Chapter 15 - Geometric Shortest Paths and Network Optimization

Abstract

This chapter discusses several forms of the geometric shortest path problem, primarily for a single point moving in a two- or three-dimensional space. An assumption is made that the map of the environment is “known,” for discussing the on-line path planning problems. Computing an optimal path in a geometric domain is a fundamental problem in computational geometry, having many applications in robotics, geographic information systems (GIS), and wire routing. One method of handling the single-source query problem is to construct a “shortest path map” (SPM), which is a decomposition of free space into regions (cells) according to the “combinatorial structure” of shortest paths from a fixed source point s to points in the regions. SPM has a particularly simple structure as the boundaries between cells in a map are simply chords of P obtained by extending appropriate edges of the visibility graph V G (P). If SPM is preprocessed for point location, then the single-source queries can be answered efficiently by locating the query point t within the decomposition.