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Additional info for An Introduction to NURBS: With Historical Perspective (The Morgan Kaufmann Series in Computer Graphics)
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.
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. The results are shown in Fig. 5. B~zier Curve Algorithm U s i n g Eqs. ( 2 . 1 ) - ( 2 . 3 ) a n d E x .
At the time, approximation methods for splines were principally interpolation based, which tended to introduce undesired undulations. Spline curve representation methods largely used piecewise power basis representations, "plus function" bases, or divided difference formulations. In early 1971 Bill went to a math meeting where he interacted with other leaders (probably including de Boor) about new directions in spline theory. He learned of a new result of de Boor's on an attractive method for evaluating splines by using an unusual basis, the B-spline basis.