By David F. Rogers
For one week after receiving this publication I agreed with an prior very severe evaluate. I replaced my brain. the topic isn't effortless yet written via somebody who is familiar with his enterprise. Having acquired used to his notation i locate this booklet progressively more worthwhile and refer again to it every time an issue arises and typically uncover the answer or a few pointer to the reply.
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Additional resources for An Introduction to NURBS: With Historical Perspective (The Morgan Kaufmann Series in Computer Graphics)
The second limiting characteristic is due to the global nature of the Bernstein b~is. This means that the value of the blending function Y~,i(t) given by Eq. 2) is nonzero for all parameter values over the entire curve. Because any point on a B(~zier curve is a result of blending the values of all control vertices, a change in one vertex is felt throughout the entire curve. This eliminates the ability to produce a local change within a curve. For example, because the end slopes of a Bdzier curve are established by the directions of the first and last polygon spans, it is possible to change the middle vertex of a five-point polygon without changing the direction of the end slopes.
The curve is invariant under an affine transformation. Several four-point B6zier polygons and the resulting cubic curves are shown in Fig. 3. With just the information given above, a user quickly learns to predict the shape of a curve generated by a B6zier polygon. (n-i)! 3 B@zier polygons for cubics. w h e r e , as i n d i c a t e d , we define (0) ~ -- 1 a n d 0] - 1. Jn,~(t) is t h e i t h n t h o r d e r B e r n s t e i n basis function. Here, n, t h e degree of t h e B e r n s t e i n basis f u n c t i o n a n d t h u s of t h e p o l y n o m i a l c urve s e g m e n t , is one less t h a n t h e n u m b e r of p o i n t s in t h e B~zier p o l y g o n .
1)(1 '~'u)n-~ = 1 i - 0 Jn,0(0) -n! n! (0)i(1 O) n-i i! (n - i)! = 0 - and Jn, i(0) - Thus P(0)- i # 0 BOJn, o(0) - Bo and the first point on the B6zier curve a n d on its control polygon are coincident, as previously claimed. (1)n(o) n-n n! (1) = 1 i - n n! t~(1 - 1) n - / - - 0 Jn, i(1) -- i! ( n - i)! Thus i ~ n P(1) -- B n J n , n(1) -- B n and the last point on the B6zier curve a n d the last point on its control polygon are coincident. T h e blending functions shown in Fig. 4 illustrate these results.