579 (119905) =11989011990511988811198882

1198881

+ 1198883 120572 = 0 (26)

where 1198881 1198882 and c

3are arbitrary constants Without loss

of generality we can omit 1198881as it does not change the

overall shape of the curve Note that (26) can be obtained bytranslating or scaling 119905 by 119888

1in the following equation

Ω (119905) = 119861119890119905 120572 = 0 (27)

We let 1198882= minus119861 and 119888

3= minusΩ(119905

119900) such that 120579(119905) = Ω(119905

119900) minusΩ(119905)

Thuswe haveDefinition 1 as followsNote that (27) has a fixedLCG and LTG gradient of 0 thus this curve will always havethe same basic shape despite influences of 119861 and V

perp Figure 7

shows the parametric plots of magnetic curves with (27) and(16) when 120573 = 120572 = 1 while Figure 8 shows plots of LM spacecurve with inputs given in Table 1

Definition 1 The equation of log-aesthetic magnetic curves(LMC) in complex plane is

119862LMC (119905) = 1198750+ Vperpint

119905

119905119900

119890119894120579(119905)

119889119905 (28)

where V||= 0 and 119875

0is the initial position of the particle and

the equation of log-aesthetic magnetic curve curves (LMSC)is

119862LMSC (119905) = Vperpint

119905

119905119900

cos 120579 (119905) 119889119905 Vperpint

119905

119905119900

sin 120579 (119905) 119889119905 int119905

119905119900

V||119889119905

(29)


0 5 10

0

5

10

minus10

minus10

minus5

minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

5

10

15

20

25

30

(a)

0 5 10 15

0

5

10

minus10

minus5

minus5t

(i) (ii) (iii)120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

1

2

3

4

(b)

0 5 10 15

0

5

minus10

minus5

minus5

t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

05

10

15

20

25

30

35

(c)

0 5 10

0

5

minus10

minus5

minus10 minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

2

4

6

8

(d)

Figure 6 Continued


0 5 10

0

5

minus10

minus5

minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

2

4

6

8

(e)

0 5 10

0

5

minus10 minus5

minus10

minus5

t

(i) (ii) (iii)120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 200

minus10

minus8

minus6

minus4

minus2

(f)

Figure 6 Column from left (i) magnetic spirals (ii) its LCG and (iii) curvature profile of magnetic spirals of various 120573 values with 119902(119905) lt 0From top to bottom (a) 120573 = minus3 (b) 120573 = minus1 (c) 120573 = 008 (d) 120573 = 05 (e) 120573 = 078 and (f) 120573 = 2

minus1

minus2

minus3

minus4

minus2minus4minus6minus8

y

x

(a)

minus10

minus10

minus05 05

minus04

minus02

minus06

minus08

y

x

(b)

Figure 7 (a) Logarithmic spiral with 119905119900= 2 119861 = 2 119905 isin [0 17] (b) Nielsenrsquos spiral with 119905

119900= minus05 119861 = 1 119905 isin [minus2 3]

with

120579 (119905) =

119861 (119890119905119900 minus 119890119905) 120572 = 0

119861 (log (119905119900) minus log (119905)) 120572 = 120573 = 1

119861 (1199051minus120573

119900minus 1199051minus120573

)

(1 minus 120573) 120572 = (

1

120573) = 1

(30)

The particle charge is

119902 (119905) =

119905minus120573 120572 =

1

120573isin R

119890119905 120572 = 0

(31)

119861 is the magnitude of themagnetic field Vperpge 0 is the particle

velocity 1199050is the time when the velocity vector is ⟨V

perp 0⟩ and


(a) (b)

(c) (d)

Figure 8 LMSC plots with torsion (a) (c) and curvature (b) and (d) profiles

120573 isin R is a parameter used for varying the gyro-frequencyto produce trajectories of various shapesThese curves have atotal arc length of (13) The curvature and torsion function isgiven by

120581 (119905) =

minus119861119905minus120573Vperp

V2perp+ V2||

120572 =1

120573isin R

minus119861119890119905Vperp

V2perp+ V2||

120572 = 0

(32)

120591 (119905) =

minus119861119905minus120573V||

V2perp+ V2||

120572 =1

120573isin R

minus119861119890119905V||

V2perp+ V2||

120572 = 0

(33)

As magnetic curves share the same general form of equationand differential geometries (8)ndash(11) the influence of 119861 and V

perp

on the curversquos curvature holds for all cases of (30) for LMCNote that Definition 1 is the formulation of magnetic curvesto represent LACs One of the differences of LMC and LACequation proposed in [5] is that LMC is unable to form circlewith any other 120573 except for 120573 = 0 (120572 rarr infin) However thisformulation allows various parameterization and equationmanipulation for different application The reparameteriza-tion of LMC is discussed in Section 5 Figure 7 illustrates two

030

025

020

015

010

005

005 010 015 020 025 030 035

x

y

120572 = 0

120572 = minus1

120572 = 1

120572 = minus1

6120572 =

1

3

Figure 9 End curvature parameterized LMC with 1205810= 167 120581

1=

352 120579 = 712058718 From left 120572 = minus1 minus16 0 13 1

configurations of LMC representing logarithmic spiral andNielsenrsquos spiral

Since 119902(119905) can be any arbitrary function there aremany possibilities of magnetic curves that can be generatedAnother particle charge function which produces LAC is119902(119905) = (120573119905)

minus1120573 However it has a LCG gradient of 120573 which


120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(a)

120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(b)

1205881 = infin

120579P0

P1

P21205880 =

1

1205810

(c)

Figure 10 Examples ofG2Hermite interpolation problem (a)120573 = 1997474 error = 139144times10minus5 (b)120573 = 13319512 error = 269981times10

minus13and (c) 120573 = 0985866 error = 832667 times 10

minus17

means it is essentially the same as 119902(119905) = 119905minus120573 in terms of curve

shapes Note that LMC does include curves with nonlinearLCGs An example of this case is when 119902(119905) = 2119905 + 2119890

119905Figure 8 depicts examples of LMSC with inputs as stated inTable 1

4 Properties of Magnetic Curves

This section discusses that the properties which are round-ness monotone curvature and torsion extensionality andlocality hold for LMC or LMSC These properties were firstintroduced byHarary andTal [20] and Levien and Sequin [14]for CAD applications However we need first to determinethe bound for 119905 or 119904 such that the resulting curve is regularand well-behaved as shown in Table 2

Table 1 Inputs for Figures 8(a)ndash8(d)

Figure 119905 119905119900

119861 V||

Vperp

120573

Figure 8(a) (0 4] 1 4 1 2 078Figure 8(b) [1 3] 1 4 1 2 minus1Figure 8(c) (0 4] 1 4 1 2 1Figure 8(d) [minus2 15] 05 4 1 2 0

The properties of LMC or LMSC are as follows

Proposition 2 LMC and LMSC exhibit self-affinity property

Self-similarity is an important fractal geometry charac-teristic in which the geometry is invariance under uniform


Table 2 Boundaries for both 119905 and 119904 of magnetic curves lowastThe pointwhere inflection points occur

Boundaries for 119905 and 1199041120572 = 120573 lt 0 120573 isin Z (minusinfin 0

lowast] cup [0

lowastinfin)

1120572 = 120573 lt 0 120573 notin Z [0infin)

120573 = 0 (minusinfininfin)

0 lt 1120572 = 120573 lt 1 [0infinlowast)

120572 = 1 [0infinlowast)

120572 = 0 (minusinfinlowastinfin)

1120572 = 120573 gt 1 (0infinlowast)

scaling whereas self-affinity preserves the geometry undernonuniform scaling operations

Proof These characteristics are inspected mathematically viaomitting a front portion of the curve and scaling it by 119886119905 + 119887Equation (30) now becomes

120579 (119886119905 + 119887) =

119861(119890119905119900 minus 119890119886119905+119887

) 120572 = 0

119861 (log (119905119900) minus log (119886119905 + 119887)) 120572 = 1

119861 (1199051minus120573

119900minus (119886119905 + 119887)

1minus120573)

(1 minus 120573) 120572 = (

1

120573) = 1

(34)

Inspecting 120582(119886119905 + 119887) and 120595(119886119905 + 119887) for each case we can seethat 120582(119886119905 + 119887) = 120582(119905) and 120595(119886119905 + 119887) = 120595(119905) Thus the originalshape is preserved and (30) is proven to be self-affine Notethat (30) becomes a logarithmic spiral when 120572 = 1 thereforeit is self-similar and inline as proven by Miura [4]

Proposition 3 LMC or LMCS forms circles (roundness prop-erty)

If an interpolation problem involves interpolating a circlea desirable interpolation spline should form an exact circleThus given any two tangent points on a circle LMC will beable to form a circular arc and fit into these tangent points

Proof LMC readily forms a circular trajectory when 120573 = 0

Proposition 4 LMC or LMSC has monotonically increasingor decreasing curvature 120581(t) and torsion 120591(t) (note set v

||= 0

to restrict the curve to 119909-119910 plane to obtain LMC)

The monotonicity of curvature is the most basic criteriafor a fair curve as suggested by many researchers and design-ers Since human eyes are very sensitive towards curvatureextrema the extrema should not appear on any point of thecurve segment except for its end points [14]

Table 3 Values of 1205811015840(119905) and 1205911015840(119905) with respective 120572 and 120573 values forall 119905 gt 0 or 119905 lt 0

Values of 120572 or 120573 1205811015840(119905) 120591

1015840(119905)

119905 lt 0 119905 gt 0 119905 lt 0 119905 gt 0

1(120572 = 1120578) = 120573 lt 0 120573 isin 2119885 lt0 gt0 gt0 lt01(120572 = 1120578) = 120573 lt 0 120573 notin 119885 mdash gt0 mdash lt01(120572 = 1120578) = 120573 lt 0 120573 isin 2119885 + 1 gt0 gt0 lt0 lt0120573 = 0 =0 =0 =0 =00 lt 1120572 = 1120578 = 120573 lt 1 mdash lt0 mdash gt0120572 = 120578 = 1 lt0 lt0 gt0 gt0120572 = 120578 = 0 mdash gt0 mdash lt01120572 = 1120578 = 120573 gt 1 mdash lt0 mdash gt0

Proof The rate of change of curvature of (30) is given by

1205811015840(119905) =

119861119890119905Vperp

(V2perp+ V2||) 120572 = 0

minus119861119905minus1minus120573

120573Vperp

(V2perp+ V2||) 120572 =

1

120573

(35)

1205911015840(119905) =

minus119861119890119905V||

(V2perp+ V2||) 120572 = 0

119861119905minus1minus120573V||120573

(V2perp+ V2||) 120572 =

1

120573

(36)

The values of 1205811015840(119905) and 1205911015840(119905) for all 119905 gt 0 or 119905 lt 0 are as

given in Table 3 by inspecting (33) and (34) As the shapesof the curve on the interval 119905 lt 0 are either nonexisting ormirror images of those on 119905 gt 0 except for 1120572 = 1120578 =

120573 lt 0 120573 isin Z thus we consider only the interval where119905 gt 0 Since 1205811015840(119905) and 120591

1015840(119905) are always negative or positive

in the interval 119905 gt 0 the proposition above holds Note thatthe sign of 1205811015840(119905) and 120591

1015840(119905) may change if the sign of 119902(119905) is

inverted

Proposition 5 LMCs have inflection points

Inflection points are important in achieving G2-continuous S-shaped splines and connecting a curve to astraight line or vice versa in CAD applications

Proof From the inspecting the curvature function the inflec-tion is most obvious for (1120572) = 120573 lt 0 120573 isin Z as thesecurvesrsquo inflection points occur when 119905 = 0 and 120581

1015840(119905) = 0

elsewhere (see (32) and (35)) Note that the circle (120573 = 0)does not have any inflection points For the rest of cases ofLMC the inflection points may occur at 119905 rarr infin See Table 2for more detail


Proposition 6 Increasing the scaling factor V1of anymagnetic

curves (including LMC) by dwill scale the original curvature bya factor of V

1(V1+ 119889)

Proof Assume a magnetic curve is originally scaled to afactor of V

1 We have 120581

1(119905) = (119861119902(119905))V

1 Increasing the

scaling factor (V1) of magnetic curves by 119889 the new curvature

at 119905 is

1205812(119905) =

119861119902 (119905)

(V1+ 119889)

(37)

Rearranging both 1205811(119905) and 120581

2(119905) such that it forms a relation-

ship we obtain

1205812(119905) =

V1

(V1+ 119889)

1205811(119905) (38)

This relationship aids the process of designingG2-continuousaesthetic splines as we may anticipate how scaling affects thecurvatures at the end points of the splines

Proposition 7 LMC is extensible

When additional data point is placed on a LMC the shapeof the curve does not change if it satisfies the extensionalityproperty

Proof Given LMC interpolating points 1199010and 119901

1with tan-

gents 0and

1 for any data point 119901

119899and respective tangent

119899taken from the curve segment the LMC interpolating

the point-tangent pairs (11990100) and (119901

119899119899) or (119901

119899119899) and

(11990111) coincides with original LMC which interpolates

(1199010 0) and (119901

11) provided that the shape parameter 120573 =

1120572 of the smaller segment is the same as the originalsegment

Proposition 8 LMC has global characteristicsGood locality is a property where a small local change in

the position of one data point will affect only the curve shapenear the local change position

Proof Similar to log-aesthetic curve LMC is determined bythree control points with two being the start and end pointsand the other being the intersection point of the tangents atstart and end points Local changes at any one of the controlpoint change almost the entire curve shape

5 Reparameterization of LMC

In this section the reparameterization of LMC for variousdesign application in two-dimensional spaces is discussedThese parameterizations are tangential angles and end cur-vature parameterization

Table 4 Boundaries for 119861 in set notation

1120572 = 120573 = 1 (0 120579 lowast (1 minus 120573))

120572 = 120573 = 1 (0infin)

120572 = 0 (0infin)

The LMC in the form of tangential angles is parameter-ized as follows

119862TA (120579)

=

Vperpint

120579

0

(1

119861119890119905119900 + 120579) 119890119894120579119889120579 120572 = 0

Vperpint

120579

120579119900

119905119900119890120579119861

119890119894120579119889120579 120572 = 120573 = 1

Vperpint

120579

120579119900

1

119861(119905119900

1minus120573+120579 (1 minus 120573)

119861)

120573(1minus120573)

times 119890119894120579119889120579 120572 =

1

120573= 1

(39)

where 119905119900is a user defined time parameter which satisfies

the range of 119905 presented in Table 4 which is the same as119905119900stated in the first section of this paper Fixing 119905

119900to a

real value the equation can be used to solve G1 Hermiteinterpolation problem using Yoshida and Saitorsquos [5] curvegeneration algorithm For simplicity we set 119905

119900= 1 Instead

of searching for the shape parameter Λ in original LACequation119861 in (39) is searchedThe boundaries for119861 are givenin Table 4 Note that 119902(119905)rsquos sign is changed to negative in (39)so that the curve is on quadrants I and II of the x-y plane

In order to achieve G2-continuity it is required to manip-ulate the end curvature of a curve LMC is parameterized todirectly manipulate both end curvatures while satisfying theuser defined tangential angles of the curve The equation ofthe end curvature parameterized curve is

119862119870(120581)

=

Vperpint

120581

1205810

120581minus1119890119894Vperp(120581minus1205810)119889120579 120572 = 0

119861int

120581

1205810

120581minus2119890119894119861((ln 1120581)minus(ln 1120581

0))119889120579 120572 = 120573 = 1

minusV2perp

119861120573

timesint

120581

1205810

119890119894119861((119861V

perp120581)(1minus120573)120573

minus(119861Vperp1205810)(1minus120573)120573

)(1minus120573)

times(1

(119861Vperp120581)minus(120573+1)120573

)119889120581 120572 =1

120573= 1

(40)


