ELSEVIER

PII: SOOlO-4485(98)00016-S

Computer-Aided Des,gn, Vol. 30, No. 8, pp. 615-620, 1998 Bs 1998 Elsevler Science Ltd. All rights reserved

Printed in Great Britain

ooio-4485/98/519.00+0.00

G’ arc spline approximation of quadratic B6zier curves Young Joon Ahn, Hong Oh Kim* and Kyoung-Yong Lee

In describing the tool path for a certain CNC machine, it is often

required to approximate Bezier curves by arc splines with the number of arc segments as small as possible. We give a new method for approximating quadratic BCzier curves by G’ arc splines with smaller number of arc segments than the biarc method.

Numerical results are given to illustrate the efficiency of the algorithm for the previously used examples of quadratic Bezier curves. 0 1998 Elsevier Science Ltd. All rights reserved.

Keywords: CNC machine, quadratic BCzier curves, arc splines

INTRODUCTION

Arc splines, piecewise curves made of circular arcs and straight line segments, are frequently used as the tool path for CNC machinery’. Many papers have been published on the G’ arc spline approximations which are based on the biarc methodle6. Walton and Meek6 described some reasons why the G’ arc splines consisting of circular arcs are better than the polygons consisting of line segments as the tool path of CNC machinery. For example, the tangent vector of an arc spline is continuous, which avoids sudden changes in the direction of tool path. In particular, Walton and Meek6 approximated the quadratic BCzier curves by G’ arc splines based on the biarc method. In programming the tool path of CNC machinery, the number of arc segments in approximation should be decreased in order to reduce the number of instructions and tool motions. In this paper, we modify the biarc method to reduce the number of arcs in approximating the quadratic Bezier curves by G’ arc splines within the given tolerance. Instead of matching a biarc (two arcs in G’ manner) to each subdivision of the Bezier curve. we can match one arc to each subdivision except for the last one and combine all the circular arcs in G ‘-manner. We use the geometric properties of curvature-monotone quadratic BCzier curves to combine circular arcs as our G’-arc spline approximations of the quadratic Bdzier curves. We can thus substantially reduce the number of arc segments in OUI

approximation as is shown in Table 1. In the following sections we describe the basic facts of the

geometric properties for the curvature-monotone quadratic Bezier curves. We then give the G’ arc spline approxima-

*To whom correspondence should be addressed. Fax: 0082 42 869 27 IO: E-mail: hkim@ftn.kaist.ac.kr Department of Mathematics, KAIST, Taejon, Korea Paper Received: 31 July 1997. Rwised: 26 Jawat-y 1998. Accepted: 26

Junuun’1998 * Partly supported by TGRC-KOSEF

tion of the quadratic Bezier curves and its error analysis. We finally give our algorithm for a G’ arc spline approximation of the given quadratic Bezier curve within a given tolerance. Finally, we present numerical results to compare our method with those of the Walton and Meekh.

PRELIMINARIES

A quadratic Bezier curve is defined as

q(t) = (I - t)‘b,, + 2t( 1 - t)b, + f2b2, 0 5 t 5 1 (1)

where b,, i = 0,1,2, are the control points of q(t). The geo- metric condition that a quadratic Bezier curve has monotone curvature was described as follows’.

Theorem 1. The qnadratic Bezier curve q(t) has monotone curvature if and only if bl lies on or inside one of the two circles having as diameters born and mb2 where m is the midpoint of b. and b2 as shown in Figure 1.

Proof. See Ref. ‘. q

If a quadratic BCzier curve q(t) has a local maximum curvature at fA in the open interval (O,l), it will be divided at iA to give two Bezier curves b,(t) and bO(t) with mono- tone curvature. Thus we may and do assume that any quad- rat’c Bezier curve has increasing curvature, by reflecting the curve about the bisector of the points bo and b2 if necessary. We also may assume that the counterclockwise angle from the vector b2 - b0 to the vector b, - b. is less than by reflecting the curve about the line b& if necessary. There- fore, in this paper, b I is assumed to lie in the closed region bounded by the thick-lines as in Figure I. Under this assumption, we will develop our approximation algorithm.

