Comparing Perfect and 2nd Voronoi decompositions: the matroidal locus

We compare two rational polyhedral admissible decompositions of the cone of positive definite quadratic forms: the perfect cone decomposition and the 2nd Voronoi decomposition. We determine which cones belong to both the decompositions, thus providing a positive answer to a conjecture of V. Alexeev and A. Brunyate. As an application, we compare the two associated toroidal compactifications of the moduli space of principal polarized abelian varieties: the perfect cone compactification and the 2nd Voronoi compactification.Comment: 27 pages, 2 figures, final version, to appear in Mathematische Annale

Classifying Voronoi graphs of hex spheres

A hex sphere is a singular Euclidean sphere with four cones points whose cone angles are (integer) multiples of 2*pi/3 but less than 2*pi. Given a hex sphere M, we consider its Voronoi decomposition centered at the two cone points with greatest cone angles. In this paper we use elementary Euclidean geometry to describe the Voronoi regions of hex spheres and classify the Voronoi graphs of hex spheres (up to graph isomorphism).Comment: 14 pages, 9 figure

Path planning algorithm for a car like robot based on Coronoi Diagram Method

The purpose of this study is to develop an efficient offline path planning algorithm that is capable of finding optimal collision-free paths from a starting point to a goal point. The algorithm is based on Voronoi diagram method for the environment representation combined with Dijkstra’s algorithm to find the shortest path. Since Voronoi diagram path exhibits sharp corners and redundant turns, path tracking was applied considering the robot’s kinematic constraints. The results has shown that the Voronoi diagram path planning method recorded fast computational time as it provides simpler, faster and efficient path finding. The final path, after considering robot’s kinematic constraints, provides shorter path length and smoother compared to the original one. The final path can be tuned to the desired path by tuning the parameter setting; velocity, v and minimum turning radius, Rmin. In comparison with the Cell Decomposition method, it shows that Voronoi diagram has a faster computation time. This leads to the reduced cost in terms of time. The findings of this research have shown that Voronoi Diagram and Dijkstra’s Algorithm are a good combination in the path planning problem in terms of finding a safe and shortest path