Representing oriented piecewise C2 surfaces

摘要

To reason about shape, one must first represent it. We deseribe a representation scheme for a large class of surfaces. Our primary focus is the formalism; implementation issues receive little attention. A good representation must be more than just minimally sufficient for the task; it must also be unambiguous, stable, local, and convenient. We first discuss four well-known schemes in view of these requirements, and then turn to differential geometric representations and argue that they, in general, are inadequate by themselves. Practicability and robustness require the incorporation of global geometric information. As representation on the Gaussian sphere offers considerable convenience, it is proposed that surfaces be represented by their Gaussian images, associating with each point the value of its support function, that is, the signed distance of the oriented tangent plane at its preimage from the origin. Such a representation is locally generative. Consequently, it is unambiguous; it is unique up to the orientation and position of the represented surface. It is local in directions of nonzero surface curvature. Also, it is additive; hence, although it is not stable, this problem seems circumventable. Several examples are provided.