APPROXIMATION METHOD

In this section, we describe a method for approximating a given quadratic Bezier curve by a G’ arc spline. For an arbitrary positive integer N, we subdivide b(t) at t, = il(N + I), i = 1,N. Then we have N + 1 quadratic Bezier curve segments b;(t), i = 0,l , ,N, with corresponding control points bz, bt+ I and bg+2. The control points are given by

bt’i = b( &I>

hi.+, =(I - I+’ Nfl)W &)bo f &,I

+ $$,U - &I + &)bl

615

Arc spline approximation of BBzier curves: Y. J. Ahn et a/.

Figure 1 The quadratic BCzier curve q(r) has monotone curvature if and only if b, lies inside one of these circles.

Figure 2 The angle 6, is increasing since o(, and /3, are

b$+,=b(&) for i = 0 1 , , . . . ,N, by the de Casteljau algorithm. (see Chapter 3 in Ref. ‘)

We then prove the following lemma which will be used in the proof of Theorem 3.

Lemma 2. Let 01~ and /3,, i = O;..,N, be the angles

&+&b:+ 1 and Lb;, ,b$+*bE, respectively, as shown in Figure 2. Then

0 < cy() < p() < CY] < p, < .‘. < PN < ;

and for all i = l,...,N,

(2)

O<cu;+/3i<zandO<$;: =cY~+/~-I<; , (3)

Proof. Since the subdivision B6zier curve by(t), i = 0,. .,N, has the increasing curvature, we have 0 < (Y; < pi by Theorem 1. First, we show that /3_, < oli f: each i. Consider a curvature-increasing B&$er cur;e Bi (t), i =

1 ,. . . ,N, having control points bTi _ ?, I; and bpiA 7 where 1” is the intersection of the line bt _ *b$ ~ 1 and bg + , bz. + 2 as shown in Figure 2. Then bN. is the shoulder point of the quadratic BCzier curve B!(t) since b; = By(1/2) and therefore

(Y; = Lbgbg+zbE_z and pi-1 = Lb~b~-~b~+Z

Since the Bkzier curve BY(t) is curvature-increasing, the shoulder point bi is located on the right side of the bisector

N of the points btp2 and b2;+2. Therefore,

O(j-1 <@,-I <ai<Pl

for i = l,..., N. Thus we get eqn (2).

Table 1 The number of line or arc segments for various error tolerances

TOL Polygonal Walton/Meek Our algorithm

EXl 0.0005 22 14 10

Ex2 0.01 5 4 3 0.001 14 8 6 0.0001 43 16 13

EX3 0.0 I 8 12 I 0.001 32 18 15 0.000 1 69 46 29

Since the Bkzier curve br(tkis cprvature-increasing for each i, the angle Lbgb;, , bzi+ 2 IS larger than 7r/2, by Theorem 1. Thus

O<Cdj+fli< 5 By eqn (2), we have

()<($i=a,+Pj_, <(-yi+&< ;

for each i. Thus we also get eqn (3).

Remark. Let two points a,bcR* and a unit tangent vector t, at a be given and assume the counterclockwise angle 0 from the vector b - a to t, lies in 0 < 0 < 71-12 as shown in Figure 3. Then there exists a unique circular arc passing through a and b with the unit tangent vector t, at a, and the counterclockwise angle from the tangent vector of the circular arc at b to the vector b - a equals 8. The circular arc has the center and the radius given by

a+b c = 2 + (cot@

ilb - ali r = (csc0) 2

We now sketch our scheme for approximating the given quadratic B6zier curve b(t) by a G arc spline. For each N, we derive N + 1 quadratic BCzier curve segments

e , \

>’ ‘\

<’ ‘\

,’ ‘\

\ ,-

.c----::;-.:‘-o \ \

\ ’ /

I

\ ,

\ I

\ I

\ I

\ I

\ ,

r , \ I

/

\ ,

\

\ I K I

,

\ I

\ ,

c

Figure 3 The circular arc is determined uniquely by the two points a, b and a unit tangent t, at a.

