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 info for Computer Graphics and Geometric Modeling Using Beta-splines**

**Sample text**

1), and rearranging, these computational requirements become m(23 + 4d)(p + 1) multiplications, m(2 + 3(9 + d) x (p + 1)) additions/subtractions, and dm(p + 1) divisions. It should be noted that this algorithm naturally computes the general curve evaluated at a different set of values of the domain parameter on each segment, with no loss of efficiency. 3) where S;g(ß1(u), ß2(u)) is given by S;g(ß1(u), ß2(u)) = 1 I dgr(ß1(u), ß2(u))Vi+r r=-2 for g = 0, 1, 2, 3 . 4) Using these equations, the following algorithm computes a general curve composed of m segments where the i 1h segment is evaluated at P; + 1 values of the domain parameter: for i := 1 to m do begin jump1 := alpha1;- alpha1;_ 1 ; jump2 := alpha2; - alpha2;_ 1 ; for each u in {u;klk = 0, 1, ...

This assumption can be exploited to efficiently evaluate a Beta-spline curve. Observe that all the coefficient functions have a constant denominator of ~- Thus, all the divisions can be performed prior to the actual computation of the Beta-spline basis functions. The following algorithm evaluates the basis functions at p + 1 given values of the domain parameter u, for a given value of each uniform shape parameter, ß1 and ß2, and requires 7 + 9(p + 1) multiplications, 12 + 2 + 9(p + 1) additionsjsubtractions, and 8 divisions.

Since a single control vertex influences only four curve segments and has no effect on the other segments, the consequences of moving one vertex are limited to four segments. Computationally, this implies that the movement of a control vertex requires the re-evaluation of only four segments. Moreover, even the four affected segments need not be completely recomputed. Although each of these segments is controlled by four vertices, only one of these vertices has changed position. Therefore, the change in each of these segments is due only to the modification of the position of one control vertex.