The following equations can be incorporated into (40) to fixthe end tangents so that the angle is stated by the user

Vperp(120579) =

120579

1205811minus 1205810

120572 = 0

119861 (120579) =120579

(ln (11205811) minus ln (1120581

0)) 120572 = 120573 = 1

(41)

Vperp(120579) = (

120579 (1 minus 120573)

119861 ((1198611205810)(1minus120573)120573

minus (1198611205811)(1minus120573)120573

)

)

minus120573(1minus120573)

120572 =1

120573= 1

(42)

where 1205810 1205811 and 120579 are the user defined start curvature end

curvature and end tangential angles Note that the tangentialangle at the origin (starting point of the curve) is always 0as discussed in Section 1 Figure 9 shows LMC curves plottedwith (40) and (41) However 119861will not affect the curve shapewhen (41) is substituted in (40) This is because the 120573 termsin (41) will cancel out in (40) when (40) is substituted with(41) and the only parameters left in the resulting equation are120573 = 1120572 120579 120581

0 and 120581

We solve G2 Hermite interpolation for C-shaped curveswith the method shown in Algorithm 1

Note

1198621198961(1205811) is 119862119896(1205811) when 120572 = 0

1198621198962(1205811) is 119862119896(1205811) when 120572 = 1

1198621198963(1205811) is 119862119896(1205811) when 120572 = 1120573 = 0 = 1

Consider 1198621198963(1205811)= minusRe[119862

1198963(1205811)] Im[119862

1198963(1205811)]

120581119878and 120581119864are substituted and used in the equations in

the entire algorithm

The following section presents implementation examples

6 Numerical Examples

In this section three examples of G2 Hermite interpolationproblem are presented G2-continuous LMCs are also shownto indicate the possibilities of joining segments with different120572 matching end curvatures to produce S-shaped curvesAt the end of this section LM surfaces are presented anddiscussed

Using the method proposed in the previous section twoexamples of implementation are provided in Figure 10 Theinputs for these figures are provided in Table 5

We used Mathematica built-in minimization function(Find Minimum) which employs principal axis method [21]for the optimization process as it does not require thecomputation of derivatives The two initial search points areset as 120573 = 6 and 120573 = 50 for 120573 = 1120572 gt 0 and 120573 =

minus50 and 120573 = minus6 for 120573 = 1120572 lt 0 These initial searchpoints do not determine the boundary of the search area

x

y

06

05

04

03

02

01

02 04 06 08

P0

P2

Figure 11 Solution of G2 Hermite interpolation problem does notexist

120581 = 14120581 = 08 120581 = 0 120581 = minus11

Figure 12 A G2-continuous LM spline where left segment 120581 isin

[14 08] 120573 = 08 119861 = 1 Vperp= 1 middle segment 120581 isin [08 0]

120573 = minus1 119861 = 1 Vperp= 1 right segment 120581 isin [08 0] 120573 = minus04 119861 = 3

Vperp= 2

There are cases where the solution does not exist due to thenumber of constraints imposed on the curve For exampleit is notable that even though Figure 10(a) seems to coincidewith the given end points in fact it has approximately 10minus5error (distance between the given end point and the end pointof the curve) which if we set the user tolerance to be below10minus12 this curve is not acceptable leading to a conclusion that

a solution does not exist Another obvious example is whenthe given inputs are 120581

0= 12 120581

1= 09 119875

2= 0849 0621

and 120579 = 1205873 the solution is 120573 = 000904244 with error =

00950918 as shown in Figure 11LMC can be modified to control both start and end

curvatures and tangent angle directly using Algorithm 1 Theproposed method is to solve G2 Hermite data with only asingle segment of LAC which has not been achieved beforeIt also preserves curvature monotonicity as only one segmentis used

Figure 12 is a G2-continuous LM spline joined togetherby matching the data points at the joints without using anyinterpolationmethods It is to show the possibility of creatinga G2-continuous spline with different 120572 values It also showsthe possibility of creating G2-continuous C-shaped and S-shaped LM spline with different 120572 values

Algorithm 1 solves G2 Hermite data with only a singlesegment of LAC which has not been achieved before for


Three control points start and end curvatures and user tolerance are given Using Principal Axisminimization method shape parameter 120572 = 1120573 is searched Using the new parameter a new curveis transformed back into the original orientation and position and be plottedInput 119875

119886 119875119887 119875119888 1205810 1205811 tol

Output 120572 = 1120573

BeginStep 0 set 120581

119878= |1205810| 120581119864= |1205811| and 120573

1= 1205732= 1

Step 1 if 120581119864gt 120581119878

1198750larr 119875119888 1198752larr 119875119886

else 120581119864le 120581119878

1198750larr 119875119886 1198752larr 119875119888

Step 2 Translate 1198750to (0 0)

Step 3 if 120581119864gt 120581119878

Reflect triangle over 119910-axisStep 4 Rotate triangle such that tangent vector at 119875

0(1198751) coincides with 119909-axis

Step 5 120579 larr 120587 minus cosminus1 (100381710038171003817100381711987501198752

1003817100381710038171003817

2

+100381710038171003817100381711987501198751

1003817100381710038171003817

2

minus100381710038171003817100381711987511198752

1003817100381710038171003817

2

2100381710038171003817100381711987501198751

1003817100381710038171003817100381710038171003817100381711987511198752

1003817100381710038171003817

)

Step 6 if 120581119864= 120581119878

Set 120573 = 0 Vperp= 1120581

119864 119861 = 1

if min 1003817100381710038171003817119862TA(120579) 1198752

1003817100381710038171003817 lt tol120573 larr 0

elseif 10038171003817100381710038171198621198961(120581119864)1198752

1003817100381710038171003817 le tol120572 larr 0 go to Step 8

else if 10038171003817100381710038171198621198962(120581119864)11987521003817100381710038171003817 le tol

120572 larr 1 go to Step 8else

if 120581119864

= 0

1205731larr 10

1205732larr argmin

120573gt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817

else1205731larr argmin

120573gt0

10038171003817100381710038171198621198963 (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817

1205732larr argmin

120573lt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817

Find 120573 from 1205731 1205732 such that 119890119903119903119900119903 = min119890119903119903119900119903

1 119890119903119903119900119903

2

end ifStep 7 if 120573 = 1

Output ldquoSolution does not exist or method failed to convergerdquoelse

Transform LMC back to 119875119886 119875119887 119875119888

Plot LMCEnd

Algorithm 1

Table 5 Inputs for Figures 10(a)ndash10(c)

Transformed end point 1198752

120581119900

1205811

120579

Figure 10(a) 1966000000000 0655000000000 2000 0125 712058718

Figure 10(b) 0265969303943 0109176561142 3400 1700 21205879Figure 10(c) 0145992027985 0084314886280 9000 0000 1205874

LA curve design The feature of controlling end curvaturesopens up to new possibilities in creating more variation ofG2 splines such as forming S-shaped G2 spline with two C-shaped LACs

Figure 13 shows the two-dimensional LMC profile andreference curve and Figure 14 shows a surface generated using

Frenet sweeping method with the curves in Figure 13 TheLM surface is generated by sweeping a C-shaped LM profilecurve along S-shaped reference curve This is achieved bytranslating the profile curve along the reference curve whilerotating the profile curve to match the reference curversquosFrenet frame The inputs of the profile and reference LMC


minus01

minus02

minus03

minus04

05 10 15

y

x

(a)

minus005

minus010

minus015

01 02 03 04 05

x

y

(b)

Figure 13 LMC reference curve (a) and profile curve (b) Note that the reference curversquos acceleration direction is in the opposite directionby changing the sign of 119902(119905)

(a) (b)

Figure 14 A Frenet swept LM surface (a) and its surface analysis using horizontal (top (b)) and vertical (bottom (b)) zebra mapping

minus05


x

y

05 10 15 20 25

(a)

minus05


x

y

05 10 15 20 25

(b)

(i)

(iii) (ii)

(c)

Figure 15 LM reference curve (a) and profile curve (b) (c) is the representation of (a) and (b) in space where (i) is the space representationof (a) (ii) is the space representation of (b) and (iii) is created by reflecting (ii)


(a) (b)

Figure 16 A Frenet swept LM surface rendered in Mathematica (a) its surface analysis using horizontal (upper (b)) and vertical (bottom(b)) zebra mapping

Table 6 Inputs for Figures 13 and 14

119905 119905119900

119861 V||

Vperp

120573

Profile curve (C-shaped) [1 16] 1 1 1 0 09

Reference curve (S-shaped) [minus07 1] minus07 2 1 0 minus1


119905 119905119900

119861 V||

Vperp

120573

Profile curve [1 16] 1 1 1 0 0

Reference curve [minus3 minus17] 0 2 1 0 minus2

are given inTable 6 Another examplewhere two symmetricalsurfaces generated using the samemethod are joined togetherwith G2-continuity in designing a car hood is provided Thereference and profile curves are shown in Figure 15 while theinputs are shown in Table 7 The surface plot is provided inFigure 16

7 Conclusion and Future Work

This paper reformulates log-aesthetic curves under the influ-ence of a magnetic field and denotes it as log-aesthetic mag-netic curves The physical analysis provides an insight intovarious parameters previously regarded as shape parametersWe derived an end curvature controllable LAC with theformulation of LMC We have also presented the possibilityof interpolating given G2 Hermite data with a single segmentof LMC The characteristics LMC indicate high potential forCAD applications Two examples of surface generation usingLMC segments illustrated in the final section along with itszebra maps are indicating LMC surfaces are of high quality

Future work includes in-depth analysis of the drawableregion of G2 LMC and the study of the generalization of mag-netics curveswith various possibilities of 119902(119905) representations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors acknowledge Ministry of Education Malaysiafor providing financial aid (FRGS 59265) to carry out thisresearch Special thanks are due to to anonymous review-ers for providing constructive comments which helped toimprove the presentation of this paper

References

[1] J Ruskin The Elements of Drawing Dover Publications NewYork NY USA 1971

[2] R U Gobithaasan and K T Miura ldquoLogarithmic curvaturegraph as a shape interrogation toolrdquo Applied MathematicalSciences vol 8 no 16 pp 755ndash765 2014

[3] T Harada F Yoshimoto andMMoriyama ldquoAn aesthetic curvein the field of industrial designrdquo in Proceedings of the IEEESymposium on Visual Languages pp 38ndash47 1999

[4] K T Miura ldquoA general equation of aesthetic curves and its self-affinityrdquo Computer-Aided Design and Applications vol 3 no 1ndash4 pp 457ndash464 2006

[5] N Yoshida and T Saito ldquoInteractive aesthetic curve segmentsrdquoThe Visual Computer vol 22 no 9ndash11 pp 896ndash905 2006

[6] R U Gobithaasan and K T Miura ldquoAesthetic spiral for designrdquoSains Malaysiana vol 40 no 11 pp 1301ndash1305 2011

[7] R U Gobithaasan R Karpagavalli and K T Miura ldquoDrawableregion of the generalized log aesthetic curvesrdquo Journal ofApplied Mathematics vol 2013 Article ID 732457 7 pages 2013

[8] R U Gobithaasan R Karpagavalli and K T Miura ldquoShapeanalysis of generalized log-aesthetic curvesrdquo International Jour-nal of Mathematical Analysis vol 7 no 36 pp 1751ndash1759 2013


[9] R U Gobithaasan Y M Teh A R M Piah and K T MiuraldquoGeneration of Log-aesthetic curves using adaptive RungendashKutta methodsrdquo Applied Mathematics and Computation vol246 pp 257ndash262 2014

[10] K T Miura D Shibuya R U Gobithaasan and S UsukildquoDesigning log-aesthetic splineswithG2 continuityrdquoComputer-Aided Design and Applications vol 10 no 6 pp 1021ndash1032 2013

[11] N Yoshida and T Saito ldquoQuasi-aesthetic curves in rationalcubic Bezier formsrdquo Computer-Aided Design and Applicationsvol 4 no 1ndash4 pp 477ndash486 2007

[12] G Farin ldquoClass A Bezier curvesrdquo Computer Aided GeometricDesign vol 23 no 7 pp 573ndash581 2006

[13] R I Nabiyev and R Ziatdinov ldquoA mathematical design andevaluation of Bernstein-Bezier curvesrsquo shape features using thelaws of technical aestheticsrdquo Mathematical Design amp TechnicalAesthetics vol 2 no 1 pp 6ndash13 2014

[14] R Levien and C H Sequin ldquoInterpolating splines which is thefairest of them allrdquo Computer-Aided Design and Applicationsvol 6 no 1 pp 91ndash102 2009

[15] 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

[16] J D Jackson Classical Electrodynamics Wiley New York NYUSA 1962

[17] J Bittencourt ldquoCharged particle motion in constant anduniform electromagnetic fieldsrdquo in Fundamentals of PlasmaPhysics pp 33ndash58 Springer New York NY USA 2004

[18] L Xu and D Mould ldquoMagnetic curves curvature-controlledaesthetic curves using magnetic fieldsrdquo in Proceedings of the 5thEurographics conference on Computationals Visualization andImaging Aesthetics in Graphic pp 1ndash8 2009

[19] R U Gobithaasan and K T Miura ldquoLogarithmic curvaturegraph as a shape interrogation toolrdquo Applied MathematicalSciences vol 8 no 13ndash16 pp 755ndash765 2014

[20] G Harary and A Tal ldquo3D Euler spirals for 3D curve comple-tionrdquo Computational Geometry vol 45 no 3 pp 115ndash126 2012

[21] R Brent Algorithms for Minimization without Drivatives Pren-tice Hall Englewood Cliffs NJ USA 1972

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


(a) Car model (side view)

(b) Original curve (c) Triple LA curve

Figure 1 Application of LAC circular arc (1(b)) replaced by a LAspline to achieve G2-continuity (1(c)) [10]

CAGD purposes but are highly compute intensive and timeconsuming Recent progresses in this subject matter includethe development of log-aesthetic spline and its application inautomobile design (see Figure 1) Recently Gobithaasan et al[9] proposed Runge Kutta methods to compute LACGLACproven to achieve tremendous speedup

There are also curves whose LCGs are approximatelylinear These curves are known as quasi-aesthetic curvescoined by Yoshida and Saito [11] They proposed quasi-aesthetic curve segments in the form of rational cubic BezierThe resulting curve has monotone curvatures and has ahigher aesthetic value compared to the usual rational Beziersplines Class A Bezier [12] is another curve with monotoniccurvature known to have approximately linear LCG and canbe extended to space curves However its shape is limited tothat of a logarithmic spiral as the degree of the polynomialincreases Furthermore it is currently difficult to employClass A Bezier space curve for surface design Recent studyby Nabiyev and Ziatdinov [13] shows that Bezier curves withmonotonic curvatures are not always aesthetic in terms ofthe law of technical aesthetics A study by Levien and Sequin[14] suggested that LAC is the most promising family of faircurves Readers are referred to Miura and Gobithaasan [15]for a comprehensive review on aesthetic curves for variousdesign feats

In 2009 Xu and Mould proposed a particle tracingmethod to produce magnetic curve which is categorized asfair curve due to the monotonicity of its curvature functionIt was inspired by artistic motives and they further showed itspracticalities in computer graphics by rendering trees hairswater and fire Magnetic curves are defined as a particletracing method that creates curves with constantly varyingcurvature by utilizing the effect of amagnetic fieldThis paperaims to reformulate magnetic curves for CAD practicalitiesand further reformulates to construct splines which satisfyG2Hermite data

