## Technical Reports (CIS)

#### Document Type

Technical Report

#### Date of this Version

July 1992

#### Abstract

The motion planning problem for an object with two degrees of freedom moving in the plane can be stated as follows: Given a set of polygonal obstacles in the plane, and a two-dimensional mobile object *B* with two degrees of freedom, determine if it is possible to move *B* from a start position to a final position while avoiding the obstacles. If so, plan a path for such a motion. Techniques from computational geometry have been used to develop exact algorithms for this fundamental case of motion planning. In this paper we obtain optimal mesh implementations of two different methods for planning motion in the plane. We do this by first presenting optimal mesh algorithms for some geometric problems that, in addition to being important substeps in motion planning, have numerous independent applications in computational geometry.

In particular, we first show that the *Voronoi diagram* of a set of *n* nonintersecting (except possibly at endpoints) *line segments* in the plane can be constructed in *O(√n)* time on a *√n x √n* mesh, which is optimal for the mesh. Consequently, we obtain an optimal mesh implementation of the sequential motion planning algorithm described in [14]; in other words, given a disc *B* and a polygonal obstacle set of size *n*, we can plan a path (if it exists) for the motion of *B* from a start position to a final position in *O (√n*) time on a mesh of size *n*. Next we show that given a set of *n* line segments and a point *p*, the set of segment endpoints that are visible from *p* can be computed in *O (√n*) mesh-optimal time on a *√n x √n* mesh. As a result, the *visibility graph* of a set of n line segments can be computed in *O(n)* time on an *n x n* mesh. This result leads to an *O(n)* algorithm on an *n x n* mesh for planning the shortest path motion between a start position and a final position for a convex object *B* (of constant size) moving among convex polygonal obstacles of total size *n*.

#### Recommended Citation

Sanguthevar Rajasekaran and Suneeta Ramaswami, "Optimal Mesh Algorithms for the Voronoi Diagram of Line Segments, Visibility Graphs and Motion Planning in the Plane", . July 1992.

**Date Posted:** 21 August 2007

## Comments

University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-92-57.