Linear precision of rational Bézier curves

COMPUTER AIDED GEOMETRIC DESIGN

N

EI£EVIER

Computer Aided Geometric Design 12 (1995)431-433

Short Communication

Linear precision of rational B6zier curves G e r a l d F a r i n *, D o n g h a k J u n g Computer Science and Engineering, Arizona State University, Tempe, AZ 8528Z USA Received September 1994; revised January 1995

Abstract We discuss the linear precision property of rational B6zier curves and give an algorithm for finding appropriate weights.

Keywords: Rational B6zier curves; Linear precision; Degree elevation

An nth-degree rational B6zier curve x ( t ) is given by

x ( t ) = ~-~in-~wibiB~ ( t ) ~in__owiBn(t ) ,

x(t),biEE

2,

(1)

see (Farin, 1995) for details. If all control points are spaced evenly on a straight line bo,bn, and if the weights are unity, then the rational B6zier curve is in fact polynomial, tracing out the straight line in a linear fashion. This property is known as the linear precision property of B6zier curves. In the rational case, there are many ways of reproducing a straight line: suppose the control points are given by

bi = (1 - cei)b 0 + c r i b n,

i=0 ..... n

where ce0 = 0, an = 1, i.e., they are distributed arbitrarily on a straight line. I Can we find weights wi such that x ( t ) traces out the line bo, bn linearly? We can determine the unknown weights wi from * Corresponding author. Email: [email protected]. l The fact that all control points must be on the line follows from the lindear independenceof the Bemstein polynomials. 0167.-8396/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDI 0167-8396(95) 00009-7

G. Farin, D. Jung/Computer Aided Geometric Design 12 (1995) 431-433

432

alpha: weights:

0.00 1.00

0,20 1.00

0.40

1.00

0.60 1.00

0.80 1.00

1.00 1.00

alpha:

0.00 1.00

0.13 1.54

0.27 2.48

0.55 3.29

0.95 3.12

0.78

weights:

b4

b5

bl

~

©

1.00

b3©

©

alpha:

0.00

-0.50 0.35

2.00

1.50

1.00

weights:

1.00

-0.40

-0.28

0.37

-0.93

-0.86

Fig. 1. Quintic rational straight lines: top, the integral case; middle and bottom: truly rational. The solid circles denote points on the straight line corresponding to equal parameter spacing.

~--0 wibiBn(t)

= (1 -

t)bo+tbn.

Since w e can assume bo = [o] and obtain

7--o wi iB? ( t ) n ~i--o wiB7(t)

bn =

[~], and consider only the x-component, we

=t

and hence

wicriB~( t) = t Z wiB7 ( t). i=O

i=O

Degree elevation and reindexing yields

Xn+,[ i

-'~-'~Wi--lOli--I •

i--O

n + 1- i -

]

n+l

n+ - 1 wiai ] BT+l(t) = Z

.

n+t

1 w`-l°i"

Dn+l(t).

i=O

We n o w compare coefficients of B~ +l on both sides and obtain the recursion formula

G. Farin, D. Jung/Computer Aided Geometric Design 12 (1995) 431-433

i W i "~

-

-

n+l

-i

(1 - a i - l ) o~i

Wi--l,

i = 1 . . . . . n, wo ~ O.

433 (2)

It is interesting that the only restrictions on the ai are 0 ~ ai v~ 1. For ai > 1 or ~i < 0, the corresponding bi are placed outside bo,bn, but x ( t ) still traces out the line bo, b, linearly. In this case, we get some negative weights from (2). Fig. 1 shows some examples. A similar result holds if we do not consider control points, but a mix of control points and control vectors. This is outlined by Fiorot and Jeannin (1995) in this issue.

Acknowledgements This work was supported in part by NSF grant DDM 9123527 to Arizona State University. The work of the second author was also supported by a scholarship from the Republic of Korea Air Force.

References Farin.. G. (1995) NURB Curves and Surfaces, A K Peters, Wellesley, MA. Fiorot, J. and Jeannin, P. (1995), Linear precision of BR-curves, Computer Aided Geometric Design 12, 435-438, this issue.