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Show 108 XIII.3 Perturbation Due to the Movement of a Control Vertex Analogous to curve perturbation, the modification of an already-eXisting surface does not require the recomputation of the entire surface; rather, the modification of an existing surface can be accomplished much more efficiently by exploiting various properties of the Beta-spline representation. Consider the consequences to an existing surface when the position of one control vertex is modified. Since a single control vertex affects only sixteen surface patches, the consequences of moving one vertex are confined to sixteen patches, and hence the movement of a control vertex requires the re-evaluation of only sixteen patches. Furthermore, even these sixteen patches do not need to be completely recomputed. Al though each of these patches is controlled by sixteen vertices, only one vertex has changed position. Therefore, the change in each of these patches is due only to the movement of one control vertex. Recalling the mathematical formulation for the (i,j)th surface patch given in equation (XII.4), the change in this patch j(u,v) can be written as .(u,v) -1J 1 1 = [br(beta1, beta2; u) r j s bs(beta1, beta2; v)] r=-2 s=-2 -1+ , + (XIII.7) where . is the change in position of the control vertex -1J YiJ·· Assuming that the modified control vertex is the only one that has moved, |
