Problemas de proximidade e de caminhos minimos em superficies poliedricas

AUTOR(ES)
DATA DE PUBLICAÇÃO

1998

RESUMO

Shortest Path Planning is the field of Computational Geometry that concerns the determination of feasible shortest paths from a point to another in a given environment. We deal with a directed shortest path problem (DFGP) that minimizes the total work spent to move a body on a polyhedral surface with constant friction coefficient and constant slope in each face. Its importance is due to the fact that it generalizes several others. In order to characterize geodesic paths and shortest paths according to the constraints of the problem, we identify the corresponding local optimality criterion and we demonstrate the strict convexity of the directed frictioned geodesic distance function (DFGF) using convex function theory. We develop an algorithm, based on the continuous Dijkstra methodology, to solve the DFGP problem. The algorithm contains some details related to the directed nature of the paths which is due to the distance function DFGF being dependent on the direction of motion and to the characterization of the geodesic paths. We prove the correctness of the proposed algorithm and analyze its complexity. Furthermore, we identify some details omitted in a few continuous Dijkstra algorithms found in the literature and fill them in. We extend the DFGP problem and obtain an algorithm to construct a shortest path Voronoi diagram (DFGV) on a polyhedral surface according to the distance function DFGF. We reduce some generalizations of proximity problems to the construction of this diagram and, therefore, the DFGV diagram solves these proximity problems. We implement an external module to the Geomview program to be able to visualize a shortest path tree and a Voronoi diagram on polyhedral surfaces to the discreet geodesic problem (DGP) that is a special case of DFGF

ASSUNTO(S)

poliedros metodos de caminho critico movimento planejamento

ACESSO AO ARTIGO

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