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Download Computer Graphics and Geometric Modeling Using Beta-splines by Brian A. Barsky PDF

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

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1 Evaluation Method I ß1 and ß2 have been assumed tobe uniform shape parameters, each assuming a single value over the entire curve. 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.

4 Perturbation Due to the Movement of a Control Vertex lf an already-existing curve is to be modified, it is not necessary to recompute the entire curve. Careful consideration of the properties of the Beta-spline representation enables an existing curve to be modified in a manner that is more efficient than a complete recomputation. Consider the consequences to an existing curve when the position of one control vertex is modified. 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.

Now the basis functions have been evaluated for a set of p + 1 given values ofthe domain parameter u. The values of the functions at each of these parametric values can then be used to compute a corresponding set of points on a single curve segment. 1) can be used in the design of the following algorithm which requires 4d multiplications and 3d additionsjsubtractions to compute a curve point at the domain parametric value u, where d is the dimension of the curve. procedure compute_Q (i; b; V; Q); begin (* compute_Q *) Q; := b_2 * V;-2; for r := -1 to 1 do Q; := Q; end(* compute_Q *); + b, * vi+r Now consider the following algorithm which computes a general curve composed of m segments where the i 1h segment is evaluated at P; + 1 values of the domain parameter: 8 Curve Evaluation and Perturbation with Uniform Shape Parameters 44 compute_d (beta1, beta2, d); compute_delta (d, delta); compute_c (delta, d, c); for i := 1 to m do for each u in {u;klk = 0, 1, ...

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