Star-shaped polyhedron point location with orthogonal walk algorithm

The point location problem is one of the most frequent tasks in computational geometry. The walking algorithms are one of the most popular solutions for finding an element in a mesh which contains a query point. Despite their suboptimal complexity, the walking algorithms are very popular because they do not require any additional memory and their implementation is simple. This paper describes the modifications of two walking algorithms for point location on a surface of a star-shaped polyhedron, a generalization of the Remembering Stochastic walk algorithm for a star-shaped polyhedron and a modification of the planar Orthogonal walk algorithm. The latter uses spherical coordinates to transfer the spatial point location problem to the planar point location problem. This way, the problem can be solved by the traditional planar algorithms. Along with the modifications, the paper proposes new methods for finding a proper starting triangle for the walking process with or without preprocessing