Intrinsic surface properties from surface triangulation

Abstract

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.

A tool for semi-automatic linear feature detection based on DTM

“…The code quantifies the non-planarity by means of principal curvature values associated to the DSM/DTM vertices, cal-22 culated implementing the method proposed by Chen and Schmitt (1992) to the adjacent vertices; similarly, a valley is an edge formed by vertices which correspond to negative values of mini-26 mum principal curvature and which represent, in absolute value, local maxima respect to the adjacent vertices. 27…”

A comparison of Gaussian and mean curvature estimation methods on triangular meshes of range image data

“…This algorithm from [2,11] uses the Meusnier's 2.1 and Euler's 2.2 theorems for the estimation of the principal curvatures. The justification for this algorithm comes from Equation (2.2) that is equivalent to κn = 1 2 (κ1 + κ2) − 1 2 (κ1 − κ2)(cos 2θ0 cos 2α + sin 2θ0 sin 2α), (3.6) where θ0 is the angle between some arbitrary chosen reference direction T0 in the tangent plane of vertex v and the principal direction that corresponds to κ1.…”

“…The method proposed in [2,11] constructs the fitted circles through the vertex v and a pair of v's neighbors, vi and vj, such that the angle between vectors (vi − v) and (vj − v) is close to π. Thus, by selecting k ≥ 3 pairs of such neighbors, the k constructed circles can prescribe using Meusnier's theorem (see Equation (2.1)) the values of A, B, C, and consequently, κ1 and κ2.…”

“…In [2,11], circular cross sections, near the examined vertex, are fitted to the surface. Then, the principal curvatures are computed using Meusnier's and Euler's theorems (see [3,15]).…”

Surface patch reconstruction by touching

“…They are not robust when applied on sparse tactile data. In (17), more than three normal curvatures were used to set up an overdetermined system to be solved for principal curvatures in a least-squares fashion. However, errors could still be significant because a circle approximation was used in estimating the normal curvature.…”