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 by way of somebody who is familiar with his enterprise. Having bought used to his notation i locate this publication an increasing number of worthy and refer again to it every time an issue arises and customarily uncover the answer or a few pointer to the reply.
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Additional info for An Introduction to NURBS: With Historical Perspective (The Morgan Kaufmann Series in Computer Graphics)
If, however, the curve is only G 1 continuous at the join, then as the point transitions across the join the direction of motion does not change; but there is an instantaneous change in speed (magnitude of the velocity) t h a t represents an instantaneous acceleration as the point transitions across the join. If the curve, for example, represents a camera p a t h or is used for inbetweening in animation, the a b r u p t velocity change is quite disconcerting. , automobile bodies, G 1 or even G 2 continuity is not adequate.
N o t i c e t h e s y m m e t r y of t h e functions. E a c h of t h e b l e n d i n g f u n c t i o n s is of d e g r e e t This notation and control polygon numbering scheme is chosen to be consistent with the vast body of existing literature on B@zier curves and Bernstein basis functions. e. (Y-I)! The control polygon vertices are numbered from 1 to N. The transformations n -- N and i - I - 1 convert between the two notations. 1 / 1 J20 0 . 4 J5,4 // t (d) B~zier/Bernstein blending functions.
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 . 765] P(1) = Ba = [3 1] Notice that the basis functions, Jn, i, determine how much of each polygon vertex is used to construct a point on the curve.
An Introduction to NURBS: With Historical Perspective (The Morgan Kaufmann Series in Computer Graphics) by David F. Rogers
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