Smooth surface and triangular mesh: Comparison of the area, the normals and the unfolding

Replacing a smooth surface with a triangular mesh (i.e., a polyedron) "close to it" leads to some errors. The geometric properties of the triangular mesh can be very different from the geometric properties of the smooth surface, even if both surfaces are very close from one another. In this paper, we give examples of "developable" triangular meshes (the discrete Gaussian curvature is equal to 0 at each interior vertex) inscribed in a sphere (whose Gaussian curvature is equal to 1 at every point). However, if we make assumptions on the geometry of the triangular mesh, on the curvature of the smooth surface and on the Hausdorff distance between both surfaces, we get an estimate of several properties of the smooth surface in terms of the properties of the triangular mesh. In particular, we give explicit approximations of the normals and of the area of the smooth surface. Furthermore, if we suppose that the smooth surface is developable (i.e., "isometric" to a surface of the plane), we give an explicit approximation of the "unfolding" of this surface. Just notice that in some of our approximations, we do not suppose that the vertices of the triangular mesh belong to the smooth surface. Oddly, the upper bounds on the errors are better when triangles are right-angled (even if there are small angles): we do not need every angle of the triangular mesh to be quite large. We just need each triangle of the triangular mesh to contain at least one angle whose sine is large enough. Besides, approximations are better if the triangles of the triangular mesh are quite small where the smooth surface has a large curvature. Some proofs will be omitted.