Boris Thibert

Developable surfaces from arbitrary sketched boundaries

Developable surfaces are surfaces that can be unfolded into the plane with no distortion. Although ubiquitous in our everyday surroundings, modeling them using existing tools requires significant geometric expertise and time. Our paper simplifies the modeling process by introducing an intuitive sketch-based approach for modeling developables. We develop an algorithm that given an arbitrary, user specified 3D polyline boundary, constructed using a sketching interface, generates a smooth discrete developable surface that interpolates this boundary. Our method utilizes the connection between developable surfaces and the convex hulls of their boundaries. The method explores the space of possible interpolating surfaces searching for a developable surface with desirable shape characteristics such as fairness and predictability. The algorithm is not restricted to any particular subset of developable surfaces. We demonstrate the effectiveness of our method through a series of examples, from architectural design to garments.

Approximation of the Normal Vector Field and the Area of a Smooth Surface

Abstract This paper deals with the comparison of the normal vector field of a smooth surface S with the normal vector field of another surface differentiable almost everywhere. The main result gives an upper bound on angles between the normals of S and the normals of a triangulation T close to S. This upper bound is expressed in terms of the geometry of T, the curvature of S and the Hausdorff distance between both surfaces. This kind of result is really useful: in particular, results of the approximation of the normal vector field of a smooth surface S can induce results of the approximation of the area; indeed, in a very general case (T is only supposed to be locally the graph of a lipschitz function), if we know the angle between the normals of both surfaces, then we can explicitly express the area of S in terms of geometrical invariants of T, the curvature of S and of the Hausdorff distance between both surfaces. We also apply our results in surface reconstruction: we obtain convergence results when T is the restricted Delaunay triangulation of an ε-sample of S; using Chew’s algorithm, we also build sequences of triangulations inscribed in S whose curvature measures tend to the curvatures measures of S.

On the approximation of a smooth surface with a triangulated mesh

We approximate the normals and the area of a smooth surface with the normals and the area of a triangulated mesh whose vertices belong to the smooth surface. Both approximations only depend on the triangulated mesh (which is supposed to be known), on an upper bound on the smooth surfaces curvature, on an upper bound on its reach (which is linked to the local feature size) and on an upper bound on the Hausdorff distance between both surfaces.We show in particular that the upper bound on the error of the normals is better when triangles are right-angled (even if there are small angles). We do not need every angle to be quite large. We just need each triangle of the triangulated mesh to contain at least one angle whose sinus is large enough.

Shape smoothing using double offsets

It has been observed for a long time that the operation consisting of offsetting a solid by a quantity r and then offsetting its complement by d < r produces, in some cases, a new solid with the same topology but with a smooth boundary. While this fact has been widely used in Computer Aided Geometric Design or in the field of image processing, we provide here for the first time a tight and robust condition that guarantees the smoothness of the new solid and gives a lower bound on its reach (distance to the medial axis). This condition is based on the general properties of the distance function to a compact set and relies on the recently introduced critical function and μ-reach.

Folded Paper Geometry from 2D Pattern and 3D Contour

Folded paper exhibits very characteristic shapes, due to the presence of sharp folds and to exact isometry with a given planar pattern. Therefore, none of the physically-based simulators developed so far can handle paper-like material. We propose a purely geometric solution to generate static folded paper geometry from a 2D pattern and a 3D placement of its contour curve. Fold lines are explicitly identified and used to control a recursive, local subdivision process, leading to an efficient procedural modeling of the surface through a fold-aligned mesh. Contrary to previous work, our method generates paper-like surfaces with sharp creases while maintaining approximate isometry with the input pattern.

Geodesic as limit of geodesics on PL-surfaces

We study the problem of convergence of geodesics on PL-surfaces and in particular on subdivision surfaces. More precisely, if a sequence (Tn)n∈N of PL-surfaces converges in distance and in normals to a smooth surface S and if Cn is a geodesic of Tn (i.e. it is locally a shortest path) such that (Cn)n∈N converges to a curve C, we wonder if C is a geodesic of S. Hildebrandt et al. [11] have already shown that if Cn is a shortest path, then C is a shortest path. In this paper, we provide a counter example showing that this result is false for geodesics. We give a result of convergence for geodesics with additional assumptions concerning the rate of convergence of the normals and of the lengths of the edges. Finally, we apply this result to different subdivisions surfaces (such as Catmull-Clark) assuming that geodesics avoid extraordinary vertices.