Section 1 elaborates in detail the physics behindmagneticcurves The contribution of this paper is threefold the firstpart elucidates the connection between magnetic curves andLACs and the second part reformulates magnetic curves inthe form of 2D and 3D log-aesthetic curve and denotes itas log-aesthetic magnetic curves (LMC) and log-aesthetic

magnetic space curves (LMSC) for CAD applications finallythe third part proposes an algorithm for developing G2continuous LMC

2 Formulation of Magnetic Curves

In a uniformmagnetic field themotion of a particle of charge119902 and mass 119898 travelling with velocity V under magneticinduction is the result of Lorentz Force [16]

= 119902 (V times ) (1)

which can be rewritten as

119898119889V119889119905

= 119902 (V times ) (2)

where times represents the cross product operation It describesthe motion of charged particles experiencing Lorentz forceFor convenience V is separated into two components withthe first V

||parallel to and the second V

perpperpendicular to

The steps on detailed derivation are omitted for brevityalbeit readers are referred to [17] To vary the curvature of thecurve 119902(119905) is set to be an arbitrary real function of 119905 instead ofa constant 119902 Note that the magnitude of the magnetic field119861 should not be confused with its vector and 119861 ge 0 Weobtain the components of magnetic curves as follows

119889V119909

119889119905=119902 (119905) 119861

119898V119910

119889V119910

119889119905= minus

119902 (119905) 119861

119898V119909

119889V119911

119889119905= 0

(3)

Note that V119911(119905) = V

119911(119905119900) = V|| and 119905

119900is the initial time 119905 of the

trajectory To simplify the expression we take (119902(119905)119861)119898 =

120596(119905) By using separation of variables method we arrive atthe following equations

V119909(119905) = 119860 cosΩ (119905) + 119863 sinΩ (119905) (4)

V119910(119905) = 119863 cosΩ (119905) minus 119860 sinΩ (119905) (5)

V119911(119905) = V

|| (6)

where Ω1015840(119905) = 120596(119905) and 119860 and 119863 are arbitrary constants Inorder to set the point at the origin the initial conditions areset to be V

119909(119905119900) = Vperpge 0 and V

119910(119905119900) = 0 which are the 119909 and

119910 components of Vperp These values are then substituted to (4)

These initial values dictate the direction andmagnitude of thevelocity vector at the origin of the Cartesian plane which can


120596(+)

rr

120596(minus)

B



minus1

rarr t0

rarr t1




V119909(119905119900) = 119860 cosΩ(119905

119900) + 119863 sinΩ(119905

119900)

V119910(119905119900) = 119863 cosΩ(119905

119900) minus 119860 sinΩ(119905

119900)

(7)



119909 (119905) = 1199090+ Vperpint

119905

119905119900

cos (Ω (119905119900) minus Ω (119905)) 119889119905 (8)

119910 (119905) = 1199100+ Vperpint

119905

119905119900

sin (Ω (119905119900) minus Ω (119905)) 119889119905 (9)

119911 (119905) = 1199110+ int

119905

119905119900

V||119889119905 (10)





perp 0 0⟩ to fix







120588 (119905) =V2perp+ V2||

Vperp |120596 (119905)|

=119898 (V2perp+ V2||)

1003816100381610038161003816119902 (119905)1003816100381610038161003816 119861Vperp

120596 (119905) =119902 (119905) 119861

119898

(11)


119900)minusΩ(119905)


120583 (119905) =V2perp+ V2||

V|| |120596 (119905)|

=119898 (V2perp+ V2||)

1003816100381610038161003816119902 (119905)1003816100381610038161003816 119861V||

(12)







perpis a constant


119902 (119905) = 119905minus120573 120573 isin R (14)


1 2 3 4 5 6

5

10

15

20

25

30

t

120581(t)

(a)

1 2 3 4 5 6

1

2

3

4

5

t

120581998400 (t)

(b)


1 15 2 25 3 35 4 45 5 119902(119905) lt 0

00 05 10 15 20 2500

05

10

15

20

25 120573 = minus1

120573 = minus2

120573 = minus3

120573 = minus4

(a)

minus10 minus05

minus05

05

05

120573 = 09

120573 = 08

120573 = 07120573 = 06

(b)

000005 00001

000002

000004

000006

120573 = 09

120573 = 08

120573 = 07

120573 = 06

minus000004


(c)



119909 (119905) = 1199090+

Vperpint

119905

119905119900

cos(119861 (1199051minus120573

119900minus 1199051minus120573

)

1 minus 120573)119889119905 120573 = 1

Vperpint

119905

119905119900

cos (119861 (ln (119905119900) minus ln (119905))) 119889119905 120573 = 1

119910 (119905) = 1199100+

Vperpint

119905

119905119900

sin(119861 (1199051minus120573

119900minus 1199051minus120573

)

1 minus 120573)119889119905 120573 = 1

Vperpint

119905

119905119900

sin (119861 (ln (119905119900) minus ln (119905))) 119889119905 120573 = 1

(15)


119862119883119872

(119905) = 1198750+

Vperpint

119905

119905119900

119890119894(119861(1199051minus120573

119900minus1199051minus120573

)(1minus120573))119889119905 120573 = 1

Vperpint

119905

119905119900

119890119894119861(ln(119905

119900)minusln(119905))

119889119905 120573 = 1

(16)


120596 (119905) = 119861119905minus120573 120573 isin R

120588 (119905) =V2perp+ V2||

1003816100381610038161003816119905minus1205731003816100381610038161003816 119861Vperp

(17)









LCG (119905) = log 120588 (119905) log(120588 (119905) 119904

1015840(119905)

1205881015840 (119905)) (18)

LTG (119905) = log120583 (119905) log(120583 (119905) 119904

1015840(119905)

1205831015840 (119905)) (19)

120582 (119905) = 1 +120588 (119905)

1205881015840(119905)2(1205881015840(119905) 11990410158401015840(119905)

1199041015840 (119905)minus 12058810158401015840(119905)) (20)

120595 (119905) = 1 +120583 (119905)

1205831015840(119905)2(1205831015840(119905) 11990410158401015840(119905)

1205831015840 (119905)minus 12058310158401015840(119905)) (21)


LCG (119905) =

logV2perp+ V2||

1198611003816100381610038161003816119905minus1205731003816100381610038161003816 Vperp


120573)

120582 (119905) = 120572 =1

120573

(22)


LTG (119905) =

logV2perp+ V2||

1198611003816100381610038161003816119905minus1205731003816100381610038161003816 V||


120573)

120595 (119905) = 120572 =1

120573

(23)


perphas







120588 (119905) =119889119904119889119905

119889120579119889119905=

Vperp

119889120579119889119905=119889119904

119889120579 (24)


120582 (119905) =1205791015840(119905) 120579(3)

(119905)

12057910158401015840(119905)2

minus 1 = 0 (25)


120579 (119905) =11989011990511988811198882

1198881

+ 1198883 120572 = 0 (26)

where 1198881 1198882 and c





Ω (119905) = 119861119890119905 120572 = 0 (27)

We let 1198882= minus119861 and 119888

3= minusΩ(119905

119900) such that 120579(119905) = Ω(119905

119900) minusΩ(119905)


perp Figure 7



119862LMC (119905) = 1198750+ Vperpint

119905

119905119900

119890119894120579(119905)

119889119905 (28)

where V||= 0 and 119875




119905

119905119900

cos 120579 (119905) 119889119905 Vperpint

119905

119905119900

sin 120579 (119905) 119889119905 int119905

119905119900

V||119889119905

(29)


0 5 10

0

5

10

minus10

minus10

minus5

minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

5

10

15

20

25

30

(a)

0 5 10 15

0

5

10

minus10

minus5

minus5t

(i) (ii) (iii)120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

1

2

3

4

(b)

0 5 10 15

0

5

minus10

minus5

minus5

t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

05

10

15

20

25

30

35

(c)

0 5 10

0

5

minus10

minus5

minus10 minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

2

4

6

8

(d)

Figure 6 Continued


0 5 10

0

5

minus10

minus5

minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

2

4

6

8

(e)

0 5 10

0

5

minus10 minus5

minus10

minus5

t

(i) (ii) (iii)120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 200

minus10

minus8

minus6

minus4

minus2

(f)


minus1

minus2

minus3

minus4


y

x

(a)

minus10

minus10

minus05 05

minus04

minus02

minus06

minus08

y

x

(b)


119900= minus05 119861 = 1 119905 isin [minus2 3]

with

120579 (119905) =

119861 (119890119905119900 minus 119890119905) 120572 = 0

119861 (log (119905119900) minus log (119905)) 120572 = 120573 = 1

119861 (1199051minus120573

119900minus 1199051minus120573

)

(1 minus 120573) 120572 = (

1

120573) = 1

(30)


119902 (119905) =

119905minus120573 120572 =

1

120573isin R

119890119905 120572 = 0

(31)



perp 0⟩ and


(a) (b)

(c) (d)



120581 (119905) =


V2perp+ V2||

120572 =1

120573isin R

minus119861119890119905Vperp

V2perp+ V2||

120572 = 0

(32)

120591 (119905) =

minus119861119905minus120573V||

V2perp+ V2||

120572 =1

120573isin R

minus119861119890119905V||

V2perp+ V2||

120572 = 0

(33)


perp


030

025

020

015

010

005

005 010 015 020 025 030 035

x

y

120572 = 0

120572 = minus1

120572 = 1

120572 = minus1

6120572 =

1

3


1=






120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(a)

120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(b)

1205881 = infin

120579P0

P1

P21205880 =

1

1205810

(c)



minus17







Figure 119905 119905119900

119861 V||

Vperp

120573








lowast] cup [0

lowastinfin)

1120572 = 120573 lt 0 120573 notin Z [0infin)





1120572 = 120573 gt 1 (0infinlowast)



120579 (119886119905 + 119887) =

119861(119890119905119900 minus 119890119886119905+119887

) 120572 = 0

119861 (log (119905119900) minus log (119886119905 + 119887)) 120572 = 1

119861 (1199051minus120573

119900minus (119886119905 + 119887)

1minus120573)

(1 minus 120573) 120572 = (

1

120573) = 1

(34)






||= 0




Values of 120572 or 120573 1205811015840(119905) 120591

1015840(119905)

119905 lt 0 119905 gt 0 119905 lt 0 119905 gt 0



1205811015840(119905) =

119861119890119905Vperp

(V2perp+ V2||) 120572 = 0


120573Vperp

(V2perp+ V2||) 120572 =

1

120573

(35)

1205911015840(119905) =

minus119861119890119905V||

(V2perp+ V2||) 120572 = 0

119861119905minus1minus120573V||120573

(V2perp+ V2||) 120572 =

1

120573

(36)







inverted




1015840(119905) = 0





1(V1+ 119889)


1 We have 120581

1(119905) = (119861119902(119905))V

1 Increasing the


at 119905 is

1205812(119905) =

119861119902 (119905)

(V1+ 119889)

(37)



ship we obtain

1205812(119905) =

V1

(V1+ 119889)

1205811(119905) (38)





1with tan-

gents 0and





119899119899) or (119901

119899119899) and


(1199010 0) and (119901









1120572 = 120573 = 1 (0 120579 lowast (1 minus 120573))

120572 = 120573 = 1 (0infin)

120572 = 0 (0infin)


119862TA (120579)

=

Vperpint

120579

0

(1

119861119890119905119900 + 120579) 119890119894120579119889120579 120572 = 0

Vperpint

120579

120579119900

119905119900119890120579119861

119890119894120579119889120579 120572 = 120573 = 1

Vperpint

120579

120579119900

1

119861(119905119900

1minus120573+120579 (1 minus 120573)

119861)

120573(1minus120573)

times 119890119894120579119889120579 120572 =

1

120573= 1

(39)



119900to a


119900= 1 Instead



119862119870(120581)

=

Vperpint

120581

1205810


119861int

120581

1205810

120581minus2119890119894119861((ln 1120581)minus(ln 1120581

0))119889120579 120572 = 120573 = 1

minusV2perp

119861120573

timesint

120581

1205810

119890119894119861((119861V

perp120581)(1minus120573)120573


)(1minus120573)

times(1

(119861Vperp120581)minus(120573+1)120573

)119889120581 120572 =1

120573= 1

(40)



Vperp(120579) =

120579

1205811minus 1205810

120572 = 0

119861 (120579) =120579

(ln (11205811) minus ln (1120581

0)) 120572 = 120573 = 1

(41)

Vperp(120579) = (

120579 (1 minus 120573)

119861 ((1198611205810)(1minus120573)120573

minus (1198611205811)(1minus120573)120573

)

)


120572 =1

120573= 1

(42)



0 and 120581


Note

1198621198961(1205811) is 119862119896(1205811) when 120572 = 0

1198621198962(1205811) is 119862119896(1205811) when 120572 = 1

1198621198963(1205811) is 119862119896(1205811) when 120572 = 1120573 = 0 = 1


1198963(1205811)] Im[119862

1198963(1205811)]









x

y

06

05

04

03

02

01

02 04 06 08

P0

P2


120581 = 14120581 = 08 120581 = 0 120581 = minus11




Vperp= 2



0= 12 120581

1= 09 119875

2= 0849 0621








119886 119875119887 119875119888 1205810 1205811 tol

Output 120572 = 1120573


119878= |1205810| 120581119864= |1205811| and 120573

1= 1205732= 1

Step 1 if 120581119864gt 120581119878

1198750larr 119875119888 1198752larr 119875119886

else 120581119864le 120581119878

1198750larr 119875119886 1198752larr 119875119888


Step 3 if 120581119864gt 120581119878




1003817100381710038171003817

2

+100381710038171003817100381711987501198751

1003817100381710038171003817

2

minus100381710038171003817100381711987511198752

1003817100381710038171003817

2

2100381710038171003817100381711987501198751

1003817100381710038171003817100381710038171003817100381711987511198752

1003817100381710038171003817

)

Step 6 if 120581119864= 120581119878

Set 120573 = 0 Vperp= 1120581

119864 119861 = 1

if min 1003817100381710038171003817119862TA(120579) 1198752

1003817100381710038171003817 lt tol120573 larr 0

elseif 10038171003817100381710038171198621198961(120581119864)1198752


else if 10038171003817100381710038171198621198962(120581119864)11987521003817100381710038171003817 le tol


if 120581119864

= 0

1205731larr 10

1205732larr argmin

120573gt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


120573gt0

10038171003817100381710038171198621198963 (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817

1205732larr argmin

120573lt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


1 119890119903119903119900119903

2




Plot LMCEnd

Algorithm 1



120581119900

1205811

120579

Figure 10(a) 1966000000000 0655000000000 2000 0125 712058718

Figure 10(b) 0265969303943 0109176561142 3400 1700 21205879Figure 10(c) 0145992027985 0084314886280 9000 0000 1205874





minus01

minus02

minus03

minus04

05 10 15

y

x

(a)

minus005

minus010

minus015

01 02 03 04 05

x

y

(b)


(a) (b)


minus05


x

y

05 10 15 20 25

(a)

minus05


x

y

05 10 15 20 25

(b)

(i)

(iii) (ii)

(c)



(a) (b)



119905 119905119900

119861 V||

Vperp

120573




119905 119905119900

119861 V||

Vperp

120573









Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120596(+)

rr

120596(minus)

B



minus1

rarr t0

rarr t1




V119909(119905119900) = 119860 cosΩ(119905

119900) + 119863 sinΩ(119905

119900)

