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

By Brian A. Barsky

Special effects and Geometric Modeling utilizing Beta-splines (Computer technology Workbench) [Hardcover] [May 03, 1988]

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Quotation details: Mimesis: Desarticulations (Paris: Flammarion, 1975), pp. 166-275.

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This notion can be formalized quite elegantly by drawing on graph theory. The set of control vertices can be considered as a graph {V, E} whose vertices form the set V= {Vuli = 0, 1, ... , m; j = 0, 1, ... , n} and with the set of edges E = {(Vii' Vi,j+dli = 0, 1, ... , m; j = 0, 1, ... , n- 1} u {(Vu, Vi+t)li = 0, 1, ... , m- 1; j = 0, 1, ... , n} The interior vertices are the vertices Vu where 1~ i ~ m - 1 and and the boundary vertices are 1~j ~ n- 1 7 Derivation of the Beta-spline Curve Representation 30 'end control - - - - - - - - - vertices Fig.

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.

Moreover, the combination coefficients of this linear combination are unique since the basis functions are linearly independent. Therefore, every Beta-spline curve segment with ß1 > 0 and ß2 ~ 0 has a unique representation as a linear combination of these basis functions, where the combination coefficients are the associated control vertices. The Beta-spline basis functions are derived in the following section. The Beta-spline curve segment Qi(u) is controlled by the control vertices Vi+" r = -2, -1, 0, 1.

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