Intrinsic Surface Properties from Surface Triangulation

Intrinsic surface properties are those properties which are not affected by the choice of the coordinate system, the position of the viewer relative to the surface, and the particular parameterization of the surface. In [2], Besl and Jain have argued the importance of the surface curvatures as such intrinsic properties for describing the surface. But such intrinsic properties may be useful only when they can be stably computed. Most of the techniques proposed so far for computing surface curvatures can only be applied to range data represented in image form (see [5] and references therein). But in practice, it is not always possible to represent the sampled data under this form, as in the case of closed surfaces. So other representations must be used. Surface triangulation refers to a computational structure imposed on the set of 3D points sampled from a surface to make explicit the proximity relationships between these points [1]. Such structure has been used to solve many problems [1]. One question concerning such structure is what properties of the underlying surface can be computed from it. It is obvious that some geometric properties, such as area, volume, axes of inertia, surface normals at the vertices, can be easily estimated [1]. But it is less clear how to compute some other intrinsic surface properties. In [8], a method for computing the minimal (geodesic) distance on a triangulated surface has been proposed. Lin and Perry [6] have discussed the use of surface triangulation to compute the Gaussian curvature and the genus of surface. In this paper, we propose a scheme for computing the principal curvatures at the vertices of a triangulated surface.