approximation of circular arcs using quartic bezier … of circular arcs using quartic bezier curves...

Applied Mathematical Sciences, Vol. 11, 2017, no. 26, 1271 - 1286

HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.73102

Approximation of Circular Arcs Using Quartic

Bezier Curves with Barycentric Coordinates

Satisfying G2 Data

Azhar Ahmad

Faculty of Science and Mathematics

Universiti Pendidikan Sultan Idris

35900 Tanjung Malim

Perak, Malaysia

R.U. Gobithaasan

Dept.of Mathematics, FST

Universiti Malaysia Terengganu

21030 Kuala Terengganu

Terengganu, Malaysia

Copyright © 2017 Azhar Ahmad and R.U. Gobithaasan. This article is distributed under the

Creative Commons Attribution License, which permits unrestricted use, distribution, and

reproduction in any medium, provided the original work is properly cited.

Abstract

This paper proposes four methods to approximate circular arcs using quartic

Bezier curves. Barycentric coordinates of two/three combination of control points

are used to obtain an optimal approximation. Interior control points of quartic

Bezier curves are found by satisfying G2 data from given circular arcs. The

maximum errors between circular arcs and approximated curves are calculated

using Hausdorff distance approach. Four different approaches are discussed which

varies in terms of accuracy, computation cost and approximation order.

Keywords: Circular Arc, Barycentric coordinate, Hausdorff distance, Quartic

Bezier

1272 Azhar Ahmad and R.U. Gobithaasan

1 Introduction

The representation of circular arcs using polynomial parametric spline is an

important study and it is essential to the field of Computer Aided Geometric

Design (CAGD). Circular arcs play an important role in geometric modeling,

where it is used along with other geometric primitive, such as straight lines,

conics curves and spirals. Circular arcs have variety of applications in CAD/CAM.

For example, they are an important design tool for automobile, aircraft industry as

well as in various product design environments. Indeed, circular arcs are also used

in areas such as font design.

The practical advantage of using a polynomial parametric form has been

widely discussed in literature. Apart from free-form curve design which gives fast

and reliable computation, it also used in interpolation and approximation of

algebraic functions and transcendental functions such as Generalization Cornu

Spiral (GCS). The Bezier-Bernstein representation is known as one of most

common types of polynomial parametric curve. Bezier splines are an excellent

and preferred method to draw smooth curves. Since they can easily converted into

NURBS which is the standard form in most of CAD/CAM systems, therefore

Bezier curves has become a de facto standard in geometric design.

Although a circular arc can be exactly represented by a low degree of rational

Bezier such as quadratic and cubic form, some CAD/CAM systems require

non-rational curve form. Major factors considered for curve representation are;

the computation must be fast and reliable, consistent calculation, flexibility and

the algorithm can be easily implemented which directly effects user’s experience.

A suitable method is not perceived as an ’extra work’, however it would be of

great benefit if the user has good control, and the computation power and time can

be reduced significantly.

There are numerous efforts made by researchers to find the best

approximation of circular arcs using polynomial parametric curves. Using low

degree n, n ≤ 5. Circular arcs approximation by using cubic Bézier curves was

proposed by deBoor et al. [3] and Dokken et al. [4]. Goldapp [7] proposed an

approximation to an arc of a unit circle, with a spanning angle / 2

satisfying geometric continuity kG for 0,1,2k .

Ahn and Kim [2] proposed the 3G quartic Bezier approximation to arcs of

angle / 2. In 2007, Kim and Ahn [8] presented quartic Bezier

approximations of circular arcs by using subdivision scheme with equi-distance of

the circular arcs. Both results have optimal approximation of order 8. Azhar et al.

[1] proposed a 2G approximation by applying Barycentric coordinates. Similarly,

their approach has the optimal approximation of order 8. Fang [5] approximated

circular arcs using quintic Bezier curves with seven different approximation

methods with kG continuities, where 2,3,4k at the joints.

This paper proposes numerous improvement of work proposed by Azhar et al.

