A topological coordinate system for the diamond cubic grid

Abstract

© 2016 International Union of Crystallography. Topological coordinate systems are used to address all cells of abstract cell complexes. In this paper, a topological coordinate system for cells in the diamond cubic grid is presented and some of its properties are detailed. Four dependent coordinates are used to address the voxels (triakis truncated tetrahedra), their faces (hexagons and triangles), their edges and the points at their corners. Boundary and co-boundary relations, as well as adjacency relations between the cells, can easily be captured by the coordinate values. Thus, this coordinate system is apt for implementation in various applications, such as visualizations, morphological and topological operations and shape analysis.The diamond cubic grid is one of the usual cubic crystal structures. The Voronoi cells are triakis truncated tetrahedra having four hexagon and 12 triangle faces, 30 edges and 16 vertices. A symmetric coordinate system is presented that addresses not only the voxels, but also the faces, edges and vertices between them. In this way, not only volumes, but also surfaces, or paths containing edges can be easily described and visualized.