Technical SectionTermination criterion for subdivision of triangular Bézier patch
Introduction
Triangular Bézier patch was first introduced to CAGD by de Casteljau in internal technical reports (de Casteljau, 1959–1963). Since then, a great deal of work has been reported in these fields. Sabin [1] studied the triangular Bézier patch independently and his representation scheme is well known.
Subdivision of curves and surfaces are important operations in CAD/CAM because they produce sequences of linear segments or planar facets approximating the curves and surfaces within a given tolerance. Lane and Riesenfeld [2] proposed a termination criterion for subdivision of the Bézier curve and Wang [3] addressed the convergence problem regarding piecewise linear approximation of rational Bézier curve. Goldman [4] studied subdivision algorithms for triangular Bézier patch in 1983 and Chang, Davis [5] gave a practical formula for calculating the control vertices of the triangular sub-patches in 1984.
One interesting problem for subdivision operations is how to determine the number of subdivision times of a triangular Bézier patch a prior based on a given tolerance. In this paper, we present a theorem (Theorem 3) to deduce such a termination criterion.
Section snippets
Preliminaries
Assume a barycentric coordinate system over a domain triangle T:=ΔT1T2T3 such that T1=(1,0,0), T2=(0,1,0) and T3=(0,0,1), a triangular Bézier patch of degree n over domain triangle T can be defined aswhere denotes the control vertices of the triangular Bézier patch whose number amounts to (n+1)(n+2)/2 and are the respective Bernstein basic functions.
Let be the 3D
Estimating the deviation of a triangular Bézier patch from its base triangle
Theorem 3 Suppose that be a triangular Bézier patch which is defined over domain triangle T(T=ΔT1T2T3) and ΔP1P2P3 be a triangle inside T with its edges parallel to that of ΔT1T2T3 (see Fig. 1, Fig. 2 and the barycentric coordinates of its vertices be (ui,vi,wi), i=1,2,3. Let be a sub-patch of corresponding to ΔP1P2P3, then the maximum normal distance of to its base triangle will be no larger than a given tolerance ε, i.e.,
Geometric interpretations of Theorem 3
Assume that a triangular Bézier patch is subdivided into r2(r⩾r0)sub-patches, then we can use 3D triangular facets defined by the corner vertices of these sub-patches to approximate the original triangular Bézier patch within the given tolerance ε. Therefore, Theorem 3 can be applied to intersection calculation, FEM mesh generation and STL output for triangular Bézier patch-based CAD systems.
An example is shown in Fig. 3. In this figure, a quartic triangular Bézier patch approximating a
Acknowledgements
The work is jointly supported by the National Advanced Technology Project of China (No. 863-511-942-018) and the Research Fund for the Doctoral Program of Higher Education (No. 98033532).
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