On The Approximation Of The Normal Vector Field Of A Smooth Surface

In this paper, we compare the normal vector field of a compact (oriented) smooth surface S with the normals of a triangulated mesh T whose vertices belong to S. As a corollary, we deduce an approximation of the area of S by the area of T. We apply this result to the restricted Delaunay triangulation obtained with a sample of S. Using Chew's algorithm, we build sequences of triangulations inscribed on S, whose curvature measures tend to the curvature measures of S

Unfolding Of Surfaces

This paper deals with the approximation of the unfolding of a smooth developpa- ble surface with a triangle mesh. First of all, we give an explicit approximat- ion of the unfolding of a smooth developable surface with the unfolding of a developable triangle mesh close to the smooth surface. The quality of the approximation depends on the maximal angle between the normals of the two surfaces and the relative curvature distance of the smooth surface (which is linked to the curvature of the smooth surface and the Hausdorff distance between the two surfaces). We give examples of sequences of developable triangle meshes inscribed on a sphere of radius 1, with a number of vertices and edges tending to infinity

On the Approximation of the Area of a Surface

We compare the area of a smooth surface S with the area of a surface M which is close to it, in terms of the local structure of S and the distance between S and M. We obtain convergence results, in particular when the surface approaching S is the restricted Delaunay complex associated to an epsilon-sample S of S

Approximation of Normal Cycles

This report deals with approximations of geometric data defined on a hypersurf- ace of the Euclidean space E^n. Using geometric measure theory, we evaluate an upper bound on the flat norm of the difference of the normal cycle of a compact subset of E^n whose boundary is a smooth (closed oriented embedded) hypersurface, and the normal cycle of a compact geometric subset of E^n "close to it". We deduce bounds between the difference of the curvature measures of the smooth hypersurface and the curvature measures of the geometric compact subset