[1] to increase the order of approximation of circular arcs. First, we consider

quartic Bezier curves that interpolate two points on the circumference of a circle

Approximation of circular arcs using quartic Bezier curves 1273

with equal unit vector tangents and curvatures at end points. The barycentric

coordinates are applied to find the location of interior control points that

approximates a circular arc. The advantage of using barycentric coordinate is that

they are not affected by affine transformations.

The rest of the paper is organized as follows. Section 2 elaborates

preliminary background and notation of Bezier curves and error analysis. Section

3 discusses on the method used to construct 2G quartic Bezier curves to

approximate circular arcs in the range of the central angle [0, ). In Section 4,

we introduce three distinct approaches to determine the location of interior control

points. An alternative approach is also presented in Section 5, along with the

discussion on error analysis based on Hausdorff distance and approximation order.

In the last section, we compare the proposed methods with other best known

methods to show the significance of the study.

2 Background

2.1 Bezier curve

A general planar Bezier curve Z: [0,1] R2 is represented by

0

( ) 1

nn i i

i

i

nZ t P t t

i , (1)

where iP , 0,1,2,...,i n are the control points and expression 1

n i in

t ti

is the Bernstein polynomial [6]. Local parameter is denoted by 0,1t . 0P and

nP are end points of the curve, also jP , 1,2,..., 1 j n are the interior points.

An important shape interrogation tool which is frequency used in the studies of

parametric curve is curvature. The signed curvature of Z t is defined as

3

'( ) "( )( )

'( )

Z t Z tt

Z t. (2)

where '( )Z t and ''( )Z t are first and second derivatives of (1). The radius of

osculating circle at t is the reciprocal of (2). It is known that ( ) t has positive

sign when the curve segment bends to the left and it is negative if it bends to the

right as t increases.

2.2 Error Analysis

To see how well the approximation, we may compare the curvature or the

sector of the approximated curve against the actual circular arc. However, the

obvious method is by finding the maximum difference of distance between set of


points on given curves, i.e., a circular arc and a quartic curve in this case. We

employed Hausdroff distance, since it is known as one of the most important and

reliable scheme to measure how far two subsets of a Euclidean space are located

from each other. The Hausdorff distance can be described as the maximum

distance between two curves but we must admit that it is not easy to find the

Hausdorff distance between two circular arcs and Bezier curves generated with

different parameters. Let us denoted b as the unit circular arc and Z as the

approximation curve, it is natural to writes radical error as follows if we assign the

quartic Bezier curve interpolating circular arc at end points

2 2 1 t x t y t (3)

It is clear that this function is the distance of a set of planar points ,x t y t

on Bezier curve to the origin. Therefore, the general mathematical definition for

Hausdorff distance between these two set of points can be derived as follows

0,1

, max ,

Ht

d b Z t (4)

Another alternative error function that is useful for the analysis is

2 21 ( ) t x t y t . (5)

Thus, t can be written in terms of the alternative error as follows

1 1 t t (6)

Since equations (4) and (5) have their zero sets and extreme values at the same

locations, we prefer to use (5) for analysis, furthermore there is no square root

term involved in (5). It can be shown that when t is small, then 2 t t

as shown by Fang [5]. However, these two error functions are extremely useful in

this study to find barycentric coordinates and to determine the approximated

errors. We also employed Taylor expansion to investigate the convergent order of

,Hd b Z corresponding to angle changes.

3 Quartic Bezier curve approximation

In 2014, Azhar et al. proposed quartic Bezier curve to approximate a circular

arc of the central angle 0, shown in (7) and satisfy 2G continuity at the

end points denoted as 0

P , 4

P and end unit tangents as 0

T , 1

T .


4

4

0

4( ) 1

i

i i

i

i

Z t P t t , 0 1 t . (7)

where three interior control points of quartic Bezier curve is defined by

1 0 4

0 4

0 11 sec2 2 4 2

P P P

P PT T ,

2 0 4

0 4 2 3

0 1

1 13 8 3cos sec

2 2 24 2

P P P

P PT T (8)

3 0 4

0 4

0 11 sec2 2 4 2

P P P

P PT T .

