Subdivision Algorithms and Convexity Analysis for Rational Bézier Triangular Patches

Triangular surfaces are important because in areas where the geometry is not similar to rectangular domain, the rectangular surface patch will collapse into a triangular patch. In such a case, one boundary edge may collapse into a boundary vertex of the patch, giving rise to geometric dissimilarities (e.g. shape parameters, Gaussian curvature distribution, cross boundary continuities etc.) and topological inconsistency. Furthermore, since a triangular patch (i.e. defined as a closed polygon) is a basic figure in algebraic topology; hence any fairly irregular complex geometry can be efficiently modeled/designed/ regenerated with rational triangular patches. But, still triangular surfaces over triangular domain remain relatively unexplored as compared to rectangular surfaces over rectangular domain. In the present paper, we present subdivision algorithms for rational Bézier triangular patches for arbitrary subdivision. The algorithms are generated by constructing degenerate Bézier triangles and tetrahedron and using their edges to subdivide the original curve. Using the ‘classical probability theory’, we derive the algorithms by constructing higher dimensional degenerate Bézier simplices and incorporate their triangular sub-simplices. Additionally, we analyze the algorithms for convexity and derive conditions that preserve convexity. AMS MSC 2000: 65D07, 65D10, 65D17, and 68U07. ACM CCS: G.1.1, G.1.2, and I.3.5.