V119910(119905119900) = 119863 cosΩ(119905

119900) minus 119860 sinΩ(119905

119900)

(7)



119909 (119905) = 1199090+ Vperpint

119905

119905119900

cos (Ω (119905119900) minus Ω (119905)) 119889119905 (8)

119910 (119905) = 1199100+ Vperpint

119905

119905119900

sin (Ω (119905119900) minus Ω (119905)) 119889119905 (9)

119911 (119905) = 1199110+ int

119905

119905119900

V||119889119905 (10)





perp 0 0⟩ to fix







120588 (119905) =V2perp+ V2||

Vperp |120596 (119905)|

=119898 (V2perp+ V2||)

1003816100381610038161003816119902 (119905)1003816100381610038161003816 119861Vperp

120596 (119905) =119902 (119905) 119861

119898

(11)


119900)minusΩ(119905)


120583 (119905) =V2perp+ V2||

V|| |120596 (119905)|

=119898 (V2perp+ V2||)

1003816100381610038161003816119902 (119905)1003816100381610038161003816 119861V||

(12)







perpis a constant


119902 (119905) = 119905minus120573 120573 isin R (14)


1 2 3 4 5 6

5

10

15

20

25

30

t

120581(t)

(a)

1 2 3 4 5 6

1

2

3

4

5

t

120581998400 (t)

(b)


1 15 2 25 3 35 4 45 5 119902(119905) lt 0

00 05 10 15 20 2500

05

10

15

20

25 120573 = minus1

120573 = minus2

120573 = minus3

120573 = minus4

(a)

minus10 minus05

minus05

05

05

120573 = 09

120573 = 08

120573 = 07120573 = 06

(b)

000005 00001

000002

000004

000006

120573 = 09

120573 = 08

120573 = 07

120573 = 06

minus000004


(c)



119909 (119905) = 1199090+

Vperpint

119905

119905119900

cos(119861 (1199051minus120573

119900minus 1199051minus120573

)

1 minus 120573)119889119905 120573 = 1

Vperpint

119905

119905119900

cos (119861 (ln (119905119900) minus ln (119905))) 119889119905 120573 = 1

119910 (119905) = 1199100+

Vperpint

119905

119905119900

sin(119861 (1199051minus120573

119900minus 1199051minus120573

)

1 minus 120573)119889119905 120573 = 1

Vperpint

119905

119905119900

sin (119861 (ln (119905119900) minus ln (119905))) 119889119905 120573 = 1

(15)


119862119883119872

(119905) = 1198750+

Vperpint

119905

119905119900

119890119894(119861(1199051minus120573

119900minus1199051minus120573

)(1minus120573))119889119905 120573 = 1

Vperpint

119905

119905119900

119890119894119861(ln(119905

119900)minusln(119905))

119889119905 120573 = 1

(16)


120596 (119905) = 119861119905minus120573 120573 isin R

120588 (119905) =V2perp+ V2||

1003816100381610038161003816119905minus1205731003816100381610038161003816 119861Vperp

(17)









LCG (119905) = log 120588 (119905) log(120588 (119905) 119904

1015840(119905)

1205881015840 (119905)) (18)

LTG (119905) = log120583 (119905) log(120583 (119905) 119904

1015840(119905)

1205831015840 (119905)) (19)

120582 (119905) = 1 +120588 (119905)

1205881015840(119905)2(1205881015840(119905) 11990410158401015840(119905)

1199041015840 (119905)minus 12058810158401015840(119905)) (20)

120595 (119905) = 1 +120583 (119905)

1205831015840(119905)2(1205831015840(119905) 11990410158401015840(119905)

1205831015840 (119905)minus 12058310158401015840(119905)) (21)


LCG (119905) =

logV2perp+ V2||

1198611003816100381610038161003816119905minus1205731003816100381610038161003816 Vperp


120573)

120582 (119905) = 120572 =1

120573

(22)


LTG (119905) =

logV2perp+ V2||

1198611003816100381610038161003816119905minus1205731003816100381610038161003816 V||


120573)

120595 (119905) = 120572 =1

120573

(23)


perphas







120588 (119905) =119889119904119889119905

119889120579119889119905=

Vperp

119889120579119889119905=119889119904

119889120579 (24)


120582 (119905) =1205791015840(119905) 120579(3)

(119905)

12057910158401015840(119905)2

minus 1 = 0 (25)


120579 (119905) =11989011990511988811198882

1198881

+ 1198883 120572 = 0 (26)

where 1198881 1198882 and c





Ω (119905) = 119861119890119905 120572 = 0 (27)

We let 1198882= minus119861 and 119888

3= minusΩ(119905

119900) such that 120579(119905) = Ω(119905

119900) minusΩ(119905)


perp Figure 7



119862LMC (119905) = 1198750+ Vperpint

119905

119905119900

119890119894120579(119905)

119889119905 (28)

where V||= 0 and 119875




119905

119905119900

cos 120579 (119905) 119889119905 Vperpint

119905

119905119900

sin 120579 (119905) 119889119905 int119905

119905119900

V||119889119905

(29)


0 5 10

0

5

10

minus10

minus10

minus5

minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

5

10

15

20

25

30

(a)

0 5 10 15

0

5

10

minus10

minus5

minus5t

(i) (ii) (iii)120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

1

2

3

4

(b)

0 5 10 15

0

5

minus10

minus5

minus5

t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

05

10

15

20

25

30

35

(c)

0 5 10

0

5

minus10

minus5

minus10 minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

2

4

6

8

(d)

Figure 6 Continued


0 5 10

0

5

minus10

minus5

minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

2

4

6

8

(e)

0 5 10

0

5

minus10 minus5

minus10

minus5

t

(i) (ii) (iii)120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 200

minus10

minus8

minus6

minus4

minus2

(f)


minus1

minus2

minus3

minus4


y

x

(a)

minus10

minus10

minus05 05

minus04

minus02

minus06

minus08

y

x

(b)


119900= minus05 119861 = 1 119905 isin [minus2 3]

with

120579 (119905) =

119861 (119890119905119900 minus 119890119905) 120572 = 0

119861 (log (119905119900) minus log (119905)) 120572 = 120573 = 1

119861 (1199051minus120573

119900minus 1199051minus120573

)

(1 minus 120573) 120572 = (

1

120573) = 1

(30)


119902 (119905) =

119905minus120573 120572 =

1

120573isin R

119890119905 120572 = 0

(31)



perp 0⟩ and


(a) (b)

(c) (d)



120581 (119905) =


V2perp+ V2||

120572 =1

120573isin R

minus119861119890119905Vperp

V2perp+ V2||

120572 = 0

(32)

120591 (119905) =

minus119861119905minus120573V||

V2perp+ V2||

120572 =1

120573isin R

minus119861119890119905V||

V2perp+ V2||

120572 = 0

(33)


perp


030

025

020

015

010

005

005 010 015 020 025 030 035

x

y

120572 = 0

120572 = minus1

120572 = 1

120572 = minus1

6120572 =

1

3


1=






120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(a)

120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(b)

1205881 = infin

120579P0

P1

P21205880 =

1

1205810

(c)



minus17







Figure 119905 119905119900

119861 V||

Vperp

120573








lowast] cup [0

lowastinfin)

1120572 = 120573 lt 0 120573 notin Z [0infin)





1120572 = 120573 gt 1 (0infinlowast)



120579 (119886119905 + 119887) =

119861(119890119905119900 minus 119890119886119905+119887

) 120572 = 0

119861 (log (119905119900) minus log (119886119905 + 119887)) 120572 = 1

119861 (1199051minus120573

119900minus (119886119905 + 119887)

1minus120573)

(1 minus 120573) 120572 = (

1

120573) = 1

(34)






||= 0




Values of 120572 or 120573 1205811015840(119905) 120591

1015840(119905)

119905 lt 0 119905 gt 0 119905 lt 0 119905 gt 0



1205811015840(119905) =

119861119890119905Vperp

(V2perp+ V2||) 120572 = 0


120573Vperp

(V2perp+ V2||) 120572 =

1

120573

(35)

1205911015840(119905) =

minus119861119890119905V||

(V2perp+ V2||) 120572 = 0

119861119905minus1minus120573V||120573

(V2perp+ V2||) 120572 =

1

120573

(36)







inverted




1015840(119905) = 0





1(V1+ 119889)


1 We have 120581

1(119905) = (119861119902(119905))V

1 Increasing the


at 119905 is

1205812(119905) =

119861119902 (119905)

(V1+ 119889)

(37)



ship we obtain

1205812(119905) =

V1

(V1+ 119889)

1205811(119905) (38)





1with tan-

gents 0and





119899119899) or (119901

119899119899) and


(1199010 0) and (119901









1120572 = 120573 = 1 (0 120579 lowast (1 minus 120573))

120572 = 120573 = 1 (0infin)

120572 = 0 (0infin)


119862TA (120579)

=

Vperpint

120579

0

(1

119861119890119905119900 + 120579) 119890119894120579119889120579 120572 = 0

Vperpint

120579

120579119900

119905119900119890120579119861

119890119894120579119889120579 120572 = 120573 = 1

Vperpint

120579

120579119900

1

119861(119905119900

1minus120573+120579 (1 minus 120573)

119861)

120573(1minus120573)

times 119890119894120579119889120579 120572 =

1

120573= 1

(39)



119900to a


119900= 1 Instead



119862119870(120581)

=

Vperpint

120581

1205810


119861int

120581

1205810

120581minus2119890119894119861((ln 1120581)minus(ln 1120581

0))119889120579 120572 = 120573 = 1

minusV2perp

119861120573

timesint

120581

1205810

119890119894119861((119861V

perp120581)(1minus120573)120573


)(1minus120573)

times(1

(119861Vperp120581)minus(120573+1)120573

)119889120581 120572 =1

120573= 1

(40)



Vperp(120579) =

120579

1205811minus 1205810

120572 = 0

119861 (120579) =120579

(ln (11205811) minus ln (1120581

0)) 120572 = 120573 = 1

(41)

Vperp(120579) = (

120579 (1 minus 120573)

119861 ((1198611205810)(1minus120573)120573

minus (1198611205811)(1minus120573)120573

)

)


120572 =1

120573= 1

(42)



0 and 120581


Note

1198621198961(1205811) is 119862119896(1205811) when 120572 = 0

1198621198962(1205811) is 119862119896(1205811) when 120572 = 1

1198621198963(1205811) is 119862119896(1205811) when 120572 = 1120573 = 0 = 1


1198963(1205811)] Im[119862

1198963(1205811)]









x

y

06

05

04

03

02

01

02 04 06 08

P0

P2


120581 = 14120581 = 08 120581 = 0 120581 = minus11




Vperp= 2



0= 12 120581

1= 09 119875

2= 0849 0621








119886 119875119887 119875119888 1205810 1205811 tol

Output 120572 = 1120573


119878= |1205810| 120581119864= |1205811| and 120573

1= 1205732= 1

Step 1 if 120581119864gt 120581119878

1198750larr 119875119888 1198752larr 119875119886

else 120581119864le 120581119878

1198750larr 119875119886 1198752larr 119875119888


Step 3 if 120581119864gt 120581119878




1003817100381710038171003817

2

+100381710038171003817100381711987501198751

1003817100381710038171003817

2

minus100381710038171003817100381711987511198752

1003817100381710038171003817

2

2100381710038171003817100381711987501198751

1003817100381710038171003817100381710038171003817100381711987511198752

1003817100381710038171003817

)

Step 6 if 120581119864= 120581119878

Set 120573 = 0 Vperp= 1120581

119864 119861 = 1

if min 1003817100381710038171003817119862TA(120579) 1198752

1003817100381710038171003817 lt tol120573 larr 0

elseif 10038171003817100381710038171198621198961(120581119864)1198752


else if 10038171003817100381710038171198621198962(120581119864)11987521003817100381710038171003817 le tol


if 120581119864

= 0

1205731larr 10

1205732larr argmin

120573gt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


120573gt0

10038171003817100381710038171198621198963 (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817

1205732larr argmin

120573lt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


1 119890119903119903119900119903

2




Plot LMCEnd

Algorithm 1



120581119900

1205811

120579

Figure 10(a) 1966000000000 0655000000000 2000 0125 712058718

Figure 10(b) 0265969303943 0109176561142 3400 1700 21205879Figure 10(c) 0145992027985 0084314886280 9000 0000 1205874





minus01

minus02

minus03

minus04

05 10 15

y

x

(a)

minus005

minus010

minus015

01 02 03 04 05

x

y

(b)


(a) (b)


minus05


x

y

05 10 15 20 25

(a)

minus05


x

y

05 10 15 20 25

(b)

(i)

(iii) (ii)

(c)



(a) (b)



119905 119905119900

119861 V||

Vperp

120573




119905 119905119900

119861 V||

Vperp

120573









Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6

5

10

15

20

25

30

t

120581(t)

(a)

1 2 3 4 5 6

1

2

3

4

5

t

120581998400 (t)

(b)


1 15 2 25 3 35 4 45 5 119902(119905) lt 0

00 05 10 15 20 2500

05

10

15

20

25 120573 = minus1

120573 = minus2

120573 = minus3

120573 = minus4

(a)

minus10 minus05

minus05

05

05

120573 = 09

120573 = 08

120573 = 07120573 = 06

(b)

000005 00001

000002

000004

000006

120573 = 09

120573 = 08

120573 = 07

120573 = 06

minus000004


(c)



119909 (119905) = 1199090+

Vperpint

119905

119905119900

cos(119861 (1199051minus120573

119900minus 1199051minus120573

)

1 minus 120573)119889119905 120573 = 1

Vperpint

119905

119905119900

cos (119861 (ln (119905119900) minus ln (119905))) 119889119905 120573 = 1

119910 (119905) = 1199100+

Vperpint

119905

119905119900

sin(119861 (1199051minus120573

119900minus 1199051minus120573

)

1 minus 120573)119889119905 120573 = 1

Vperpint

119905

119905119900

sin (119861 (ln (119905119900) minus ln (119905))) 119889119905 120573 = 1

(15)


119862119883119872

(119905) = 1198750+

Vperpint

119905

119905119900

119890119894(119861(1199051minus120573

119900minus1199051minus120573

)(1minus120573))119889119905 120573 = 1

Vperpint

119905

119905119900

119890119894119861(ln(119905

119900)minusln(119905))

119889119905 120573 = 1

(16)


120596 (119905) = 119861119905minus120573 120573 isin R

120588 (119905) =V2perp+ V2||

1003816100381610038161003816119905minus1205731003816100381610038161003816 119861Vperp

(17)









LCG (119905) = log 120588 (119905) log(120588 (119905) 119904

1015840(119905)

1205881015840 (119905)) (18)

LTG (119905) = log120583 (119905) log(120583 (119905) 119904

1015840(119905)

1205831015840 (119905)) (19)

120582 (119905) = 1 +120588 (119905)

1205881015840(119905)2(1205881015840(119905) 11990410158401015840(119905)

1199041015840 (119905)minus 12058810158401015840(119905)) (20)

120595 (119905) = 1 +120583 (119905)

1205831015840(119905)2(1205831015840(119905) 11990410158401015840(119905)

1205831015840 (119905)minus 12058310158401015840(119905)) (21)


LCG (119905) =

logV2perp+ V2||

1198611003816100381610038161003816119905minus1205731003816100381610038161003816 Vperp


120573)

120582 (119905) = 120572 =1

120573

(22)


LTG (119905) =

logV2perp+ V2||

1198611003816100381610038161003816119905minus1205731003816100381610038161003816 V||


120573)