The turning angle were obtained from given end tangents, 1

0 1cos T T

and the radius of the circle is 0 4 / 2sin / 2 r P P . It is clear that the

interior control points are obtained from the given data. Here, the only unknown is

parameter , which will be the major factor in determining how good the

approximation. The rest of this section describes on identifying the locations of

interior control points as stated in (8). Firstly, let us denote 0B as the center of a

circle, SP as the intercept points of straight lines parallel to end unit vector that is

tangential to two endpoints. It also restricted that the circular arc segment has

similar end points as quartic Bezier; 0P and

4 ,P and symmetrical to 0SP B . We

denote 0N and

1N as unit normal vectors of 0T and

1T at 0P and

4P ,

respectively (Figure 1).

P0

P1

P2

P3

P4

PS

B0

N0

T0

N1T1

Figure 1 The setup of quartic Bezier’s of control points

Length from endpoints 0P and

4P to SP are 0 4 S SP P P P is a

constant. Here, we only consider the construction of curve segment that is

negative curvature, whereas the approximation of a circular arc with positive cur-


vature can be constructed in a similar manner. Since we are considering circular

arc of negative curvature thus 0 0 0 T N and

1 1 0 T N . So we re-write the

control points as follows,

0 0 0

4 0 1

0 4 0 1

- ,

- ,

( ) ( - ).

2S

P B rN

P B rN

P P T TP

(9)

By using the barycentric coordinate, the interior control points 1 2 3, ,P P P may

now be written in the terms of 0 4, ,

SP P P as

1 01

SP P P ,

3 41

SP P P , (10)

2 1 31 2

SP P P P .

Point 1P is constrained along 0 SP P with coordinate ,1 and point 3P

along line 4SP P with coordinate 1 , . While ( , ,1 2 ) is coordinate of

2P with respect to 1 2,P P and SP . Without any loss of generality in terms of

translation, rotation, reflection and uniform scaling, let the center of the circle

0 (0,0)B at the origin while SP on positive y axis and points 0P and 4P be

on the same ordinate, thus 0SP B perpendicular to

0 4P P . Secondly, since our

objective is to obtain sufficient condition of a single segment of quartic Bezier

satisfying end curvatures, hence let 0 1 1/ r . 1G tangent continuity

is satisfied by letting 0 1 4 0 0 T T P P where 0P and 4P is located on the

circular arc. Let 1r since this do not give any effect on and in general

analysis. The of the equations below shows in detail how the end curvatures are

satisfied. We first substitute (10), (9) into (7) to obtain Z t in the terms of unit

vectors as

1 0 2 1 3 0 4 1 Z t a T a T a N a N (11)

where

2

1

3 6 1 41 ,

3 6 1 4 2

ta t t

t

2 1 a a (12)


2

3 1 ( 2 4 ) (1 2 ) 1) a r t t t ,

4 3. a a

The first derivative of Z t is

1 0 2 1 3 0 4 1' ' ' ' ' Z t a T a T a N a N . (13)

And the second derivative of Z t is

1 0 2 1 3 0 4 1'' '' '' '' '' . Z t a T a T a N a N (14)

Here, 'ja and ''ja are first and second derivatives of ,ja 1,2,3,4j written

in the terms of t as

2

1 ' 2 1 2 3 6 1 4 3 6 1 4 ,a t t t

2 1' 'a a (15)

2

3 ' 2 ( 3 6 ) (3 6 ) ) a r t t ,

4 3' '. a a

And

2

1 " 6 2 3 6 1 4 2 3 6 1 4 1 2 1 2 ,a t t

2 1" " a a (16)

3 " 6 ( 1 2 )( 1 2 ) a r t ,

4 3" ". a a

By using standard product vectors of iT and

iN , 0,1i , we obtain (17).

0 0 0 0 1 1 1 1 0 0 1 1

0 0 1 1 0 0 0 0 1 1 1 1

0 0 1 0 0 1 0 1

0 1 0 0 0 1 1 0

1

0

sin ,

cos , tan / 2 .

N N T T N N T T N T N T

N T N T N N T T N N T T

N T N T T T N N

T T N N N T N T r

(17)

Substituting (16), (15) into (13) and (14), with (17), the numerator and 2

Z

curvature, can be written as


2

3 1 2 3' '' 4 ' " ' " sin , Z Z a a a a (18)

2 2 2

1 3' ' 4 ( ') ( ') sin Z Z a a . (19)

Here, we introduce a new notation / 2 for algebraic simplicity.