616

Arc spline approximation of BBzier curves: Y. J. Ahn et al.

b. = bf a; = 00

Figure 4 B,,, = $,+, - 0, and 0, 10 for i = O;,,.N- 1.

by(t) with corresponding control points bg, bg + , and bg + *. According to most of the previous works9.‘~3,‘0.5,“,‘2,6, a biarc, made of ‘two’ circular arcs, is associated with each segment br(t) to match both end points and the corresponding tangent vectors simultaneously. Instead, we match ‘one’ circular arc C”(t) with each segment b;(t), recursively, except for the last segment. The first arc C:(t) interpolates both end points of ho(t) and matches the unit tangent vector of b(t) at b0 = bt by such a wa as described

NY in the Remark above. The subsequent arcs, Ci (t), z = 1,. . .,N - 1, interpolate both end points of b”(t) and match the unit tangent vector ti of Cj? ‘(t) at CF_ ‘(1) =bg as shown in Figure 4. We define the angle 8; to be (Y~ if i = 0 and to be the counterclockwise angle from the vector bE+* - b: to t, ifi= 1 ,. . .,N. Since BN # PN in general, the last Btzier segment b;(t) should be approximated by the biarc which interpolates both end points of b;(t) and matches the unit tangent vectors of Ci _ , (t) at C,"_ , (1) = b& and of b;(t) at bc( 1) = b{ + 2. The biarc approximation of the last segment in our algorithm will be the same as Walton and Meek6. Thus, for each N, we obtain a G’ arc spline approximation of the given quadratic Btzier curve. We show that the arc spline approximation by our method is shape-preserving, i.e. it turns always to the right, in the following theorem.

Theorem 3. Let b(r) be a curvature-increasing quadratic B6zier curve with control points bO, bi and b2. Then for each positive integer N, there exists a shape-preserving arc splines consisting of N + 2 arcs with G’-continuity, that interpolates b(i/(N + l)), i = O,l,..., N f 1, and matches the unit tangent vectors of b(t) at b0 and b?.

Proof. The approximate arc spline of the quadratic Bizier curve constructed above consisting of N + 2 arcs, has G’- continuity, interpolates b(i/(N + l)), i = 0,l.. .,N + 1, and matches the unit tangent vectors of b(t) at b. and b2. Thus it suffices to show that 0 < 8; <: 7r/2 for all i. Let (Y~, pi and 4; be as in Lemma 2. By eqn (2),

a o<e,=cY,< 7

L

Assume that 0 < 0, < ~12 for i = 0,. . .,n. By the definition of 8, and Figure 4, the positive quantities 0, satisfy en+’ + 8,1 = 4,,+’ (or 4,,+’ + 2n) for each IZ. Since 0 < $n+l, t9n < n/2 we get

0 < d,, + I < :or ?iB,,+, 527r (4)

Since f3;+’ = 4i+’ - 19, for each i and 4’ = PO + (Y;, we have

PO + a’ - %j, if n=O 0 nfl- - 4% + I - o,, =

4 N+~-4,1+0,-~, ifns 1

Since (Y’ > a0 and 4,,+’ > d,, hy eqns (2) and (3),

O<flo+(~‘--~‘)<aand (5)

()<(&+I -4J,)+4~‘<~

By eqns (4) and (5),

Hence, by induction, 0 < 0; < 7r/2 for all i. Thus, for each N 2 1 there exists a shape-preserving arc spline consisting of N $ 2 arcs with G’-continuity that interpolates b(i/(N + I)), i = O,l,. . . ,N + 1, and matches the unit tangent vectors of b(t) at b’) and bz. 0

ERROR MEASUREMENT AND ALGORITHM

In this section, we give a method for measuring the error in approximating a given quadratic Bizier curve by the G’ arc spline by our method and finally present its pseudo- algorithm. For each N and 0 5 i < N, let EN be the maximum of the distance from the point by(t) to the ith arc segment Cy with radius ry and with center I$, measured along a radial direction of the arc segment, i.e.

where p:(t) : = /lb;(t) - $I/- $. The maximum distance E! is obtainable as follows.

