# Curves and surfaces in computer aided geometric design - Yamaguchi F.

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Eq. (2.16), we have:

Ly [(y —x)^-1] —Mj M(x)

, (x-x^ff-1 (x-xJ+Mff~H

L W0(Xj) ^(xj+1) - WM(xJ+M) J

This equation shows that the B-spline function MJ>M(x) = Ly[(}/ —x)+_1] is a spline function of degree M—1 having the knots xp xJ + l, ..., xj+M (refer to Eq. (4.5)). However, since (y — x)+_1 is 0 at y = x}, xJ + 1, ..., xj+M when x^x7+JW, the divided difference with respect to ó knots are 0; in other words, for x^xj+M, MjM(x) = 0. In addition, when õ^õë (ó — x)+ ”1 = (ó — x)M~1 for

272

6. The Â-Spline Approximation

y = Xj, xj+1, ..., xj+M. Since this is a polynomial of degree M— 1 in y, the M-th order divided difference for it is 0; in other words, for x^xv MJM(x) = 0.

From the above discussion, we see that a B-spline is a C-spline; in fact, a B-spline is a special case of a C-spline.

© “B” in B-spline function stands for basis. This refers to the important property that an arbitrary C-spline of degree M— 1 having knots x0, xt, ..., xn can be uniquely expressed as a linear combination of n — M+ 1 B-splines8):

n-M

S{x)= X b}MhM{x). (6.70)

j = î

(ç) Consider the case in which M — 1 knots each are added at both ends of the knots x0, x1? ..., xn. These knots are called extended knots:

ó ^ ó»!5 ^ ó ^ y ^ Y* ^ y^ ”1

x0 õã ... XM-2 *ì- 1 XM ••• Ëï + Ì- 1 Ëï + Ì *** An+2Af-2J

= [-X'-(M-l) X-(M-2) ••• -^-1 ^0 -^1 ••• Xn Xfj -j~ i ... Xn + ë/ — j J . (6.71)

In extended knots, x0, xl9 ..., are called interior knots, and ..., x_x

and xn + i, ..., xn+M_1 are called additional knots.

n + M-1 B-spline functions are determined by the extended knots. It can be shown that an arbitrary spline function S(x) of degree M —1, having the

interior knots as its knots, can be uniquely expressed as a linear combination of

these n + M— 1 B-spline functions7*. Denoting constants by bj, we have:

n + M-2

S(x)= X bjMjM(x). (6.72)

j = î

In the following discussion, except when it is specifically necessary to make the distinction, “*” will be abbreviated in the extended knots.

6.4 Á-Spline Functions and Their Properties (2)

This book deals mainly with the use of parametric vector functions for describing curves and surfaces. Let us now define a B-spline function with knots expressed by the symbol t.

Definition 6.2. In the following 2 variable functions of t and è:

6.4 Â-Spline Functions and Their Properties (2)

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define an M-th order divided difference function M(t) with respect to the variable è based on t0, t1? tn:

The function Mj M(t) in Eq. (6.75) is called a Á-spline function of order M.

The B-spline functions used in practical applications are the normalized B-splines Nj M(t). Nj M(t) is given by the following formula:

Mj M(t) in (6.75) and Nj M(t) in (6.76) have no meaning at t = tj with M= 1. For M = 1 we define:

There is also a relation corresponding to Eq. (6.72) for the normalized B-spline functions NjM{t); it is clear that:

n + M-2

S(t)= Y CjNjtM(t). (6.81)

j=o

Let us now continue our discussion of the properties of Â-spline functions.

© The following relations hold for B-spline functions (refer to Appendix B.l).

— M[t’, tj, tJ+l, ..., tJ+M].

(6.75)

(6.76)

(6.77)

(^<^ + 1)*> (outside above range)

(6.78)

In this case:

range)

It is necessary to note that (6.78) and (6.79) imply that: MJtl{t) = Nhl{t) = 0 {tj = tJ+1).

(6.79)

(6.80)

(6.82)

*) In (6 78) and (6 79), note carefully where the equal sign is in describing the domains of t

274

6. The Á-Spline Approximation

Fig. 6.30. Relation between maximum value and knots of a Á-spline function, (a) M: even; (b) M: odd

Equation (6.82) means that for an arbitrary f, MjM(t) is the average of Mj,M-1(0 anc* Mj+ and that for tj<t<tj+M, MjM(t) is a convex

combination of ) and Mj+1M_1(t). Consequently, from Eq. (6.78),

is positive for tj^t<tj+l and zero elsewhere. Accordingly, for M>1:

Ì;,ì(0>0 (tj<t<tj+M) |

Mj'M(t) = 0 (tZtj or t^tj+M)\'

Similarly:

N,,m(0>0 (tj<t<tj+M) j

NjtM(t) = 0 or t^tj+M) )

As shown in Fig. 6.30, the graph of the Á-spline function Nj)M(t) is, in general, bell-shaped in The maximum value of the function occurs

at tj+M/2 or in its vicinity when M is even, midway between tj+(M-1)/2 and tj+(M+ d/2 when M is odd.

© As can be seen from Appendix B, in Eq. (6.81) the following relation holds:

n + M-2

I (6-86)

j = 0

Therefore, S(t) is a convex combination of c-} 0 = 0, ..., n + M — 2).

6.5 Derivation of fi-Spline Functions

Let us find the Á-spline functions of orders M= 1,2,3,4 from the definition of a divided difference.

In general, the M-th order divided difference of a function / can be expressed as a linear combination of the M + l function values /0, fu ..., fM according to the following equation:

6.5 Derivation of Á-Spline Functions

Ai(0

tj tJ+1

275

N^(t)

N,2 (t)

t j t j t j f 2

(b)

(a:

Fig. 6.31. Á-spline functions for M = 1, 2, 3, 4

(t0 ti)...(t0 tM) (fj to) (O ^2) — (^1 ^m)

+ ...+-----------------—-----------------. (6.87)

(tM — to) • • • (^Af ^M -1)

of M— 1:

From Eq. (6.79) we have:

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