Preserving geometric properties in reconstructing regions from internal and nearby points
Computational Geometry: Theory and Applications
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We consider the issue of shape approximation in kinematic mechanical systems; that is, systems of rigid solid objects whose behavior can be characterized entirely in terms of the constraints that each object moves rigidly and that no two objects overlap, without considering masses or forces. The general question we address is the following: Suppose we have calculated the behavior of some kinematic system using ideal descriptions of the shapes of the objects involved. Does it then follow that a real mechanism, in which the shape of the objects approximates this ideal will have a similar behavior? In addressing this question, we present various possible definitions of what it means (a) for one shape to approximate another and (b) for the behavior of one mechanism to be similarto the behavior of another. We characterize the behavioral properties of a kinematic system in terms of its configuration space; that is, the set of physically feasible positions and orientations of the objects. We prove several existential theorems that guarantee that a sufficiently precise approximation of shape preserves significant properties of configuration space. In particular, we show that It is often possible to guarantee that the configuration space of system A is close to that of system B in terms of metric criteria by requiring that the shapes of A closely approximate those of B in terms of the dual-Hausdorff distance. It is often possible to guarantee further that the configuration space of A is topologically similar to that of B by requiring that the surface normals are close at corresponding boundary points of A and B.