Cutting Segments Configuration in Square Cutting

摘要

The number of equivalence classes of configured points/segments for square cutting is investigated in this paper. In Volume Computer-Aided Design (VCAD), the shape of three-dimensional objects is approximated in discrete cubic lattice. With our unique shape approximation method called Kitta cube, three-dimensional shape is approximated by triangles(cutting triangles) which are held in the cubes of a cubic lattice. The intersection between the edge of a cube of a cubic lattice and three-dimensional shape is called cutting point. Combinatorial analysis of Kitta cube is essential for better shape approximation. In Kitta square, which is a two-dimensional analogue of Kitta cube, the boundary of two-dimensional shape is approximated by segments(cutting segments). Two endpoints of the segment are cutting points located on the edge of a square of a square lattice. The purpose of this paper is to provide mathematical foundations to Kitta square by enumerating the number of configured cutting points/segments using Polya-Redfield's theory of counting. We assume that the number of cutting points on one edge of the square is at most one.