120595 (119905) = 120572 =1

120573

(23)


perphas







120588 (119905) =119889119904119889119905

119889120579119889119905=

Vperp

119889120579119889119905=119889119904

119889120579 (24)


120582 (119905) =1205791015840(119905) 120579(3)

(119905)

12057910158401015840(119905)2

minus 1 = 0 (25)


120579 (119905) =11989011990511988811198882

1198881

+ 1198883 120572 = 0 (26)

where 1198881 1198882 and c





Ω (119905) = 119861119890119905 120572 = 0 (27)

We let 1198882= minus119861 and 119888

3= minusΩ(119905

119900) such that 120579(119905) = Ω(119905

119900) minusΩ(119905)


perp Figure 7



119862LMC (119905) = 1198750+ Vperpint

119905

119905119900

119890119894120579(119905)

119889119905 (28)

where V||= 0 and 119875




119905

119905119900

cos 120579 (119905) 119889119905 Vperpint

119905

119905119900

sin 120579 (119905) 119889119905 int119905

119905119900

V||119889119905

(29)


0 5 10

0

5

10

minus10

minus10

minus5

minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

5

10

15

20

25

30

(a)

0 5 10 15

0

5

10

minus10

minus5

minus5t

(i) (ii) (iii)120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

1

2

3

4

(b)

0 5 10 15

0

5

minus10

minus5

minus5

t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

05

10

15

20

25

30

35

(c)

0 5 10

0

5

minus10

minus5

minus10 minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

2

4

6

8

(d)

Figure 6 Continued


0 5 10

0

5

minus10

minus5

minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

2

4

6

8

(e)

0 5 10

0

5

minus10 minus5

minus10

minus5

t

(i) (ii) (iii)120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 200

minus10

minus8

minus6

minus4

minus2

(f)


minus1

minus2

minus3

minus4


y

x

(a)

minus10

minus10

minus05 05

minus04

minus02

minus06

minus08

y

x

(b)


119900= minus05 119861 = 1 119905 isin [minus2 3]

with

120579 (119905) =

119861 (119890119905119900 minus 119890119905) 120572 = 0

119861 (log (119905119900) minus log (119905)) 120572 = 120573 = 1

119861 (1199051minus120573

119900minus 1199051minus120573

)

(1 minus 120573) 120572 = (

1

120573) = 1

(30)


119902 (119905) =

119905minus120573 120572 =

1

120573isin R

119890119905 120572 = 0

(31)



perp 0⟩ and


(a) (b)

(c) (d)



120581 (119905) =


V2perp+ V2||

120572 =1

120573isin R

minus119861119890119905Vperp

V2perp+ V2||

120572 = 0

(32)

120591 (119905) =

minus119861119905minus120573V||

V2perp+ V2||

120572 =1

120573isin R

minus119861119890119905V||

V2perp+ V2||

120572 = 0

(33)


perp


030

025

020

015

010

005

005 010 015 020 025 030 035

x

y

120572 = 0

120572 = minus1

120572 = 1

120572 = minus1

6120572 =

1

3


1=






120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(a)

120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(b)

1205881 = infin

120579P0

P1

P21205880 =

1

1205810

(c)



minus17







Figure 119905 119905119900

119861 V||

Vperp

120573








lowast] cup [0

lowastinfin)

1120572 = 120573 lt 0 120573 notin Z [0infin)





1120572 = 120573 gt 1 (0infinlowast)



120579 (119886119905 + 119887) =

119861(119890119905119900 minus 119890119886119905+119887

) 120572 = 0

119861 (log (119905119900) minus log (119886119905 + 119887)) 120572 = 1

119861 (1199051minus120573

119900minus (119886119905 + 119887)

1minus120573)

(1 minus 120573) 120572 = (

1

120573) = 1

(34)






||= 0




Values of 120572 or 120573 1205811015840(119905) 120591

1015840(119905)

119905 lt 0 119905 gt 0 119905 lt 0 119905 gt 0



1205811015840(119905) =

119861119890119905Vperp

(V2perp+ V2||) 120572 = 0


120573Vperp

(V2perp+ V2||) 120572 =

1

120573

(35)

1205911015840(119905) =

minus119861119890119905V||

(V2perp+ V2||) 120572 = 0

119861119905minus1minus120573V||120573

(V2perp+ V2||) 120572 =

1

120573

(36)







inverted




1015840(119905) = 0





1(V1+ 119889)


1 We have 120581

1(119905) = (119861119902(119905))V

1 Increasing the


at 119905 is

1205812(119905) =

119861119902 (119905)

(V1+ 119889)

(37)



ship we obtain

1205812(119905) =

V1

(V1+ 119889)

1205811(119905) (38)





1with tan-

gents 0and





119899119899) or (119901

119899119899) and


(1199010 0) and (119901









1120572 = 120573 = 1 (0 120579 lowast (1 minus 120573))

120572 = 120573 = 1 (0infin)

120572 = 0 (0infin)


119862TA (120579)

=

Vperpint

120579

0

(1

119861119890119905119900 + 120579) 119890119894120579119889120579 120572 = 0

Vperpint

120579

120579119900

119905119900119890120579119861

119890119894120579119889120579 120572 = 120573 = 1

Vperpint

120579

120579119900

1

119861(119905119900

1minus120573+120579 (1 minus 120573)

119861)

120573(1minus120573)

times 119890119894120579119889120579 120572 =

1

120573= 1

(39)



119900to a


119900= 1 Instead



119862119870(120581)

=

Vperpint

120581

1205810


119861int

120581

1205810

120581minus2119890119894119861((ln 1120581)minus(ln 1120581

0))119889120579 120572 = 120573 = 1

minusV2perp

119861120573

timesint

120581

1205810

119890119894119861((119861V

perp120581)(1minus120573)120573


)(1minus120573)

times(1

(119861Vperp120581)minus(120573+1)120573

)119889120581 120572 =1

120573= 1

(40)



Vperp(120579) =

120579

1205811minus 1205810

120572 = 0

119861 (120579) =120579

(ln (11205811) minus ln (1120581

0)) 120572 = 120573 = 1

(41)

Vperp(120579) = (

120579 (1 minus 120573)

119861 ((1198611205810)(1minus120573)120573

minus (1198611205811)(1minus120573)120573

)

)


120572 =1

120573= 1

(42)



0 and 120581


Note

1198621198961(1205811) is 119862119896(1205811) when 120572 = 0

1198621198962(1205811) is 119862119896(1205811) when 120572 = 1

1198621198963(1205811) is 119862119896(1205811) when 120572 = 1120573 = 0 = 1


1198963(1205811)] Im[119862

1198963(1205811)]









x

y

06

05

04

03

02

01

02 04 06 08

P0

P2


120581 = 14120581 = 08 120581 = 0 120581 = minus11




Vperp= 2



0= 12 120581

1= 09 119875

2= 0849 0621








119886 119875119887 119875119888 1205810 1205811 tol

Output 120572 = 1120573


119878= |1205810| 120581119864= |1205811| and 120573

1= 1205732= 1

Step 1 if 120581119864gt 120581119878

1198750larr 119875119888 1198752larr 119875119886

else 120581119864le 120581119878

1198750larr 119875119886 1198752larr 119875119888


Step 3 if 120581119864gt 120581119878




1003817100381710038171003817

2

+100381710038171003817100381711987501198751

1003817100381710038171003817

2

minus100381710038171003817100381711987511198752

1003817100381710038171003817

2

2100381710038171003817100381711987501198751

1003817100381710038171003817100381710038171003817100381711987511198752

1003817100381710038171003817

)

Step 6 if 120581119864= 120581119878

Set 120573 = 0 Vperp= 1120581

119864 119861 = 1

if min 1003817100381710038171003817119862TA(120579) 1198752

1003817100381710038171003817 lt tol120573 larr 0

elseif 10038171003817100381710038171198621198961(120581119864)1198752


else if 10038171003817100381710038171198621198962(120581119864)11987521003817100381710038171003817 le tol


if 120581119864

= 0

1205731larr 10

1205732larr argmin

120573gt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


120573gt0

10038171003817100381710038171198621198963 (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817

1205732larr argmin

120573lt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


1 119890119903119903119900119903

2




Plot LMCEnd

Algorithm 1



120581119900

1205811

120579

Figure 10(a) 1966000000000 0655000000000 2000 0125 712058718

Figure 10(b) 0265969303943 0109176561142 3400 1700 21205879Figure 10(c) 0145992027985 0084314886280 9000 0000 1205874





minus01

minus02

minus03

minus04

05 10 15

y

x

(a)

minus005

minus010

minus015

01 02 03 04 05

x

y

(b)


(a) (b)


minus05


x

y

05 10 15 20 25

(a)

minus05


x

y

05 10 15 20 25

(b)

(i)

(iii) (ii)

(c)



(a) (b)



119905 119905119900

119861 V||

Vperp

120573




119905 119905119900

119861 V||

Vperp

120573









Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












LCG (119905) = log 120588 (119905) log(120588 (119905) 119904

1015840(119905)

1205881015840 (119905)) (18)

LTG (119905) = log120583 (119905) log(120583 (119905) 119904

1015840(119905)

1205831015840 (119905)) (19)

120582 (119905) = 1 +120588 (119905)

1205881015840(119905)2(1205881015840(119905) 11990410158401015840(119905)

1199041015840 (119905)minus 12058810158401015840(119905)) (20)

120595 (119905) = 1 +120583 (119905)

1205831015840(119905)2(1205831015840(119905) 11990410158401015840(119905)

1205831015840 (119905)minus 12058310158401015840(119905)) (21)


LCG (119905) =

logV2perp+ V2||

1198611003816100381610038161003816119905minus1205731003816100381610038161003816 Vperp


120573)

120582 (119905) = 120572 =1

120573

(22)


LTG (119905) =

logV2perp+ V2||

1198611003816100381610038161003816119905minus1205731003816100381610038161003816 V||


120573)

120595 (119905) = 120572 =1

120573

(23)


perphas







120588 (119905) =119889119904119889119905

119889120579119889119905=

Vperp

119889120579119889119905=119889119904

119889120579 (24)


120582 (119905) =1205791015840(119905) 120579(3)

(119905)

12057910158401015840(119905)2

minus 1 = 0 (25)


120579 (119905) =11989011990511988811198882

1198881

+ 1198883 120572 = 0 (26)

where 1198881 1198882 and c





Ω (119905) = 119861119890119905 120572 = 0 (27)

We let 1198882= minus119861 and 119888

3= minusΩ(119905

119900) such that 120579(119905) = Ω(119905

119900) minusΩ(119905)


perp Figure 7



119862LMC (119905) = 1198750+ Vperpint

119905

119905119900

119890119894120579(119905)

119889119905 (28)

where V||= 0 and 119875




119905

119905119900

cos 120579 (119905) 119889119905 Vperpint

119905

119905119900

sin 120579 (119905) 119889119905 int119905

119905119900

V||119889119905

(29)


0 5 10

0

5

10

minus10

minus10

minus5

minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

5

10

15

20

25

30

(a)

0 5 10 15

0

5

10

minus10

minus5

minus5t

(i) (ii) (iii)120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

1

2

3

4

(b)

0 5 10 15

0

5

minus10

minus5

minus5

t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

05

10

15

20

25

30

35

(c)

0 5 10

0

5

minus10

minus5

minus10 minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

2

4

6

8

(d)

Figure 6 Continued


0 5 10

0

5

minus10

minus5

minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

2

4

6

8

(e)

0 5 10

0

5

minus10 minus5

minus10

minus5

t

(i) (ii) (iii)120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 200

minus10

minus8

minus6

minus4

minus2

(f)


minus1

minus2

minus3

minus4


y

x

(a)

minus10

minus10

minus05 05

minus04

minus02

minus06

minus08

y

x

(b)


119900= minus05 119861 = 1 119905 isin [minus2 3]

with

120579 (119905) =

119861 (119890119905119900 minus 119890119905) 120572 = 0

119861 (log (119905119900) minus log (119905)) 120572 = 120573 = 1

119861 (1199051minus120573

119900minus 1199051minus120573

)

(1 minus 120573) 120572 = (

1

120573) = 1

(30)


119902 (119905) =

119905minus120573 120572 =

1

120573isin R

119890119905 120572 = 0

(31)



perp 0⟩ and


(a) (b)

(c) (d)



120581 (119905) =


V2perp+ V2||

120572 =1

120573isin R

minus119861119890119905Vperp

V2perp+ V2||

120572 = 0

(32)

120591 (119905) =

minus119861119905minus120573V||

V2perp+ V2||

120572 =1

120573isin R

minus119861119890119905V||

V2perp+ V2||

120572 = 0

(33)


perp


030

025

020

015

010

005

005 010 015 020 025 030 035

x

y

120572 = 0

120572 = minus1

120572 = 1

120572 = minus1

6120572 =

1

3


1=






120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(a)

120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(b)

1205881 = infin

120579P0

P1

P21205880 =

1

1205810

(c)



minus17







Figure 119905 119905119900

119861 V||

Vperp

120573








lowast] cup [0

lowastinfin)

1120572 = 120573 lt 0 120573 notin Z [0infin)





1120572 = 120573 gt 1 (0infinlowast)



120579 (119886119905 + 119887) =

119861(119890119905119900 minus 119890119886119905+119887

) 120572 = 0

119861 (log (119905119900) minus log (119886119905 + 119887)) 120572 = 1

119861 (1199051minus120573

119900minus (119886119905 + 119887)

1minus120573)

(1 minus 120573) 120572 = (

1

120573) = 1

(34)






||= 0




Values of 120572 or 120573 1205811015840(119905) 120591

1015840(119905)

119905 lt 0 119905 gt 0 119905 lt 0 119905 gt 0



1205811015840(119905) =

119861119890119905Vperp

(V2perp+ V2||) 120572 = 0


120573Vperp

(V2perp+ V2||) 120572 =

1

120573

(35)

1205911015840(119905) =

minus119861119890119905V||

(V2perp+ V2||) 120572 = 0

119861119905minus1minus120573V||120573

(V2perp+ V2||) 120572 =

1

120573

(36)







inverted




1015840(119905) = 0





1(V1+ 119889)


1 We have 120581

1(119905) = (119861119902(119905))V

1 Increasing the


at 119905 is

1205812(119905) =

119861119902 (119905)

(V1+ 119889)

(37)



ship we obtain

1205812(119905) =

V1

(V1+ 119889)

1205811(119905) (38)





1with tan-

gents 0and





119899119899) or (119901

119899119899) and


(1199010 0) and (119901









1120572 = 120573 = 1 (0 120579 lowast (1 minus 120573))

120572 = 120573 = 1 (0infin)

120572 = 0 (0infin)


119862TA (120579)

=

Vperpint

120579

0

(1

119861119890119905119900 + 120579) 119890119894120579119889120579 120572 = 0

Vperpint

120579

120579119900

119905119900119890120579119861

119890119894120579119889120579 120572 = 120573 = 1

Vperpint

120579

120579119900

1

119861(119905119900

1minus120573+120579 (1 minus 120573)

119861)

120573(1minus120573)

times 119890119894120579119889120579 120572 =

1

120573= 1

(39)



119900to a


119900= 1 Instead



119862119870(120581)

=

Vperpint

120581

1205810


119861int

120581

1205810

120581minus2119890119894119861((ln 1120581)minus(ln 1120581

0))119889120579 120572 = 120573 = 1

minusV2perp

119861120573

timesint

120581

1205810

119890119894119861((119861V

perp120581)(1minus120573)120573


)(1minus120573)

times(1

(119861Vperp120581)minus(120573+1)120573

)119889120581 120572 =1

120573= 1

(40)