Consequently, we fixed 0 1 1/ r to impose 2G continuity at the

endpoints, hence the following equation is obtained

2

2

3 1 cos1

2

. (20)

can now be written in the terms of and / 2 as

2 22 sec.

3 1

(21)

Finally, by substituting from (21), SP from (9) and 0 4 S SP P P P

onto (10), the control points of explicit 2G quartic approximation can be stated as

(8). Hence, we only need to determine parameter in order to find the optimal

approximation.

For simplicity, the error analysis approximating two sets of points is

independent on their location so we will consider a unit circle with center on

0 0,0B and with radius 1r . First let the endpoints of circular arc located at

0 sin ,cos P and 4 sin ,cos P . By considering the minor circular

arc with negative curvature, the unit tangent vectors at 0 4, P P are

0 cos ,sin T and 1 ,cos , sin T respectively. Thus, the normal

vector at the end points are 0 sin , cos N and 1 .sin , cos N By

substituting tan , r 2 and (8) into (7), the components x and y of

quartic curve ( )Z t can be simplified as

4 3 4

2 2

3

( ) (-1 ) - 4(-1 ) cos 2

6(-1 ) -1 (-1 )cos 2

4(1- ) cos 2 - cos 2 ,

x t t t t t

t t

t t

(22)

and


3 3

2

2

sin 2 - 4(-1 ) tan

( ) 6(-1 ) -1 (-1 ) cos 2 tan

4(-1 ) sin 2 - cos 2 tan

t t

y t t t t

t t

. (23)

So the error function t and it’s first derivative ' t are

32 2 31 1

1 18 2

t x y B t t t k

, (24)

2 2

2 2

' 1 '

1 1 1 1 2

2 2

t x y

B t t t t h

(25)

where

4 2sec tanB ,

1 0 1

1

4

2

w w wk

w

, (26)

1 0 1

1

3

2

w w wh

w

,

2 3

2 2 3

0

3 15 16 16

16cos 4 1 4 4 4 cos 2

1 cos 4

w

(27)

2

2 2

1 8sin 3 4 8 3 4 cos 2w (28)

Since 0t , we have zeros with multiplicity 3 at 0t , 1t , and each

at 1/ 2t k with total zeros of 8. When ' 0t , the zeros is obtained at

0t and 1t with multiplicity 2, 1/ 2t and 1/ 2t h with total zeros

are 7. These two h and k are the quantity that we are interested to obtain this

value of and . Next section discusses four approaches to obtain parameter

and it’s approximation accuracy.


4 Finding ρ

4.1 Similar curvature values at t = {0, ½, 1}

The first approach is as shown in Azhar et al. [1], where the curvatures at

0,½ ,1t for the approximation curve is considered as equal. Thus, by letting

½ 1/ r and simplifying (2) upon substituting (18) and (19), we obtain a

quadratic equation in the terms of as follows,

2 34 8sec 12 9 6sec 0 . (29)

It is clear from (29) that there exist two real values. Two solutions of will be

obtained, where one of the solution is positive in 0, / 2 denoted as 1 ,

2

1

3cos3 cos 9 8 6 sin 2 cos2

.2 8 3cos cos3

(30)

We may use 1 to obtain h and k which will be in the form of complex

number. This means for 0t there are no zeros except at 0t and 1t .

Meanwhile for ' 0,t the extreme values occur only at

0, ½ , 1 ,t indicating the maximum distance between those two curves occur

at ½ t . Figure 2 shows a graph of an example of function 1 t using

1 0.402656 for / 2 and 0,1t . Two more functions 2 t and

3 t that obtained from others approaches for comparison purpose. Now, it

follows from (5), if the uniform norm of t on 0,1t which is denoted by

0,1

maxt

t t , then at 1/ 2t we have

0 1

1

4

2048

B w wt

w

(31)

Hence, Hausdroff distance ,Hd b Z with respect to can be written as

0 1

1

4, 1 1

2048H

B w wd b Z

w

. (32)

By using Mathematica, we can get the approximation order of (32) by means of


Taylor expansion at 0 as

8 1017 1, .