Theorem 4. For each N and 0 5 i < N,

r~=max(~p~(t)~:O~t%1,A~‘+Bt’+Cf+D=0}

(6)

Figure 5 Approximation of the last segment using biarc

617

Arc spline approximation of B&ier curves: Y. J. Ahn et a/.

where

A = ]]A2b;]]2, B = 3A2bg 0 Ab;, C = A2b;. 0 (b2i - cr)

+ 2iiAb;ii2, D = Ab; 0 (bzi - c”)

and Akbi=AkP1(b,+, -hi) for k>O

and A”bj = bj

Proof. Let g”(t) = (br(t) - cy) 0 (b:(t) - cr) - rr’. Since d#(t)/dt has a zero at t0 if d#(t)/dt at to, each local extremum of 4:(t) occurs where that of p;(t) does. Since #y(t) is a quartic polynomial, d$Y(f)ldt is a cubic poly- nomial and its roots are solvable algebraically. By a routine algebra, d$N(t) = Oldt reduces to eqn (6). 0

The last BCzier spment b;(t) is approximated by biarc, made of two arcs Ci , 1 = N,N + 1, with radii ry and with centers c” as shown in Figure 5. Let r&0,1) be the parameter such that bN(tB) is the intersection of the line $cz + , and the parabola b!(t). Thus we define E; and E; + 1 by the maximum distance from the point b;(t), &[O,t,] and te[ts,l], to the arc segments Cg and Cc+ , , respectively, measured along a radial direction of the arc segment. That is,

E; = max ]]]bc(t) - c$ - r$ and E;+, O~Etg

Example 1

(4.5,2.75)

Example 2.

(1,2.75)

Example 3 = r~~~,Illb~(t)-c~+,Il-r~+,I

B Figure 6 Curves corresponding to examples.

Table 2 The unit tangent vectors of the arc splines obtained by our method and the quadratic Btzier curves for Example 1, Example 2 with TOL = 0.001 and Example 3 with TOL = 0.001. pt is the angle between two unit tangent vectors of the arc splines and the quadratic Btzier curves at b(t,)

Parameter (r ,) The unit tangent vector of p , (radian)

Btzier curve Arc spline

(a) Example 1 (TOL = 0.0005) t,r = 0 f, = 0.1 tA = 0.2 tj = 0.3333 t4 = 0.4667 rs = 0.6 I6 = 0.73333 r7 = 0.8667 28 = 1

(b) Example 2 (TOL = 0.001) t” = 0 t, = 0.2 tz = 0.4 tl = 0.6 r4 = 0.8 t5= I

(c) Example 3 (TOL = 0.001) 10 = 0 I, = 0.0795 t2 = 0.1593 t3 = 0.0.2386 t4 = 0.3181 t5 = 0.3976 th = 0.4772 t7 = 0.5519 rR = 0.6265 tg = 0.7012 f ,,, = 0.7759 t,, = 0.8056 t ,2 = 0.9253 t,,= 1

(0.ooo, 1 .OOO) (0.217.0.976) (0.447.0.894) (0.707,0.707) (0.868.0.496) (0.949,0.3 16) (0.989,O. 179) (0.997,0.077) (1.000,0.000)

(1.000,0.ooo) (0.966.0.260) (0.916,0.401) (0.844.0.537) (0.875,0.484) (0.819,0.573)

(1.000,0.000) (0.999.0.041) (0.995,0.102) (0.981.0.196) (0.934.0.357) (0.762.0.648) (0.214,0.977)