Vperp(120579) =

120579

1205811minus 1205810

120572 = 0

119861 (120579) =120579

(ln (11205811) minus ln (1120581

0)) 120572 = 120573 = 1

(41)

Vperp(120579) = (

120579 (1 minus 120573)

119861 ((1198611205810)(1minus120573)120573

minus (1198611205811)(1minus120573)120573

)

)


120572 =1

120573= 1

(42)



0 and 120581


Note

1198621198961(1205811) is 119862119896(1205811) when 120572 = 0

1198621198962(1205811) is 119862119896(1205811) when 120572 = 1

1198621198963(1205811) is 119862119896(1205811) when 120572 = 1120573 = 0 = 1


1198963(1205811)] Im[119862

1198963(1205811)]









x

y

06

05

04

03

02

01

02 04 06 08

P0

P2


120581 = 14120581 = 08 120581 = 0 120581 = minus11




Vperp= 2



0= 12 120581

1= 09 119875

2= 0849 0621








119886 119875119887 119875119888 1205810 1205811 tol

Output 120572 = 1120573


119878= |1205810| 120581119864= |1205811| and 120573

1= 1205732= 1

Step 1 if 120581119864gt 120581119878

1198750larr 119875119888 1198752larr 119875119886

else 120581119864le 120581119878

1198750larr 119875119886 1198752larr 119875119888


Step 3 if 120581119864gt 120581119878




1003817100381710038171003817

2

+100381710038171003817100381711987501198751

1003817100381710038171003817

2

minus100381710038171003817100381711987511198752

1003817100381710038171003817

2

2100381710038171003817100381711987501198751

1003817100381710038171003817100381710038171003817100381711987511198752

1003817100381710038171003817

)

Step 6 if 120581119864= 120581119878

Set 120573 = 0 Vperp= 1120581

119864 119861 = 1

if min 1003817100381710038171003817119862TA(120579) 1198752

1003817100381710038171003817 lt tol120573 larr 0

elseif 10038171003817100381710038171198621198961(120581119864)1198752


else if 10038171003817100381710038171198621198962(120581119864)11987521003817100381710038171003817 le tol


if 120581119864

= 0

1205731larr 10

1205732larr argmin

120573gt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


120573gt0

10038171003817100381710038171198621198963 (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817

1205732larr argmin

120573lt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


1 119890119903119903119900119903

2




Plot LMCEnd

Algorithm 1



120581119900

1205811

120579

Figure 10(a) 1966000000000 0655000000000 2000 0125 712058718

Figure 10(b) 0265969303943 0109176561142 3400 1700 21205879Figure 10(c) 0145992027985 0084314886280 9000 0000 1205874





minus01

minus02

minus03

minus04

05 10 15

y

x

(a)

minus005

minus010

minus015

01 02 03 04 05

x

y

(b)


(a) (b)


minus05


x

y

05 10 15 20 25

(a)

minus05


x

y

05 10 15 20 25

(b)

(i)

(iii) (ii)

(c)



(a) (b)



119905 119905119900

119861 V||

Vperp

120573




119905 119905119900

119861 V||

Vperp

120573









Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10

0

5

10

minus10

minus10

minus5

minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

5

10

15

20

25

30

(a)

0 5 10 15

0

5

10

minus10

minus5

minus5t

(i) (ii) (iii)120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

1

2

3

4

(b)

0 5 10 15

0

5

minus10

minus5

minus5

t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

05

10

15

20

25

30

35

(c)

0 5 10

0

5

minus10

minus5

minus10 minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

2

4

6

8

(d)

Figure 6 Continued


0 5 10

0

5

minus10

minus5

minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

2

4

6

8

(e)

0 5 10

0

5

minus10 minus5

minus10

minus5

t

(i) (ii) (iii)120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 200

minus10

minus8

minus6

minus4

minus2

(f)


minus1

minus2

minus3

minus4


y

x

(a)

minus10

minus10

minus05 05

minus04

minus02

minus06

minus08

y

x

(b)


119900= minus05 119861 = 1 119905 isin [minus2 3]

with

120579 (119905) =

119861 (119890119905119900 minus 119890119905) 120572 = 0

119861 (log (119905119900) minus log (119905)) 120572 = 120573 = 1

119861 (1199051minus120573

119900minus 1199051minus120573

)

(1 minus 120573) 120572 = (

1

120573) = 1

(30)


119902 (119905) =

119905minus120573 120572 =

1

120573isin R

119890119905 120572 = 0

(31)



perp 0⟩ and


(a) (b)

(c) (d)



120581 (119905) =


V2perp+ V2||

120572 =1

120573isin R

minus119861119890119905Vperp

V2perp+ V2||

120572 = 0

(32)

120591 (119905) =

minus119861119905minus120573V||

V2perp+ V2||

120572 =1

120573isin R

minus119861119890119905V||

V2perp+ V2||

120572 = 0

(33)


perp


030

025

020

015

010

005

005 010 015 020 025 030 035

x

y

120572 = 0

120572 = minus1

120572 = 1

120572 = minus1

6120572 =

1

3


1=






120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(a)

120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(b)

1205881 = infin

120579P0

P1

P21205880 =

1

1205810

(c)



minus17







Figure 119905 119905119900

119861 V||

Vperp

120573








lowast] cup [0

lowastinfin)

1120572 = 120573 lt 0 120573 notin Z [0infin)





1120572 = 120573 gt 1 (0infinlowast)



120579 (119886119905 + 119887) =

119861(119890119905119900 minus 119890119886119905+119887

) 120572 = 0

119861 (log (119905119900) minus log (119886119905 + 119887)) 120572 = 1

119861 (1199051minus120573

119900minus (119886119905 + 119887)

1minus120573)

(1 minus 120573) 120572 = (

1

120573) = 1

(34)






||= 0




Values of 120572 or 120573 1205811015840(119905) 120591

1015840(119905)

119905 lt 0 119905 gt 0 119905 lt 0 119905 gt 0



1205811015840(119905) =

119861119890119905Vperp

(V2perp+ V2||) 120572 = 0


120573Vperp

(V2perp+ V2||) 120572 =

1

120573

(35)

1205911015840(119905) =

minus119861119890119905V||

(V2perp+ V2||) 120572 = 0

119861119905minus1minus120573V||120573

(V2perp+ V2||) 120572 =

1

120573

(36)







inverted




1015840(119905) = 0





1(V1+ 119889)


1 We have 120581

1(119905) = (119861119902(119905))V

1 Increasing the


at 119905 is

1205812(119905) =

119861119902 (119905)

(V1+ 119889)

(37)



ship we obtain

1205812(119905) =

V1

(V1+ 119889)

1205811(119905) (38)





1with tan-

gents 0and





119899119899) or (119901

119899119899) and


(1199010 0) and (119901









1120572 = 120573 = 1 (0 120579 lowast (1 minus 120573))

120572 = 120573 = 1 (0infin)

120572 = 0 (0infin)


119862TA (120579)

=

Vperpint

120579

0

(1

119861119890119905119900 + 120579) 119890119894120579119889120579 120572 = 0

Vperpint

120579

120579119900

119905119900119890120579119861

119890119894120579119889120579 120572 = 120573 = 1

Vperpint

120579

120579119900

1

119861(119905119900

1minus120573+120579 (1 minus 120573)

119861)

120573(1minus120573)

times 119890119894120579119889120579 120572 =

1

120573= 1

(39)



119900to a


119900= 1 Instead



119862119870(120581)

=

Vperpint

120581

1205810


119861int

120581

1205810

120581minus2119890119894119861((ln 1120581)minus(ln 1120581

0))119889120579 120572 = 120573 = 1

minusV2perp

119861120573

timesint

120581

1205810

119890119894119861((119861V

perp120581)(1minus120573)120573


)(1minus120573)

times(1

(119861Vperp120581)minus(120573+1)120573

)119889120581 120572 =1

120573= 1

(40)



Vperp(120579) =

120579

1205811minus 1205810

120572 = 0

119861 (120579) =120579

(ln (11205811) minus ln (1120581

0)) 120572 = 120573 = 1

(41)

Vperp(120579) = (

120579 (1 minus 120573)

119861 ((1198611205810)(1minus120573)120573

minus (1198611205811)(1minus120573)120573

)

)


120572 =1

120573= 1

(42)



0 and 120581


Note

1198621198961(1205811) is 119862119896(1205811) when 120572 = 0

1198621198962(1205811) is 119862119896(1205811) when 120572 = 1

1198621198963(1205811) is 119862119896(1205811) when 120572 = 1120573 = 0 = 1


1198963(1205811)] Im[119862

1198963(1205811)]









x

y

06

05

04

03

02

01

02 04 06 08

P0

P2


120581 = 14120581 = 08 120581 = 0 120581 = minus11




Vperp= 2



0= 12 120581

1= 09 119875

2= 0849 0621








119886 119875119887 119875119888 1205810 1205811 tol

Output 120572 = 1120573


119878= |1205810| 120581119864= |1205811| and 120573

1= 1205732= 1

Step 1 if 120581119864gt 120581119878

1198750larr 119875119888 1198752larr 119875119886

else 120581119864le 120581119878

1198750larr 119875119886 1198752larr 119875119888


Step 3 if 120581119864gt 120581119878




1003817100381710038171003817

2

+100381710038171003817100381711987501198751

1003817100381710038171003817

2

minus100381710038171003817100381711987511198752

1003817100381710038171003817

2

2100381710038171003817100381711987501198751

1003817100381710038171003817100381710038171003817100381711987511198752

1003817100381710038171003817

)

Step 6 if 120581119864= 120581119878

Set 120573 = 0 Vperp= 1120581

119864 119861 = 1

if min 1003817100381710038171003817119862TA(120579) 1198752

1003817100381710038171003817 lt tol120573 larr 0

elseif 10038171003817100381710038171198621198961(120581119864)1198752


else if 10038171003817100381710038171198621198962(120581119864)11987521003817100381710038171003817 le tol


if 120581119864

= 0

1205731larr 10

1205732larr argmin

120573gt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


120573gt0

10038171003817100381710038171198621198963 (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817

1205732larr argmin

120573lt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


1 119890119903119903119900119903

2




Plot LMCEnd

Algorithm 1



120581119900

1205811

120579

Figure 10(a) 1966000000000 0655000000000 2000 0125 712058718

Figure 10(b) 0265969303943 0109176561142 3400 1700 21205879Figure 10(c) 0145992027985 0084314886280 9000 0000 1205874





minus01

minus02

minus03

minus04

05 10 15

y

x

(a)

minus005

minus010

minus015

01 02 03 04 05

x

y

(b)


(a) (b)


minus05


x

y

05 10 15 20 25

(a)

minus05


x

y

05 10 15 20 25

(b)

(i)

(iii) (ii)

(c)



(a) (b)



119905 119905119900

119861 V||

Vperp

120573




119905 119905119900

119861 V||

Vperp

120573









Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10

0

5

minus10

minus5

minus5 t

(i) (ii) (iii)

120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 20

2

4

6

8

(e)

0 5 10

0

5

minus10 minus5

minus10

minus5

t

(i) (ii) (iii)120581(t)

log[120588(t)s998400 (t)120588

998400 (t)]

log[120588(t)]

05 10 15 200

minus10

minus8

minus6

minus4

minus2

(f)


minus1

minus2

minus3

minus4


y

x

(a)

minus10

minus10

minus05 05

minus04

minus02

minus06

minus08

y

x

(b)


119900= minus05 119861 = 1 119905 isin [minus2 3]

with

120579 (119905) =

119861 (119890119905119900 minus 119890119905) 120572 = 0

119861 (log (119905119900) minus log (119905)) 120572 = 120573 = 1

119861 (1199051minus120573

119900minus 1199051minus120573

)

(1 minus 120573) 120572 = (

1

120573) = 1

(30)


119902 (119905) =

119905minus120573 120572 =

1

120573isin R

119890119905 120572 = 0

(31)



perp 0⟩ and


(a) (b)

(c) (d)



120581 (119905) =


V2perp+ V2||

120572 =1

120573isin R

minus119861119890119905Vperp

V2perp+ V2||

120572 = 0

(32)

120591 (119905) =

minus119861119905minus120573V||

V2perp+ V2||

120572 =1

120573isin R

minus119861119890119905V||

V2perp+ V2||

120572 = 0

(33)


perp


030

025

020

015

010

005

005 010 015 020 025 030 035

x

y

120572 = 0

120572 = minus1

120572 = 1

120572 = minus1

6120572 =

1

3


1=






120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(a)

120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(b)

1205881 = infin

120579P0

P1

P21205880 =

1

1205810

(c)



minus17







Figure 119905 119905119900

119861 V||

Vperp

120573








lowast] cup [0

lowastinfin)

1120572 = 120573 lt 0 120573 notin Z [0infin)





1120572 = 120573 gt 1 (0infinlowast)



120579 (119886119905 + 119887) =

119861(119890119905119900 minus 119890119886119905+119887

) 120572 = 0

119861 (log (119905119900) minus log (119886119905 + 119887)) 120572 = 1

119861 (1199051minus120573

119900minus (119886119905 + 119887)

1minus120573)

(1 minus 120573) 120572 = (

1

120573) = 1

(34)






||= 0




Values of 120572 or 120573 1205811015840(119905) 120591

1015840(119905)

119905 lt 0 119905 gt 0 119905 lt 0 119905 gt 0



1205811015840(119905) =

119861119890119905Vperp

(V2perp+ V2||) 120572 = 0


120573Vperp

(V2perp+ V2||) 120572 =

1

120573

(35)

1205911015840(119905) =

minus119861119890119905V||

(V2perp+ V2||) 120572 = 0

119861119905minus1minus120573V||120573

(V2perp+ V2||) 120572 =

1

120573

(36)







inverted




1015840(119905) = 0





1(V1+ 119889)


1 We have 120581

1(119905) = (119861119902(119905))V

1 Increasing the


at 119905 is

1205812(119905) =

119861119902 (119905)

(V1+ 119889)

(37)



ship we obtain

1205812(119905) =

V1

(V1+ 119889)

1205811(119905) (38)





1with tan-

gents 0and





119899119899) or (119901

119899119899) and


(1199010 0) and (119901









1120572 = 120573 = 1 (0 120579 lowast (1 minus 120573))

120572 = 120573 = 1 (0infin)

120572 = 0 (0infin)


119862TA (120579)

=

Vperpint

120579

0

(1

119861119890119905119900 + 120579) 119890119894120579119889120579 120572 = 0

Vperpint

120579

120579119900

119905119900119890120579119861

119890119894120579119889120579 120572 = 120573 = 1

Vperpint

120579

120579119900

1

119861(119905119900

1minus120573+120579 (1 minus 120573)

119861)

120573(1minus120573)

times 119890119894120579119889120579 120572 =

1

120573= 1

(39)



119900to a


119900= 1 Instead



119862119870(120581)

=

Vperpint

120581

1205810


119861int

120581

1205810

120581minus2119890119894119861((ln 1120581)minus(ln 1120581

0))119889120579 120572 = 120573 = 1

minusV2perp

119861120573

timesint

120581

1205810

119890119894119861((119861V

perp120581)(1minus120573)120573


)(1minus120573)

times(1

(119861Vperp120581)minus(120573+1)120573

)119889120581 120572 =1

120573= 1

(40)



Vperp(120579) =

120579

1205811minus 1205810

120572 = 0

119861 (120579) =120579

(ln (11205811) minus ln (1120581

0)) 120572 = 120573 = 1

(41)

Vperp(120579) = (

120579 (1 minus 120573)

119861 ((1198611205810)(1minus120573)120573

minus (1198611205811)(1minus120573)120573

)

