ANALYSIS AND CHARACTERIZATION OF GEOMETRIC OBJECTS ON TETRAHEDRAL GRID

Abstract

A grid usually is two or more sets of parallel lines in 2D or planes in 3D that intersect one another (called as grid points) at particular angle. It divides the plane or space into many congruent polytopes. Most common grids are square (in 2D) and cubical (in 3D). Grid is used in digital modeling of geometric objects. A digital model can be represented by a set of 2D pixels or 3D voxels. The choice of shape of pixel or of voxel is very important in digital modeling. Circular pixels and spherical voxels have this interesting property that all points on their surface is equidistant from grid point which is not the case in square pixels or cubical voxels. Spherical voxels are not stable on cubical grid. Stable self-alignment of spheres in 3D forms a tetrahedron. A simplex is the generalization of tetrahedron and triangle to n dimensions. 2- simplex grid is equivalent to triangular and 3-simplex grid is equivalent to tetrahedral grid. The dual of triangular grid is hexagonal grid. The notion of simplex grid topologically simplifies study of polytopes. Our aim is to analyze simplex grid and characterize basic geometric shapes on it.