Symmetries of Polyhedra : Detection and ApplicationsX

This paper deals with the detection of symmetries of polyhedra and its applications. It consists of two parts. First, we review seven algorithms for symmetry detection of poly-hedra. Since these algorithms supply symmetry information in quite diierent forms, we classify the three most common output forms of the symmetry detection algorithms and discuss their relationships and the transformation from one form into another. For each algorithm, the following ve aspects are considered: the output form, the computational complexity, the polyhedra class the algorithm can handle, the implementation, and the suitability for solving the related polyhedral congruity problem. Then, we compare the seven symmetry detection algorithms and give some recommendations as to which algorithm to choose for particular applications. In the second part of this paper we discuss some applications of symmetry information in robotics, geometric problem-solving, and computer vision. The conclusions of this review are twofold. On the one hand, symmetries can be very useful to resolve ambiguities or increase computational eeciency in problem-solving processes involving geometric objects. On the other hand, simple and eecient symmetry detection algorithms are available now. This makes the symmetry exploration a practical issue for many applications.