)


120572 =1

120573= 1

(42)



0 and 120581


Note

1198621198961(1205811) is 119862119896(1205811) when 120572 = 0

1198621198962(1205811) is 119862119896(1205811) when 120572 = 1

1198621198963(1205811) is 119862119896(1205811) when 120572 = 1120573 = 0 = 1


1198963(1205811)] Im[119862

1198963(1205811)]









x

y

06

05

04

03

02

01

02 04 06 08

P0

P2


120581 = 14120581 = 08 120581 = 0 120581 = minus11




Vperp= 2



0= 12 120581

1= 09 119875

2= 0849 0621








119886 119875119887 119875119888 1205810 1205811 tol

Output 120572 = 1120573


119878= |1205810| 120581119864= |1205811| and 120573

1= 1205732= 1

Step 1 if 120581119864gt 120581119878

1198750larr 119875119888 1198752larr 119875119886

else 120581119864le 120581119878

1198750larr 119875119886 1198752larr 119875119888


Step 3 if 120581119864gt 120581119878




1003817100381710038171003817

2

+100381710038171003817100381711987501198751

1003817100381710038171003817

2

minus100381710038171003817100381711987511198752

1003817100381710038171003817

2

2100381710038171003817100381711987501198751

1003817100381710038171003817100381710038171003817100381711987511198752

1003817100381710038171003817

)

Step 6 if 120581119864= 120581119878

Set 120573 = 0 Vperp= 1120581

119864 119861 = 1

if min 1003817100381710038171003817119862TA(120579) 1198752

1003817100381710038171003817 lt tol120573 larr 0

elseif 10038171003817100381710038171198621198961(120581119864)1198752


else if 10038171003817100381710038171198621198962(120581119864)11987521003817100381710038171003817 le tol


if 120581119864

= 0

1205731larr 10

1205732larr argmin

120573gt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


120573gt0

10038171003817100381710038171198621198963 (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817

1205732larr argmin

120573lt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


1 119890119903119903119900119903

2




Plot LMCEnd

Algorithm 1



120581119900

1205811

120579

Figure 10(a) 1966000000000 0655000000000 2000 0125 712058718

Figure 10(b) 0265969303943 0109176561142 3400 1700 21205879Figure 10(c) 0145992027985 0084314886280 9000 0000 1205874





minus01

minus02

minus03

minus04

05 10 15

y

x

(a)

minus005

minus010

minus015

01 02 03 04 05

x

y

(b)


(a) (b)


minus05


x

y

05 10 15 20 25

(a)

minus05


x

y

05 10 15 20 25

(b)

(i)

(iii) (ii)

(c)



(a) (b)



119905 119905119900

119861 V||

Vperp

120573




119905 119905119900

119861 V||

Vperp

120573









Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)



120581 (119905) =


V2perp+ V2||

120572 =1

120573isin R

minus119861119890119905Vperp

V2perp+ V2||

120572 = 0

(32)

120591 (119905) =

minus119861119905minus120573V||

V2perp+ V2||

120572 =1

120573isin R

minus119861119890119905V||

V2perp+ V2||

120572 = 0

(33)


perp


030

025

020

015

010

005

005 010 015 020 025 030 035

x

y

120572 = 0

120572 = minus1

120572 = 1

120572 = minus1

6120572 =

1

3


1=






120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(a)

120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(b)

1205881 = infin

120579P0

P1

P21205880 =

1

1205810

(c)



minus17







Figure 119905 119905119900

119861 V||

Vperp

120573








lowast] cup [0

lowastinfin)

1120572 = 120573 lt 0 120573 notin Z [0infin)





1120572 = 120573 gt 1 (0infinlowast)



120579 (119886119905 + 119887) =

119861(119890119905119900 minus 119890119886119905+119887

) 120572 = 0

119861 (log (119905119900) minus log (119886119905 + 119887)) 120572 = 1

119861 (1199051minus120573

119900minus (119886119905 + 119887)

1minus120573)

(1 minus 120573) 120572 = (

1

120573) = 1

(34)






||= 0




Values of 120572 or 120573 1205811015840(119905) 120591

1015840(119905)

119905 lt 0 119905 gt 0 119905 lt 0 119905 gt 0



1205811015840(119905) =

119861119890119905Vperp

(V2perp+ V2||) 120572 = 0


120573Vperp

(V2perp+ V2||) 120572 =

1

120573

(35)

1205911015840(119905) =

minus119861119890119905V||

(V2perp+ V2||) 120572 = 0

119861119905minus1minus120573V||120573

(V2perp+ V2||) 120572 =

1

120573

(36)







inverted




1015840(119905) = 0





1(V1+ 119889)


1 We have 120581

1(119905) = (119861119902(119905))V

1 Increasing the


at 119905 is

1205812(119905) =

119861119902 (119905)

(V1+ 119889)

(37)



ship we obtain

1205812(119905) =

V1

(V1+ 119889)

1205811(119905) (38)





1with tan-

gents 0and





119899119899) or (119901

119899119899) and


(1199010 0) and (119901









1120572 = 120573 = 1 (0 120579 lowast (1 minus 120573))

120572 = 120573 = 1 (0infin)

120572 = 0 (0infin)


119862TA (120579)

=

Vperpint

120579

0

(1

119861119890119905119900 + 120579) 119890119894120579119889120579 120572 = 0

Vperpint

120579

120579119900

119905119900119890120579119861

119890119894120579119889120579 120572 = 120573 = 1

Vperpint

120579

120579119900

1

119861(119905119900

1minus120573+120579 (1 minus 120573)

119861)

120573(1minus120573)

times 119890119894120579119889120579 120572 =

1

120573= 1

(39)



119900to a


119900= 1 Instead



119862119870(120581)

=

Vperpint

120581

1205810


119861int

120581

1205810

120581minus2119890119894119861((ln 1120581)minus(ln 1120581

0))119889120579 120572 = 120573 = 1

minusV2perp

119861120573

timesint

120581

1205810

119890119894119861((119861V

perp120581)(1minus120573)120573


)(1minus120573)

times(1

(119861Vperp120581)minus(120573+1)120573

)119889120581 120572 =1

120573= 1

(40)



Vperp(120579) =

120579

1205811minus 1205810

120572 = 0

119861 (120579) =120579

(ln (11205811) minus ln (1120581

0)) 120572 = 120573 = 1

(41)

Vperp(120579) = (

120579 (1 minus 120573)

119861 ((1198611205810)(1minus120573)120573

minus (1198611205811)(1minus120573)120573

)

)


120572 =1

120573= 1

(42)



0 and 120581


Note

1198621198961(1205811) is 119862119896(1205811) when 120572 = 0

1198621198962(1205811) is 119862119896(1205811) when 120572 = 1

1198621198963(1205811) is 119862119896(1205811) when 120572 = 1120573 = 0 = 1


1198963(1205811)] Im[119862

1198963(1205811)]









x

y

06

05

04

03

02

01

02 04 06 08

P0

P2


120581 = 14120581 = 08 120581 = 0 120581 = minus11




Vperp= 2



0= 12 120581

1= 09 119875

2= 0849 0621








119886 119875119887 119875119888 1205810 1205811 tol

Output 120572 = 1120573


119878= |1205810| 120581119864= |1205811| and 120573

1= 1205732= 1

Step 1 if 120581119864gt 120581119878

1198750larr 119875119888 1198752larr 119875119886

else 120581119864le 120581119878

1198750larr 119875119886 1198752larr 119875119888


Step 3 if 120581119864gt 120581119878




1003817100381710038171003817

2

+100381710038171003817100381711987501198751

1003817100381710038171003817

2

minus100381710038171003817100381711987511198752

1003817100381710038171003817

2

2100381710038171003817100381711987501198751

1003817100381710038171003817100381710038171003817100381711987511198752

1003817100381710038171003817

)

Step 6 if 120581119864= 120581119878

Set 120573 = 0 Vperp= 1120581

119864 119861 = 1

if min 1003817100381710038171003817119862TA(120579) 1198752

1003817100381710038171003817 lt tol120573 larr 0

elseif 10038171003817100381710038171198621198961(120581119864)1198752


else if 10038171003817100381710038171198621198962(120581119864)11987521003817100381710038171003817 le tol


if 120581119864

= 0

1205731larr 10

1205732larr argmin

120573gt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


120573gt0

10038171003817100381710038171198621198963 (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817

1205732larr argmin

120573lt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


1 119890119903119903119900119903

2




Plot LMCEnd

Algorithm 1



120581119900

1205811

120579

Figure 10(a) 1966000000000 0655000000000 2000 0125 712058718

Figure 10(b) 0265969303943 0109176561142 3400 1700 21205879Figure 10(c) 0145992027985 0084314886280 9000 0000 1205874





minus01

minus02

minus03

minus04

05 10 15

y

x

(a)

minus005

minus010

minus015

01 02 03 04 05

x

y

(b)


(a) (b)


minus05


x

y

05 10 15 20 25

(a)

minus05


x

y

05 10 15 20 25

(b)

(i)

(iii) (ii)

(c)



(a) (b)



119905 119905119900

119861 V||

Vperp

120573




119905 119905119900

119861 V||

Vperp

120573









Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(a)

120579P0P1

P2

1205881 =1

1205811

1205880 =1

1205810

(b)

1205881 = infin

120579P0

P1

P21205880 =

1

1205810

(c)



minus17







Figure 119905 119905119900

119861 V||

Vperp

120573








lowast] cup [0

lowastinfin)

1120572 = 120573 lt 0 120573 notin Z [0infin)





1120572 = 120573 gt 1 (0infinlowast)



120579 (119886119905 + 119887) =

119861(119890119905119900 minus 119890119886119905+119887

) 120572 = 0

119861 (log (119905119900) minus log (119886119905 + 119887)) 120572 = 1

119861 (1199051minus120573

119900minus (119886119905 + 119887)

1minus120573)

(1 minus 120573) 120572 = (

1

120573) = 1

(34)






||= 0




Values of 120572 or 120573 1205811015840(119905) 120591

1015840(119905)

119905 lt 0 119905 gt 0 119905 lt 0 119905 gt 0



1205811015840(119905) =

119861119890119905Vperp

(V2perp+ V2||) 120572 = 0


120573Vperp

(V2perp+ V2||) 120572 =

1

120573

(35)

1205911015840(119905) =

minus119861119890119905V||

(V2perp+ V2||) 120572 = 0

119861119905minus1minus120573V||120573

(V2perp+ V2||) 120572 =

1

120573

(36)







inverted




1015840(119905) = 0





1(V1+ 119889)


1 We have 120581

1(119905) = (119861119902(119905))V

1 Increasing the


at 119905 is

1205812(119905) =

119861119902 (119905)

(V1+ 119889)

(37)



ship we obtain

1205812(119905) =

V1

(V1+ 119889)

1205811(119905) (38)





1with tan-

gents 0and





119899119899) or (119901

119899119899) and


(1199010 0) and (119901









1120572 = 120573 = 1 (0 120579 lowast (1 minus 120573))

120572 = 120573 = 1 (0infin)

120572 = 0 (0infin)


119862TA (120579)

=

Vperpint

120579

0

(1

119861119890119905119900 + 120579) 119890119894120579119889120579 120572 = 0

Vperpint

120579

120579119900

119905119900119890120579119861

119890119894120579119889120579 120572 = 120573 = 1

Vperpint

120579

120579119900

1

119861(119905119900

1minus120573+120579 (1 minus 120573)

119861)

120573(1minus120573)

times 119890119894120579119889120579 120572 =

1

120573= 1

(39)



119900to a


119900= 1 Instead



119862119870(120581)

=

Vperpint

120581

1205810


119861int

120581

1205810

120581minus2119890119894119861((ln 1120581)minus(ln 1120581

0))119889120579 120572 = 120573 = 1

minusV2perp

119861120573

timesint

120581

1205810

119890119894119861((119861V

perp120581)(1minus120573)120573


)(1minus120573)

times(1

(119861Vperp120581)minus(120573+1)120573

)119889120581 120572 =1

120573= 1

(40)



Vperp(120579) =

120579

1205811minus 1205810

120572 = 0

119861 (120579) =120579

(ln (11205811) minus ln (1120581

0)) 120572 = 120573 = 1

(41)

Vperp(120579) = (

120579 (1 minus 120573)

119861 ((1198611205810)(1minus120573)120573

minus (1198611205811)(1minus120573)120573

)

)


120572 =1

120573= 1

(42)



0 and 120581


Note

1198621198961(1205811) is 119862119896(1205811) when 120572 = 0

1198621198962(1205811) is 119862119896(1205811) when 120572 = 1

1198621198963(1205811) is 119862119896(1205811) when 120572 = 1120573 = 0 = 1


1198963(1205811)] Im[119862

1198963(1205811)]









x

y

06

05

04

03

02

01

02 04 06 08

P0

P2


120581 = 14120581 = 08 120581 = 0 120581 = minus11




Vperp= 2



0= 12 120581

1= 09 119875

2= 0849 0621








119886 119875119887 119875119888 1205810 1205811 tol

Output 120572 = 1120573


119878= |1205810| 120581119864= |1205811| and 120573

1= 1205732= 1

Step 1 if 120581119864gt 120581119878

1198750larr 119875119888 1198752larr 119875119886

else 120581119864le 120581119878

1198750larr 119875119886 1198752larr 119875119888


Step 3 if 120581119864gt 120581119878




1003817100381710038171003817

2

+100381710038171003817100381711987501198751

1003817100381710038171003817

2

minus100381710038171003817100381711987511198752

1003817100381710038171003817

2

2100381710038171003817100381711987501198751

1003817100381710038171003817100381710038171003817100381711987511198752

1003817100381710038171003817

)

Step 6 if 120581119864= 120581119878

Set 120573 = 0 Vperp= 1120581

119864 119861 = 1

if min 1003817100381710038171003817119862TA(120579) 1198752

1003817100381710038171003817 lt tol120573 larr 0

elseif 10038171003817100381710038171198621198961(120581119864)1198752


else if 10038171003817100381710038171198621198962(120581119864)11987521003817100381710038171003817 le tol


if 120581119864

= 0

1205731larr 10

1205732larr argmin

120573gt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


120573gt0

10038171003817100381710038171198621198963 (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817

1205732larr argmin

120573lt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


1 119890119903119903119900119903

2




Plot LMCEnd

Algorithm 1



120581119900

1205811

120579

Figure 10(a) 1966000000000 0655000000000 2000 0125 712058718

Figure 10(b) 0265969303943 0109176561142 3400 1700 21205879Figure 10(c) 0145992027985 0084314886280 9000 0000 1205874





minus01

minus02

minus03

minus04

05 10 15

y

x

(a)

minus005

minus010

minus015

01 02 03 04 05

x

y

(b)


(a) (b)


minus05


x

y

05 10 15 20 25

(a)

minus05


x

y

05 10 15 20 25

(b)

(i)

(iii) (ii)

(c)



(a) (b)



119905 119905119900

119861 V||

Vperp

120573




119905 119905119900

119861 V||

Vperp

120573









Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












lowast] cup [0

lowastinfin)

1120572 = 120573 lt 0 120573 notin Z [0infin)





1120572 = 120573 gt 1 (0infinlowast)



120579 (119886119905 + 119887) =

119861(119890119905119900 minus 119890119886119905+119887

) 120572 = 0

119861 (log (119905119900) minus log (119886119905 + 119887)) 120572 = 1

119861 (1199051minus120573

119900minus (119886119905 + 119887)

1minus120573)

