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

Yamaguchi F. Curves and surfaces in computer aided geometric design - Tokyo, 1988. - 390 p. Previous << 1 .. 59 60 61 62 63 64 < 65 > 66 67 68 69 70 71 .. 90 >> Next 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 ő^őë (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 őă ... XM-2 *ě- 1 XM ••• Ëď + Ě- 1 Ëď + Ě *** 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 č:
6.4 Â-Spline Functions and Their Properties (2)
273
define an M-th order divided difference function M(t) with respect to the variable č 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:
Ě;,ě(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: Previous << 1 .. 59 60 61 62 63 64 < 65 > 66 67 68 69 70 71 .. 90 >> Next 