98304 4096 2Hd b Z

(33)

It evident that the proposed method has the approximation of order eight which is

in agreement with other known methods. From the calculation we get

5, 1.23356 10Hd b Z for / 2 when 1 0.402656 .

4.2 Interpolating circular arc at t = {0, ½, 1}

In order to find using this approach, we force Z to interpolate circular

arc at 0,½ ,1t and denote it by 2 . Thus we impose ½ 0,1Z from

(22) and (23). After simplification, a quadratic equation is obtained as follows

2 2 2 21cos 3 5cos cos sec 0.

8 2

(34)

Equation (34) has two real solution of indicated by the positive value of it

discriminant in the interval of 0 / 2 . A suitable solution of where we

denoted as 2 is

2

1 3cos sec cos 2 3 cos 1 cos cos

4 2 2 2

. (35)

Equation (35) ensures the approximated quartic Bezier curve lies on the circular

arc at 0, ½ and 1, which can be verified from 0t when 0k . The

extreme value of t occurs when 1/ 2h at 0,1/ 4,3 / 4,1t for arbitrary

angle in the interval of 0 / 2 . We may calculate the maximum error by

using (6) at 1/ 4t or 3/ 4t . For example, when / 2 we obtain

2 0.402599 with the maximum distance is 6, 3.55692 10 .Hd b Z In

general, the uniform norm of t on [0,1] at 1/ 4t or 3/ 4t is

0 1

1

27 16 3

524288

B w wt

w

(36)

So, the Hausdroff distance ,Hd b Z can be written as

0 1

1

27 16 3, 1 1.

524288H

B w wd b Z

w

(37)


The approximation order of Z for the circular arc b from Taylor’s expansion

at 0 is

8 10459 81,

8388608 1048576 2Hd b Z

(38)

Similar to the previous approach, the approximation order of this method is also 8,

but the absolute value of coefficient is less than the previous results, thus this

method has better approximation accuracy. The function of 2( )t with respect to

2 at / 2 it showed in Figure 2.

3

2

1

0.2 0.4 0.6 0.8 1.0t

0.000025

0.00002

0.000015

0.00001

5. 10 6

5. 10 6

Figure 2 Three function of ( )t on 0,1t with respect to 1,

2 and 3 , at

/ 2.

4.3 Equal maximum errors on both side of the circular arc

Figure 2, indicates that it would be possible to improve the approximation if

we allow the quartic Bezier curve to have equal maximum errors on both inside

and outside of the circular arc. Hence, the third approach to find is by letting the

maximum error on the extreme value of ' 0t at 1/ 2t and 1/ 2 .t h

Errors at these three parameters are equal and can be summarized by

1/ 2 1/ 2 h . For constrained , the errors at 1/ 2t and

1/ 2t h are as follows,

0 1

1

41/ 2

2048

B w w

w

,

4

0

4

1

271/ 2

2048

Bwh

w . (39)

Hence, the following equation is obtained after simplification

4 3 4

0 0 1 127 4 0w w w w (40)


A long polynomial of degree 16 in the terms of is obtained when we

substitute (27) and (28) into (20). Therefore we cannot evaluate analytically

for any . The only way to find a suitable solution is by using numerical

methods. Let us denote a desired solution which satisfy (40) as 3 , where

3 <2 <

1 . We would be able to find 3 using Newton Raphson method with

an initial value 2 . Using the obtained

3 , we get 0.138438k and

0.277261.h Hence, 0t occur at 0, 0.361562, 0.638438, 1 .t

Which concludes that the extreme value of t exists at t = {0, 0.222739, 0.5,

0.777261, 1}. Finally, uniform norm of t at 1/ 2t and 1/ 2t h is as

follows

0 1

1

4

2048

B w wt

w

. (41)

So far 3 is the best result as compared to

1 and 2 where the Hausdroff

distance ,Hd b Z can be simplified as (32). However there is no analytical

solution for 3 in the terms of , thus Taylor expansion cannot be used to

obtain Hausdroff distance ,Hd b Z . For the case of / 2 , we obtained

3 0.402587 which gives 6, 2.59233 10 .Hd b Z Notice that the above

procedure can be applied if we know 2 for any given angle 0 .

