Incremental Convex Hull as an Orientation to Solving the Shortest Path Problem
Keywords:
Motion planning, Euclidean shortest path, convex hull algorithm, convex hull.Abstract
The following problem is very classical in motion
planning: Let a and b be two vertices of a polygon and P (Q, respectively) be the polyline formed by vertices of the polygon from a to b (from b to a, respectively) in counterclockwise order. We find the Euclidean shortest path in the polygon between a and b. In this paper, an efficient algorithm based on incremental convex hulls is presented. Under some assumption, the shortest path consists of some extreme vertices of the convex hulls of subpolylines of P (Q, respectively), first to start from a,
advancing by vertices of P, then by vertices of Q, alternating until the vertex b is reached. Each such convex hull is delivered from the incremental convex hull algorithm for a subpolyline of P (Q, respectively) just before reaching Q (P, respectively). Unlike known algorithms, our algorithm does not rely upon
triangulation and graph theory. The algorithm is implemented by a C code then is illustrated by some numerical examples. Therefore, incremental convex hull is an orientation to determine the shortest path. This approach provides a contribution to the solution of the open question raised by J. S. B. Mitchell in J. R. Sack and J. Urrutia, eds, Handbook of Computational Geometry, Elsevier Science B. V., 2000, p. 642.
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