By David F. Rogers
For one week after receiving this ebook I agreed with an prior very severe evaluation. I replaced my brain. the topic isn't effortless yet written by way of a person who understands his enterprise. Having acquired used to his notation i locate this ebook an increasing number of necessary 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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Extra resources for An Introduction to NURBS: With Historical Perspective (The Morgan Kaufmann Series in Computer Graphics)
Consequently, practical design using B~zier curves results in higher-degree polynomial curves when second-derivative continuity is required. An example of second-derivative continuity is shown in Fig. 8 for two five-point adjacent B~zier curves. 5 / 14 I x ~ C3 Second-derivative continuity between quartic B6zier curves. Increasing the Flexibility of Bdzier Curves Degree Elevation If a Bdzier curve with additional flexibility is required, the degree of the defining polynomial can be increased by increasing the number of control polygon points.
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. 1 and differentiating the basis functions directly yields J3,o(t) = (1 - t) 3 => J~,o(t) = - 3 ( 1 - t) 2 Ja, l(t) = 3t(1 - t ) 2 =~ J3,1(t) = 3(1-t) 2 -6t(1-t) J3,2(t) = 3t~(1 - t) J3,3(t) = t 3 =~ J~,2(t) = 6t(1 - t) - 3t 2 =~ J~,3(t) = 3t 2 E v a l u a t i n g these results at t = 0 gives J~,0(t)(0) = - 3 J~,l (t)(0) = 3 J~,2(t)(0) --- 0 J3,3(t)(0) ~---0 S u b s t i t u t i n g these values into the e q u a t i o n for the first derivative yields P ' ( 0 ) = - 3 B 0 + 3B1 = 3(B1 - B0) Thus, t h e direction of the t a n g e n t vector at the beginning of the curve is the same as t h a t of the first polygon span (see Fig.
Steve Coons liked matrix formulations. The general case of the B-spline curve method that I was developing for my thesis did not easily fit that formulation. However, by considering only the special case of floating uniform knots, Steve was able to derive a matrix formulation. His efforts had the effect of making B-spline curves much easier to grasp and therefore more popular to use. However, this was a double-edged sword, for only the uniform floating B-splines lent themselves to this approach.
An Introduction to NURBS: With Historical Perspective (The Morgan Kaufmann Series in Computer Graphics) by David F. Rogers