(1 minus 120573) 120572 = (

1

120573) = 1

(34)






||= 0




Values of 120572 or 120573 1205811015840(119905) 120591

1015840(119905)

119905 lt 0 119905 gt 0 119905 lt 0 119905 gt 0



1205811015840(119905) =

119861119890119905Vperp

(V2perp+ V2||) 120572 = 0


120573Vperp

(V2perp+ V2||) 120572 =

1

120573

(35)

1205911015840(119905) =

minus119861119890119905V||

(V2perp+ V2||) 120572 = 0

119861119905minus1minus120573V||120573

(V2perp+ V2||) 120572 =

1

120573

(36)







inverted




1015840(119905) = 0





1(V1+ 119889)


1 We have 120581

1(119905) = (119861119902(119905))V

1 Increasing the


at 119905 is

1205812(119905) =

119861119902 (119905)

(V1+ 119889)

(37)



ship we obtain

1205812(119905) =

V1

(V1+ 119889)

1205811(119905) (38)





1with tan-

gents 0and





119899119899) or (119901

119899119899) and


(1199010 0) and (119901









1120572 = 120573 = 1 (0 120579 lowast (1 minus 120573))

120572 = 120573 = 1 (0infin)

120572 = 0 (0infin)


119862TA (120579)

=

Vperpint

120579

0

(1

119861119890119905119900 + 120579) 119890119894120579119889120579 120572 = 0

Vperpint

120579

120579119900

119905119900119890120579119861

119890119894120579119889120579 120572 = 120573 = 1

Vperpint

120579

120579119900

1

119861(119905119900

1minus120573+120579 (1 minus 120573)

119861)

120573(1minus120573)

times 119890119894120579119889120579 120572 =

1

120573= 1

(39)



119900to a


119900= 1 Instead



119862119870(120581)

=

Vperpint

120581

1205810


119861int

120581

1205810

120581minus2119890119894119861((ln 1120581)minus(ln 1120581

0))119889120579 120572 = 120573 = 1

minusV2perp

119861120573

timesint

120581

1205810

119890119894119861((119861V

perp120581)(1minus120573)120573


)(1minus120573)

times(1

(119861Vperp120581)minus(120573+1)120573

)119889120581 120572 =1

120573= 1

(40)



Vperp(120579) =

120579

1205811minus 1205810

120572 = 0

119861 (120579) =120579

(ln (11205811) minus ln (1120581

0)) 120572 = 120573 = 1

(41)

Vperp(120579) = (

120579 (1 minus 120573)

119861 ((1198611205810)(1minus120573)120573

minus (1198611205811)(1minus120573)120573

)

)


120572 =1

120573= 1

(42)



0 and 120581


Note

1198621198961(1205811) is 119862119896(1205811) when 120572 = 0

1198621198962(1205811) is 119862119896(1205811) when 120572 = 1

1198621198963(1205811) is 119862119896(1205811) when 120572 = 1120573 = 0 = 1


1198963(1205811)] Im[119862

1198963(1205811)]









x

y

06

05

04

03

02

01

02 04 06 08

P0

P2


120581 = 14120581 = 08 120581 = 0 120581 = minus11




Vperp= 2



0= 12 120581

1= 09 119875

2= 0849 0621








119886 119875119887 119875119888 1205810 1205811 tol

Output 120572 = 1120573


119878= |1205810| 120581119864= |1205811| and 120573

1= 1205732= 1

Step 1 if 120581119864gt 120581119878

1198750larr 119875119888 1198752larr 119875119886

else 120581119864le 120581119878

1198750larr 119875119886 1198752larr 119875119888


Step 3 if 120581119864gt 120581119878




1003817100381710038171003817

2

+100381710038171003817100381711987501198751

1003817100381710038171003817

2

minus100381710038171003817100381711987511198752

1003817100381710038171003817

2

2100381710038171003817100381711987501198751

1003817100381710038171003817100381710038171003817100381711987511198752

1003817100381710038171003817

)

Step 6 if 120581119864= 120581119878

Set 120573 = 0 Vperp= 1120581

119864 119861 = 1

if min 1003817100381710038171003817119862TA(120579) 1198752

1003817100381710038171003817 lt tol120573 larr 0

elseif 10038171003817100381710038171198621198961(120581119864)1198752


else if 10038171003817100381710038171198621198962(120581119864)11987521003817100381710038171003817 le tol


if 120581119864

= 0

1205731larr 10

1205732larr argmin

120573gt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


120573gt0

10038171003817100381710038171198621198963 (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817

1205732larr argmin

120573lt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


1 119890119903119903119900119903

2




Plot LMCEnd

Algorithm 1



120581119900

1205811

120579

Figure 10(a) 1966000000000 0655000000000 2000 0125 712058718

Figure 10(b) 0265969303943 0109176561142 3400 1700 21205879Figure 10(c) 0145992027985 0084314886280 9000 0000 1205874





minus01

minus02

minus03

minus04

05 10 15

y

x

(a)

minus005

minus010

minus015

01 02 03 04 05

x

y

(b)


(a) (b)


minus05


x

y

05 10 15 20 25

(a)

minus05


x

y

05 10 15 20 25

(b)

(i)

(iii) (ii)

(c)



(a) (b)



119905 119905119900

119861 V||

Vperp

120573




119905 119905119900

119861 V||

Vperp

120573









Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1(V1+ 119889)


1 We have 120581

1(119905) = (119861119902(119905))V

1 Increasing the


at 119905 is

1205812(119905) =

119861119902 (119905)

(V1+ 119889)

(37)



ship we obtain

1205812(119905) =

V1

(V1+ 119889)

1205811(119905) (38)





1with tan-

gents 0and





119899119899) or (119901

119899119899) and


(1199010 0) and (119901









1120572 = 120573 = 1 (0 120579 lowast (1 minus 120573))

120572 = 120573 = 1 (0infin)

120572 = 0 (0infin)


119862TA (120579)

=

Vperpint

120579

0

(1

119861119890119905119900 + 120579) 119890119894120579119889120579 120572 = 0

Vperpint

120579

120579119900

119905119900119890120579119861

119890119894120579119889120579 120572 = 120573 = 1

Vperpint

120579

120579119900

1

119861(119905119900

1minus120573+120579 (1 minus 120573)

119861)

120573(1minus120573)

times 119890119894120579119889120579 120572 =

1

120573= 1

(39)



119900to a


119900= 1 Instead



119862119870(120581)

=

Vperpint

120581

1205810


119861int

120581

1205810

120581minus2119890119894119861((ln 1120581)minus(ln 1120581

0))119889120579 120572 = 120573 = 1

minusV2perp

119861120573

timesint

120581

1205810

119890119894119861((119861V

perp120581)(1minus120573)120573


)(1minus120573)

times(1

(119861Vperp120581)minus(120573+1)120573

)119889120581 120572 =1

120573= 1

(40)



Vperp(120579) =

120579

1205811minus 1205810

120572 = 0

119861 (120579) =120579

(ln (11205811) minus ln (1120581

0)) 120572 = 120573 = 1

(41)

Vperp(120579) = (

120579 (1 minus 120573)

119861 ((1198611205810)(1minus120573)120573

minus (1198611205811)(1minus120573)120573

)

)


120572 =1

120573= 1

(42)



0 and 120581


Note

1198621198961(1205811) is 119862119896(1205811) when 120572 = 0

1198621198962(1205811) is 119862119896(1205811) when 120572 = 1

1198621198963(1205811) is 119862119896(1205811) when 120572 = 1120573 = 0 = 1


1198963(1205811)] Im[119862

1198963(1205811)]









x

y

06

05

04

03

02

01

02 04 06 08

P0

P2


120581 = 14120581 = 08 120581 = 0 120581 = minus11




Vperp= 2



0= 12 120581

1= 09 119875

2= 0849 0621








119886 119875119887 119875119888 1205810 1205811 tol

Output 120572 = 1120573


119878= |1205810| 120581119864= |1205811| and 120573

1= 1205732= 1

Step 1 if 120581119864gt 120581119878

1198750larr 119875119888 1198752larr 119875119886

else 120581119864le 120581119878

1198750larr 119875119886 1198752larr 119875119888


Step 3 if 120581119864gt 120581119878




1003817100381710038171003817

2

+100381710038171003817100381711987501198751

1003817100381710038171003817

2

minus100381710038171003817100381711987511198752

1003817100381710038171003817

2

2100381710038171003817100381711987501198751

1003817100381710038171003817100381710038171003817100381711987511198752

1003817100381710038171003817

)

Step 6 if 120581119864= 120581119878

Set 120573 = 0 Vperp= 1120581

119864 119861 = 1

if min 1003817100381710038171003817119862TA(120579) 1198752

1003817100381710038171003817 lt tol120573 larr 0

elseif 10038171003817100381710038171198621198961(120581119864)1198752


else if 10038171003817100381710038171198621198962(120581119864)11987521003817100381710038171003817 le tol


if 120581119864

= 0

1205731larr 10

1205732larr argmin

120573gt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


120573gt0

10038171003817100381710038171198621198963 (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817

1205732larr argmin

120573lt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


1 119890119903119903119900119903

2




Plot LMCEnd

Algorithm 1



120581119900

1205811

120579

Figure 10(a) 1966000000000 0655000000000 2000 0125 712058718

Figure 10(b) 0265969303943 0109176561142 3400 1700 21205879Figure 10(c) 0145992027985 0084314886280 9000 0000 1205874





minus01

minus02

minus03

minus04

05 10 15

y

x

(a)

minus005

minus010

minus015

01 02 03 04 05

x

y

(b)


(a) (b)


minus05


x

y

05 10 15 20 25

(a)

minus05


x

y

05 10 15 20 25

(b)

(i)

(iii) (ii)

(c)



(a) (b)



119905 119905119900

119861 V||

Vperp

120573




119905 119905119900

119861 V||

Vperp

120573









Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Vperp(120579) =

120579

1205811minus 1205810

120572 = 0

119861 (120579) =120579

(ln (11205811) minus ln (1120581

0)) 120572 = 120573 = 1

(41)

Vperp(120579) = (

120579 (1 minus 120573)

119861 ((1198611205810)(1minus120573)120573

minus (1198611205811)(1minus120573)120573

)

)


120572 =1

120573= 1

(42)



0 and 120581


Note

1198621198961(1205811) is 119862119896(1205811) when 120572 = 0

1198621198962(1205811) is 119862119896(1205811) when 120572 = 1

1198621198963(1205811) is 119862119896(1205811) when 120572 = 1120573 = 0 = 1


1198963(1205811)] Im[119862

1198963(1205811)]









x

y

06

05

04

03

02

01

02 04 06 08

P0

P2


120581 = 14120581 = 08 120581 = 0 120581 = minus11




Vperp= 2



0= 12 120581

1= 09 119875

2= 0849 0621








119886 119875119887 119875119888 1205810 1205811 tol

Output 120572 = 1120573


119878= |1205810| 120581119864= |1205811| and 120573

1= 1205732= 1

Step 1 if 120581119864gt 120581119878

1198750larr 119875119888 1198752larr 119875119886

else 120581119864le 120581119878

1198750larr 119875119886 1198752larr 119875119888


Step 3 if 120581119864gt 120581119878




1003817100381710038171003817

2

+100381710038171003817100381711987501198751

1003817100381710038171003817

2

minus100381710038171003817100381711987511198752

1003817100381710038171003817

2

2100381710038171003817100381711987501198751

1003817100381710038171003817100381710038171003817100381711987511198752

1003817100381710038171003817

)

Step 6 if 120581119864= 120581119878

Set 120573 = 0 Vperp= 1120581

119864 119861 = 1

if min 1003817100381710038171003817119862TA(120579) 1198752

1003817100381710038171003817 lt tol120573 larr 0

elseif 10038171003817100381710038171198621198961(120581119864)1198752


else if 10038171003817100381710038171198621198962(120581119864)11987521003817100381710038171003817 le tol


if 120581119864

= 0

1205731larr 10

1205732larr argmin

120573gt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


120573gt0

10038171003817100381710038171198621198963 (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817

1205732larr argmin

120573lt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


1 119890119903119903119900119903

2




Plot LMCEnd

Algorithm 1



120581119900

1205811

120579

Figure 10(a) 1966000000000 0655000000000 2000 0125 712058718

Figure 10(b) 0265969303943 0109176561142 3400 1700 21205879Figure 10(c) 0145992027985 0084314886280 9000 0000 1205874





minus01

minus02

minus03

minus04

05 10 15

y

x

(a)

minus005

minus010

minus015

01 02 03 04 05

x

y

(b)


(a) (b)


minus05


x

y

05 10 15 20 25

(a)

minus05


x

y

05 10 15 20 25

(b)

(i)

(iii) (ii)

(c)



(a) (b)



119905 119905119900

119861 V||

Vperp

120573




119905 119905119900

119861 V||

Vperp

120573









Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119886 119875119887 119875119888 1205810 1205811 tol

Output 120572 = 1120573


119878= |1205810| 120581119864= |1205811| and 120573

1= 1205732= 1

Step 1 if 120581119864gt 120581119878

1198750larr 119875119888 1198752larr 119875119886

else 120581119864le 120581119878

1198750larr 119875119886 1198752larr 119875119888


Step 3 if 120581119864gt 120581119878




1003817100381710038171003817

2

+100381710038171003817100381711987501198751

1003817100381710038171003817

2

minus100381710038171003817100381711987511198752

1003817100381710038171003817

2

2100381710038171003817100381711987501198751

1003817100381710038171003817100381710038171003817100381711987511198752

1003817100381710038171003817

)

Step 6 if 120581119864= 120581119878

Set 120573 = 0 Vperp= 1120581

119864 119861 = 1

if min 1003817100381710038171003817119862TA(120579) 1198752

1003817100381710038171003817 lt tol120573 larr 0

elseif 10038171003817100381710038171198621198961(120581119864)1198752


else if 10038171003817100381710038171198621198962(120581119864)11987521003817100381710038171003817 le tol


if 120581119864

= 0

1205731larr 10

1205732larr argmin

120573gt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


120573gt0

10038171003817100381710038171198621198963 (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817

1205732larr argmin

120573lt0

10038171003817100381710038171198621198963minus (10038161003816100381610038161205811

1003816100381610038161003816) 11987521003817100381710038171003817


1 119890119903119903119900119903

2




Plot LMCEnd

Algorithm 1



120581119900

1205811

120579

Figure 10(a) 1966000000000 0655000000000 2000 0125 712058718

Figure 10(b) 0265969303943 0109176561142 3400 1700 21205879Figure 10(c) 0145992027985 0084314886280 9000 0000 1205874





minus01

minus02

minus03

minus04

05 10 15

y

x

(a)

minus005

minus010

minus015

01 02 03 04 05

x

y

(b)


(a) (b)


minus05


x

y

05 10 15 20 25

(a)

minus05


x

y

05 10 15 20 25

(b)

(i)

(iii) (ii)

(c)



(a) (b)



119905 119905119900

119861 V||

Vperp

120573




119905 119905119900

119861 V||

Vperp

120573









Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus01

minus02

minus03

minus04

05 10 15

y

x

(a)

minus005

minus010

minus015

01 02 03 04 05

x

y

(b)


(a) (b)


minus05


x

y

05 10 15 20 25

(a)

minus05


x

y

05 10 15 20 25

(b)

(i)

(iii) (ii)

(c)



(a) (b)



119905 119905119900

119861 V||

Vperp

120573




119905 119905119900

119861 V||

Vperp

120573









Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)



119905 119905119900

119861 V||

Vperp

120573




119905 119905119900

119861 V||

Vperp

120573









Acknowledgments


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article log-aesthetic magnetic curves and their...

Documents