(-0.395.0.919) (-0.678.0.735) (-0.795,0.606) (-0.852.0.524) (-0.882,0.469) (-0.903,0.430) (-0.916,0.401)

(O.OOo,l .OOO) (0.208,0.978) (0.447,0.894) (0.715,0.700) (0.872.0.490) (0.956,0.308) (0.984,O. 176) (0.997,0.071) (1.000,0.000)

(1.000,0.000) (0.963,0.271) (0.913.0.408) (0.842.0.540) (0.874,0.486) (0.819,0.573)

( 1.000,0.000) (0.999,0.038) (0.995,0.099) (0.982,O. 187) (0.940,0.341) (0.791.0.612) (0.214,0.977)

(-0.43 I ,0.902) (-0.692,0.722) (-0.800,0.600) (-0.854.0.520) (-0.884,0.468) (-0.904,0.428) (-0.916,0.401)

0

8.915 X IO ’ 0 1.057 x IO ? 6.971 X 10 ’ 9.105 x IOYi 2.542 X I()-’ 5.374 x lo- o

0 1.15 x 10-l 7.82 X lo-’ 3.45 x 10 3 2.21 x 10-j 0

0 3.56 X IO-’ 2.62 X 10 ml 6.30 X IO-’ 1.70 x 10 Z 4.58 X 10 ’ 0 4.04 x I()_? 1.91 x IO_? 7.24 X IO ’ 5.04 x IO -? 1.39 x IO i 2.34 X IO-’ 0

618

Arc spline approximation of Bbzier curves: Y. J. Ahn et al.

(a) Example 1 (TOL = 0.0005) (b) Example 2 (TOL = 0.001)

Walton’s result( 14 segments) +

to d Our result( 10 segments) +3-

(c) Example 3 (TOL = 0.001)

Walton’s result( 18 segments) +

tlJ t1 l_,rtA

Our result( 15 segments) +

Walt on’s result (8 segments) +-

Our result (6 segments) +

Figure 7 Approximate arc splines obtained by Walton’s method and our method, for Example I, Example 2 with TOL = 0.00 1 and Example 3 with TOL = 0.001. The arc splines with knots 0 are plotted by solid-lines, and each quadratic B&zier curves are by dash-lines. I, are equidistance para- meters in [OJ,]. [rA,l] or [O,l].

Remark. For i = N, N + 1, the maximum distance E” is also obtained algebraically, using the local extremum of the quartic polynomial G”(t) as in the proof of Theorem 4. Let EN = max05iCN+ 1 hi { ci ). Thus eN is the maximum dis- tance from the quadratic Bkzier curve to the approximate G’ arc spline. Using Theorem 4 and the earlier Remark, we now give a pseudo code of our algorithm for approx- imating a quadratic Bkzier curve q(t) by a G’ arc spline as follows.

Algorithm. Shape-preserving G’ arc spline approxima- tion

(1)

(2)

Input: a quadratic Btzier curve q(f) and an error toler- ance 7. Output: an approximate G’ arc spline and the number of arc segments within the error tolerance 7.

Use Theorem 1 to check whether the curvature is mono- tone or not.

(a) If the curvature has a local extremum at t = t,t(O,l). subdivide the given quadratic BCzier curve q(r) at t = tA which yields two curvature-increasing BCzier subsegments b,(t) and ha(t). GOT0 step 2. (b) Otherwise GOT0 step 3.

I a) Arc-approxi(b,(t)) and Arc-approxi(bo(t)).

(b) Merge the two approximate arc splines obtained in Step 2(a).

619

Arc spline approximation of Bbzier curves: Y. J. Ahn et a/.

(3) Arc-approxi(q(t)).

Sub-algorithm. Arc-approxi(b(r))

(1) (2)

Initialization: N = 1. Iteration with N.

(a) Subdivide b(t) at il(N + l), i = 1,. . .,N, to obtain N + 1 peaces b:(t). i = 0 ,..., N.

