By Brian A. Barsky
Special effects and Geometric Modeling utilizing Beta-splines (Computer technological know-how Workbench) [Hardcover] [May 03, 1988]
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Extra resources for Computer Graphics and Geometric Modeling Using Beta-splines
However, an even stron ger condi tion has been satisfied by the basis functi ons; they sum to unity for all values of the doma in param eter u, 1 L r= -2 b,(ß1, ß2; u) = 1 . 17) and the coefficient functions of the const ant terms add to unity, L 1 r=-2 c0 ,(ßl, ß2) = 1 . 10). All the expressions in u are nonne gative for 0 ~ u < 1. Therefore, each of the basis functions is nonne gative for 0 ~ u < 1 as long as ß1 ~ 0 and ß2 ~ 0. Since each basis function is nonne gative for these values, and since they sum to unity, each one V;-1 ----- ----- ----- ----- -- --- --V;-2 Fig.
M . Now, the original curve representation can be treated as a special case where rx1; and ß1;(u) are both replaced with ß1; rx2; and ß2;(u) are both replaced with ß2; and Y; and b;(u) are both replaced with b. 1 Evaluation Method I Since all the coefficient functions have a common denominator of b(u), all four basis functions will have this common denominator, and therefore the expression for the curve will have a denominator of b(u). It is thus of computational interest to define corresponding sets of coefficient functions and basis functions that are scaled by a factor of b(u).
3. The effects of moving a Beta-spline control vertex are confined to four segments. 7 Derivation of the Beta-spline Curve Representation 32 The Beta-spline formulation exploits the piecewise representation, in order to achieve local control, by defining each piece in terms of only a few nearby vertices. F or Beta-spline curves, each curve segment is controlled by only four of the control vertices and is completely unaffected by all the other control vertices. Equivalently, a given control vertex only influences four curve segments and has no effect whatsoever on the remaining segments.