An Introduction to NURBS: With Historical Perspective (The by David F. Rogers

By David F. Rogers

For one week after receiving this e-book I agreed with an past very serious overview. I replaced my brain. the topic isn't really effortless yet written by way of an individual who understands his enterprise. Having bought used to his notation i locate this ebook an increasing number of beneficial and refer again to it each time an issue arises and typically uncover the answer or a few pointer to the reply.

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Extra info for An Introduction to NURBS: With Historical Perspective (The Morgan Kaufmann Series in Computer Graphics)

Sample text

Clearly, C 1 continuity implies G 1 continuity. , a curve may be G 1 continuous and may not be C 1 continuous. If the n t h derivatives of a curve, d~P(t)/dt ~, at the curve segment join are equal in both direction and magnitude, then the curve is said to have C n parametric continuity at the join. 4 illustrates the difference between geometric and parametric continuity. The vectors indicate the length of the tangent vector at the join indicated by the dot. The tangent vector at the end of the left-hand curve segment is equal in length to the longer tangent vector shown for the righthand curve segment.

T - ~/2. Here, the results for midpoint subdivision are derived for the special case of cubic B~zier curves. A cubic B~zier curve (see Ex. 1) is given by P(t) - (1 - t)3Bo + 3t(1 - t)2B1 + 3t2(1 - t)B2 + t3B3 0

2 Derivatives of B@zier Curves Consider a four-point B@zier polygon as shown, for example, in Figs. 6. Recall that the B@zier curve is given by P(t) = BoJ3,0(t) + BIJ3,1 (t) + B2J3,2(t) + BaJa, a(t) t Algebraic evaluation of Jn, i(t) and substitution before numerical evaluation yields correct results (see Ex. 2). Yi Bt~ t=0 5 B2 Br I I .... 6 B6zier curve and its derivatives. (a) P ( t ) ; (b) P ' ( t ) ; (c) P"(t). Hence, the first derivative is P'(t) = BoJ~,o(t) + B1J~,l(t) + B2J~,2(t) + BaJJ,3(t) Recalling Ex.

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