However it is tedious and it should only be used to find most accurate

approximation when previous methods fail. The graph of function 3( )t of

correspond to 3 , at / 2 is showed in Figure 2.

5 An alternative approach

Text of Section 2 Although the third approach is guaranteed to have better

approximation as compared to first two methods, the proposed numerical method

for finding 3 can be troublesome. Thus, we suggest the use of

1 23*

1

m

m

, 0.173689m (42)

where parameter 3* is close to

3 obtained from the graph of ,Hd b Z for

0,1 .t Equation (42) is derived by forming linear interpolation

2 3 1(1 )m m when / 2 . Now, if we use 3* , then the uniform

norm in [0,1] is


40 1 0

40,11 1

4 27max , .

2048 2048t

B w w Bwt

w w

(43)

The maximum distance occurs at 1/ 2t or 1/ 2t h , so the Hausdroff

distance ,Hd b Z can be found using (6). Figure 3 illustrates the comparison of

,Hd b Z for 1 ,

2 and 3* . It is evident that the result obtained by using

3*

is much superior then 1 and

2 .

1

2

3

0.5 1.0 1.5 2.0 2.5 3.0

0.0004

0.0003

0.0002

0.0001

dH

Figure 3 ,H

d b Z for 1 ,

2 and 3*

The approximation order of ,Hd b Z for 3* by Taylor expansion at 0 is

8 8 10, 5.172 10Hd b Z (44)

Similarly, we have optimal approximation of 8 eight by using this alternative

approach. Table 1 show the maximum distance of ,Hd b Z at a few simple

angle by using 1 ,

2 ,3 and

3*. The results obtained from 3* is closer to

3 , and much superior then using other two parameters. In summary

3 3* 2 1, ; , ; , ; , ;H H H Hd b Z d b Z d b Z d b Z .


Table 4 ,H

d b Z by using 1 2 3, , and

3*.

Turning

Angles θ

Maximum distance

1

, ;Hd b Z

2, ;

Hd b Z 3, ;

Hd b Z

3*, ;

Hd b Z

/6 9

1.71116 10

105.3599 10

10

3.89703 10

10

3.99775 10

/ 4 8

4.44021 10

8

1.37611 10

8

1.00093 10

8

1.02289 10

/ 3 7

4.53565 10

7

1.37796 10

7

1.00281 10

7

1.01928 10

/ 2 5

1.23356 10

6

3.55692 10

6

2.59233 10

6

2.59233 10

2 / 3 4

1.34299 10

5

3.58936 10

5

2.62097 10

5

2.8219 10

3 / 4 4

3.63702 10

5

9.2672 10 5

6.77443 10

5

7.64173 10

5 /6 4

8.98838 10

4

2.16787 10

4

1.58663 10

4

1.88854 10

Two researchers has obtained the Hausdroff distance ,Hd b Z for the

approximation of a unit quarter circle using cubic Bezier curves, namely Goldapp

[7] obtained 41.96 10 , and Ahn and Kim [2] obtained 53.50 10 . While using

quartic Bezier approximation, Ahn and Kim [2] obtained 63.55 10 , and Kim

and Ahn [8] obtained 62.03 10 . Whereas Azhar et al. [1] manage to obtain 51.23356 10 . It is evident from Table 1, that the proposed method can also be

used for circular arc approximation.

6 Conclusion

This works employs quartic Bezier curves to approximate circular arcs which

satisfies the 2G data at the end points. We showed four techniques to minimize

the maximum distance between approximated quartic Bezier curves and circular

arcs. These methods are based on a symmetrical quartic Bezier curve to avoid the

existence of cusp, loop and inflection point. All four proposed approaches have

greater accuracy and optimal approximation of order 8, which is similar to other

known methods. As a future work, we will extend the circle approximation by

quartic Bezier curve to the conic approximation in planar and spatial coordinate.

Acknowledgements. This work is financial supported by the Faculty of Science

and Mathematics, Universiti Pendidikan Sultan Idris.


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Received: April 6, 2017; Published: May 12, 2017

https://doi.org/10.4236/ojapps.2012.24b032

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