Tripod a minimalist data structure for embedded triangulations

We show that a vertex based data structure that keeps only pointers per vertex can store triangulations navigate them and maintain them under swap operations By comparison edge based structures such as the winged edge take pointers per vertex and triangle based structures take pointers per vertex Introduction Representations of planar subdivisions are central to applications of geo metric computing Triangulations are particularly important In solid modelling boundary representations are often composed of triangulations because graphics hardware is optimized for triangles and triangle strips In nite element analysis triangle meshes are fundamen tal In geographic information systems triangulated irregular meshes TINs model digital terrain data Even planar subdivisions that are not triangulations may be represented as such the Voronoi diagram for example is commonly represented through its dual Delaunay triangulation Edge based structures such as the winged edge are commonly used to support op erations for embedded graphs For each edge these structures store pointers to the two incident vertices the next edges clockwise cw and counter clockwise ccw around these two vertices and possibly pointers to faces if there is data that must be associated with faces Face based or triangle based structures store for each triangle pointers to the ad jacent triangles and incident vertices Since by Euler s relation the number of edges in a planar triangulation with n vertices is n the edge based structures take pointers per vertex and face based structures take pointers per vertex We investigate a vertex based structure suggested by Martin Heller of the University of Zurich We show that this structure with only pointers per vertex is well de ned and that it can support maintenance operations including insert and swap although not in constant time All winged edge navigation operations are supported in constant time The Tripod structure We de ne the tripod data structure for an embedded planar graph in which every face including one distinguished outer face is a triangle We assign directions Supported in part by grants from NSERC and IRIS MITACS and GEOID NCEs Supported in part by a UBC Univ Graduate Fellowship to the edges such that the outer face is a cycle with the rest of the graph to the left and every vertex not on the outer face has exactly three outgoing edges Schnyder has shown that every planar triangulation has such