(b) Construct C:(t), i = 0,. . . ,N-1, which approximates by(r), and construct C”, and Ci+r which approxi- mate b;(t). (c) Find the local errors EN, i = O,.. .,N + 1, by using Theorem 4 and the Remark in this section. (d) Compute eN = max05i5N+, {EN}. (e) If E*’ < 7, then STOP the procedure. (f) Otherwise, N = N + 1.

Return the approximate circular arcs C;” i = 0,. .,N + 1, and the optimal number N + 2.

NUMERICAL RESULTS

In this paper we present the G’ arc spline approximation of the quadratic BCzier curves by matching one arc instead of two arcs (a biarc) as in the biarc method to each subdivision except for the last one. To compare the efficiency of our method with the biarc method, we apply our algorithm to the examples as shown in Figure 6, which are used by Walton and Meek’. As shown in Table 1, for all cases, our algorithm present better results than the algorithm using the biarc method in the number of arc segments within the same error tolerance. In particular, in case Example 3 with TOL = 0.01, our method give better results than the polygonal approximation, but Walton’s method does not. For Example 1, Example 2 with TOL = 0.001 and Example 3 with TOL = 0.001, we present the results of the arc spline approximation obtained by Walton’s method and our method as shown in Figure 7. The approximate arc splines are plotted by solid lines with their knots denoted by “ 0 “, and the quadratic BCzier curves by dashed lines. In Table 2, we also see that the approximate arc splines obtained by our algorithm is a good approximation of the tangent directions of the quadratic Bezier curves at each point b(t,), where t; are the equidistance parameters in [OJ,], [tA,l] or [O,l], and pi is the angle between two tangent vectors of the arc splines and the Bezier curves at b(t;).

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Mathematics, 1994. 54, IO7- 120. Sapidis, N. and Frey. W. H., Controlling the curvature of a quadratic Bezier curve. Compufrr Aided Geometric Design, 1992. 9, 85-91,

Farin. G., Cnne.v rmd Sur@ce.s ,for Computer Aided Geometric

Design. Academic Press, San Diego, CA, 1993. Bolton, K. M., Biarc curves. Cr,mputerAidrdDesign. 1975,7. 89-92.

Nutbourne, A. W. and Martin, R. R., Differential Geornetv Applied to Curve rmtf Surf& De,yign, Vol. I : Foundations. Ellis Horwood, Chichester, 1988. Sabin, M. A.. The use of cicular arcs to form curves interpolated through empirical data points. Rep. VTO/MS/l64, British Aircraft Corporation. Weybridge, 1976. Su, B. Q. and Liu. D. Y., Computational Geometr?: - Curves and

Swfuces Motfe[ing. Academic Press, San Diego, CA, 1989.

Young Joon Ahn is u PhD student in the

Department of Muthrmcttics. KAIST. He

received the BS degree in Mathematics

from Yonsei University. Korea, in 1991,

and the MS degree in Mathematics ram

KAIST, Korea, in 1994. Hi.7 rrsrwch

interests are Approximufion Theory, Numericul Analysis and Computer Aided

Geometric Design.

Hong Oh Kim is u professor of mcuhe- matics. He received his PhD in Mnthe-

matics from University of Wisconsin in

1982. His reseurch interest are in the

areas of complex and harmonic annlysis,

and also in approximation theo9.

Kyoung-Yang Lee is u PhD student in the

Department of Muthematics, KAIST. He received the BS degree in Mathematics

Education from Korea University, Korea,

in 1995, and the MS degree in Mathr- mutics from KAIST, Korea, in 1997. His

research interests are Approximution

Theory, Computer Aided Design. and

Computer Graphics.

1

1

I Meek, D. S. and Walton, D. J., Approximation of discrete data by G’ arc splines. Computer Aided Design, 1992, 24, 301-306.

2. Meek, D. S. and Walton, D. J., Approximating quadratic NURBS curves by arc splines. Computer Aided Design, 1993, 25, 371-376.

3. Meek. D. S. and Walton, D. J., Approximating smooth planar curves

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