New Conversion between Triangular and Tensor-Product Bézier Patches

Triangular and tensor-product Bézier patches are widely used in computer-aided geometric design. It is important to study the conversion between them for modeling complex shapes, fulfilling data exchanges between CAD systems, etc. In general, a triangular Bézier patch is converted into three tensor-product Bézier patches, or a tensor-product Bézier patch is converted into two triangular Bézier patches. However, such conversions will increase the burdens of storage and display. In this paper, a new conversion scheme between two types of patches are given, i.e., a triangular Bézier patch is converted into a trimmed surface of tensor-product Bézier patch, or a tensor-product Bézier patch is converted into a trimmed surface of triangular Bézier patch. Theoretical analysis shows that the runtime and storage of new approach are only 1/2 or 1/3 of that of existing approaches, and experiments show that the numerical accuracy remains unchanged. The additional advantage of new approach is able to display the triangular Bézier patches by using OpenGL API.