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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Additional info for Computer Graphics and Geometric Modeling Using Beta-splines
1). - 1 + 1] Ym-1 2 Vm- 1 + -Vm . Ym-1 To have the curve start closer to V0 and end nearer Vm, additional curve segments can be defined at the ends. 2) in the usual way since this would reference nonexistent vertices. Various methods are available for defining these curve segments 11 Classification and Analysis of Beta-spline Curve End Conditions 60 ~. Vo ··~ Fig. 1. Interior segments naturally defined by control polygon. and will now be described. These techniques fall into two classifications, multiple vertices and phantom vertices.
A Beta-spline surface has each surface patch controlled by sixteen control vertices and is unaffected by all other control vertices. Again, this is equivalent to the fact that a given control vertex exerts influence over only sixteen surface patches and has no effect on the remaining patches. Thus, the effects of manipulating one control vertex are limited to sixteen patches. 3 Explanation In order to effect local control, a Beta-spline curve segment is completely controlled by only four of the control vertices; therefore, a point on this curve segment can be regarded as a weighted average of these four control vertices.
These techniques fall into two classifications, multiple vertices and phantom vertices. 1 Double Vertices The double vertices technique defines one additional curve segment at each end by repeating the end vertex in the Beta-spline curve formulation. This technique yields a Beta-spline curve composed of m segments, that is Q 1 (u), Q 2 (u), ... , Qm(u). Since the control polygon has m line segments, this is a corresponding number of curve segments. 2) in the usual manner except that vertices V0 and Vm are used when V_ 1 and Vm+t• respectively, are referenced.