By David F. Rogers
For one week after receiving this e-book I agreed with an previous very severe evaluation. I replaced my brain. the topic isn't effortless yet written through somebody who understands his company. Having bought used to his notation i locate this publication a growing number of necessary and refer again to it at any time when 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)
Notice in this case t h a t t h e direction of t h e t a n g e n t v e c t o r is s o m e w h a t c o u n t e r i n t u i t i v e . T h e basis functions given above, along w i t h Eqs. 14), are used to e v a l u a t e t h e derivatives along t h e curve. Specifically, t h e first derivatives are 0- 3t J~,o(t) - t(1 - t) (1 - t) 3 - - 3 ( 1 - t) 2 1 - 3t g~,~(t) = tt(1 - - - ~3~(1 - t)~ = 3(1 - 3t)(1 - t) = 3 ( 1 - 4 t J~,2(t) - + 3t ~) 2 - 3t t(1 - t) 3t2(1 - t) = 3t(2 - 3t) J~,~(t) = 3(1 - t) t~ t(1 -- t) -- 3t2 Notice t h a t t h e r e is no difficulty in e v a l u a t i n g t h e s e results at e i t h e r t -- 0 or t -- 1.
T h u s , in p r a c t i c e P(t) - P(t) + BiJn, i(t) is x(t) -- x(t) + Bi~Jn, i(t) y(t) - y(t) + BiyJ~,i(t) z(t) = z(t) + B~z&,~(t) where the Bi~, Biy, Biz are the x , y , z components of the vector-valued control polygon vertices Bi [xi(t) yi(t) zi(t)]. 6) Jn,n] and the matrix [a]T--[B0 B1 "'" Bn] contains the geometry of the curve. The specific matrix forms for low values of n are of interest. 10) is more convenient to evaluate for arbitrary values of n. Notice that for each value of n the matrix [N] is symmetrical about the main diagonal, and that the lower right triangular corner is all zeros.
4 illustrate these results. 5) i=0 An example illustrates the technique for d e t e r m i n i n g a Bdzier curve. 1 B~zier Curve Given the vertices of a B6zier polygon (see Fig. 5) B0[1 1], B112 3], B214 3], B313 determine seven points on the B6zier curve. Recall Eqs. 5 R e s u l t s for Bdzier c u r v e s e g m e n t for Ex. 1. J,~,i(t)- where (n) and (n) ti(1 - t)~-i i! (~ - i)! Here, n = 3, b e c a u s e t h e r e are four vertices. (3-~)! 1 T h e p o i n t s on t h e c u r v e are t h e n P ( 0 ) = B0 = [1 1] P ( 0 .