01.01.2019 – 31.12.2021
Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. The closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively. However, suitable assessments of the smoothness of a polyhedral surface are a very recent direction of research.
Smooth discrete surfacesare as follows:
The paper of Günther, Jiang, and Pottmann introduced notions of smoothness for polyhedral surfaces. The property they started with is that the Gauss image of a vertex star has no self-intersections. Already this property limits the possible shapes of Gauss images to convex spherical polygons in the case of positive discrete Gaussian curvature and to spherical pseudo-digons, -triangles, and -quadrilaterals in the case of negative curvature, as can be seen in the picture below. To get further analogies to the smooth theory, one has to additionally require that the Gauss image of each negatively curved vertex star is star-shaped. These conditions are easily implemented into an algorithm, as described by these authors and Wallner. Their final notion of smoothness is projectively invariant and takes into account how the Gauss images of vertex stars are arranged around a face. In this setting, projective transformations map the discrete normals, discrete tangent planes, discrete asymptotic directions, and discrete parabolic curves to the corresponding objects of the image surface.
Crucial to the above approaches is the fact that the discrete Gaussian curvature at a point of an embedded polyhedral vertex star equals the algebraic area of the image of its Gauss map. However, the literature does not seem to contain a complete proof of this fact. In the recent paper by Banchoff and Günther, the authors give a complete and elementary proof. It is based on a theorem about the critical point index of a normal vector as discussed by Banchoff in the 1970s. This allows them to derive a formula that relates the number of inflection faces in a vertex star with the numbers of positively and negatively oriented components in the Gauss image. Using this formula, they deduce all possible shapes that a Gauss image can take.
Currently, we are working on a curvature theory for general polyhedral surfaces. Together with Jiang, Müller, and Pottmann, we are investigating how the curvature theory that was developed by Bobenko, Pottmann, and Wallner can be generalized to surfaces whose face offsets do not have the same combinatorics. By splitting vertices of higher valence into trees, we can define a discrete mean curvature and a discrete Gaussian curvature on a triple of an incident vertex, edge, and face. These curvatures satisfy a discrete analog of Steiner’s formula. While the discrete mean curvature does not depend on the particular splitting of the vertices, the discrete Gaussian curvature does. Nevertheless, these discrete Gaussian curvatures are the gradients of the discrete mean curvature of the corresponding offset surfaces. In turn, the discrete mean curvature is the gradient of the surface area. These are the exact analogs to the smooth case. We are investigating if there exists a certain splitting that gives a particular suitable discretization of Gaussian curvature. Also, we want to perform experiments to see how well our discretization approximates the Gaussian curvature of a smooth surface. Moreover, these investigations go hand-in-hand with notions of smoothness that are based on the polyhedral Gauss image being free of self-intersections or on a consistent sign of the discrete Gaussian curvature.
We also study discrete mean curvature flows of polygons and polyhedra; edges and faces are shifted by a parallel offset according to their discrete mean curvature. Our original goal was to discretize Huisken’s result that smooth convex surfaces shrink to a point under the mean curvature flow and take asymptotically the form of a round sphere. This question has already been raised by Taylor while considering the crystalline curvature flow. In fact, her crystalline curvatures correspond to the discrete curvatures based on face offsets that were later discussed by Bobenko, Pottmann, and Wallner. Novaga and Paolini showed that the discrete analog of Huisken’s theorem does not hold for all polyhedra. However, their examples possess vertices of high discrete Gaussian curvature. So the statement could be true for polyhedra that are spherical enough. We consider it more promising to look at the volume-preserving discrete mean curvature flow. Huisken has also proved the corresponding smooth theorem that the flow deforms a convex surface into a sphere of the same volume. We expect the volume-preserving discrete mean curvature flow to be the gradient flow of the area functional in the space of polyhedra with the same volume. The motivation for our approach is the result of Lindelöf that a circumscribed polyhedron has the smallest surface area among all polyhedra with parallel faces that enclose the same volume. Another motivation to consider the volume-preserving discrete mean curvature flow is that its smooth anisotropic counterpart was studied by Andrews: The flow deforms a convex surface into the Wulff shape of the anisotropy of the same volume. In contrast, the anisotropic mean curvature flow without the fixed volume constraint seems to be much more difficult to handle.
The mean curvature flow for polygons is also known as the curve shortening flow. In the isotropic case, Gage and Hamilton proved that the curve shortening flow shrinks any convex curve to a point and when the curves are rescaled to enclose the same area, they converge to a circle in the limit. Grayson extended this result to general simple plane curves. The anisotropic case was studied by Gage and Li. If the Wulff shape is centrally symmetric, then the flow deforms any convex curve to an infinitesimal scale of the Wulff shape. But if the Wulff shape of the anisotropy is non-symmetric, only convergence of subsequences is known since self-similar solutions may be non-unique as was shown by Yagisita. For the symmetric case, Chou and Zhu generalized the result to general simple plane curves for the anisotropic curve shortening flow. In addition, Gage investigated the area-preserving isotropic curve shortening flow and proved convergence to the unit circle.
Quite surprisingly, the discrete curve shortening flow for polygons, also known as crystalline curvature flow, does not behave in complete analogy to the smooth setting. Stancu proved that in the case of a centrally symmetric Wulff shape, a convex polygon converges to a point under the corresponding anisotropic crystalline curvature flow and when the flow is scaled to enclosed the same area as the Wulff shape, then it converges to it. In another paper, she claimed that any anisotropic crystalline curvature flow given by weights on the edges makes convex polygons converge (in subsequence) to the shape of a self-similar solution corresponding to a discrete Wulff shape of the flow; however, her work contains serious errors. Counterexamples to this general claim were given by Andrews. In contrast, Yazaki considered the isotropic area-preserving crystalline curvature flow and proved convergence of polygons to the circumscribed polygon with parallel edges. The anisotropic case seems to be open and is now studied by us.