Abstract For a surface in 3‐space that is represented by a set S of sample points, we construct a coarse approximating polytope P that uses a subset of S as its vertices and preserves the topology of . In contrast to surface reconstruction we do not use all the sample points, but we try to use as few points as possible. Such a polytope P is useful as a ‘seed polytope’ for starting an incremental refinement procedure to generate better and better approximations of based on interpolating subdivision surfaces or e.g. Bézier patches. Our algorithm starts from an r‐sample S of . Based on S, a set of surface covering balls with maximal radii is calculated such that the topology is retained. From the weighted α‐shape of a proper subset of these highly overlapping surface balls we get the desired polytope. As there is a rather large range for the possible radii for the surface balls, the method can be used to construct triangular surfaces from point clouds in a scalable manner. We also briefly sketch how to combine parts of our algorithm with existing medial axis algorithms for balls, in order to compute stable medial axis approximations with scalable level of detail.

Level sets are isosurfaces of an implicit function F: R3rarr R K, that is the set of points satisfying F(x,y,z) = 0.In this paper we introduce an algorithm to move at interactive speed through the different level sets. Furthermore the meshes of the level sets are isotopic to the isosurface itself, as long as the surface stays away from singularities. When the surface moves close to singularities, the algorithm indicates arbitrarily small boxes where the topology is not certified. In this case, the user can decide to decrease the size of the boxes by further refinement. For special classes of functions, such as algebraic surfaces, other methods could be used to determine the topology inside the singular boxes.

The contour generator is an important visibility feature of a smooth object seen under parallel projection. It is the curve on the surface which seperates front-facing from back-facing regions. The apparent contour is the projection of the contour generator onto a plane perpendicular to the view direction. Both curves play an important role in computer graphics.Our goal is to obtain fast and robust algorithms that compute the contour generator with a guarantee of topological correctness. To this end, we first study the singularities of the contour generator and apparent contour for both generic views and generic time-dependent projections, for example, when the surface is rotated or deformed. The singularities indicate when components of the contour generator merge or split as time evolves.We present an algorithm to compute an initial contour generator by using a dynamic step size. An interval test guarantees the topological correctness. This initial contour generator can thus be maintained under a time-dependent projection by examining its singularities.