Discrete surfaces with prescribed mean curvature

Discrete surfaces have been investigated since thousands of years. The interest in those surfaces is theoretical, philosophical and also with a view towards applications. In particular in architecture and civil engineering appear discrete surfaces which can be seen as surfaces composed of individual facets. These facets can be planar triangles, quadrilaterals or other polygons. In some cases it makes even sense to consider non-planar facets. A surface in three dimensional space, discrete or otherwise, has some curvature. There are several notions of curvature. In our project we are interested in the so called mean curvature. For example, a plane has mean curvature 0 whereas a sphere of radius r has mean curvature 1/r. A large radius implies a small curvature. In the last decades discrete analogs of smooth curvature notions have been investigated. There are several discrete mean curvature notions which illustrates that the process of discretization is not always unique. The branch of mathematics that studies these objects and in which the present proposal is located is called discrete differential geometry. It now seems natural to analyze a given discrete surface and to determine its curvature. Our primary objective is the reverse question. Suppose we are given any arbitrary mean curvature function. Is there a surface which has exactly that curvature? In smooth differential geometry this type of questions is by now very well studied even though there are still open questions. In the setting of discrete surfaces this question has been addressed only for two special cases which are geometrically important, though. The question is, how does a surface, smooth or discrete, look like if the curvature is zero or any other constant everywhere. Both cases appear in nature. For example, as soap films in equilibrium. If the air pressure is equal on both sides of the surface then we have vanishing mean curvature and otherwise the mean curvature is any other constant number. In our project we prescribe a mean curvature function which is not constant. A typical question can be the following. Suppose we are given a function which maps a value to every point in three dimensional space. This function will be our prescribed mean curvature function. Further we choose a curve in space which can be thought as a wire loop. Now investigate the existence of a surface which passes at the end through that curve such that in between the mean curvature value of that surface is exactly equal to the value of our prescribed function.

Discrete surfaces have been investigated since thousands of years. The interest in those surfaces is theoretical, philosophical and also with a view towards applications. In particular in architecture and civil engineering appear discrete surfaces which can be seen as surfaces composed of individual facets. These facets can be planar triangles, planar quadrilaterals or other planar polygons. Modern fabrication methods make it simpler to include even facets which are not planar. However the manufacturing of such double curved surface panels is costly and potentially not sustainable. In the last decades discrete analogs of smooth curvature notions have been investigated a lot. There are several discrete curvature notions which illustrates that the process of discretization is not always unique. The branch of mathematics that studies these objects and in which the present project is located is called "discrete differential geometry". A typical question is the following. How does a surface, smooth or discrete, look like if the mean curvature is zero or any other constant everywhere. Both cases appear in nature. For example, as soap films in equilibrium. If the air pressure is equal on both sides of the surface then we have vanishing mean curvature and otherwise the mean curvature is any other constant number. A typical task in mathematics or geometry is to look at a (potentially abstract) smooth object and compute something about it like lengths, areas, volumes, or even curvatures. The same can be done for the above described discrete surfaces. In our case we could determine its discrete curvature. However, a primary objective can also be the reverse task. Suppose we are given some properties like a curvature function. Is there a surface which has exactly that curvature? In smooth differential geometry this type of questions is by now very well studied even though there are still open questions. In the setting of discrete surfaces this question has been addressed only for a few special cases which are geometrically important, though. Another question is, for example, how should a designer shape the geometry of a double curved facade such that the necessary molds for the fabrication can be re-used several times. We propose a Method which solves such a problem. Discrete structures like the meshes from above can also appear visually realized, for example, as support structure of freeform shaped facades. An aesthetically pleasing distribution of vertices and edges of such a mesh is often related to an equal distribution of angles between their edges. We invented two methods to easily generate such meshes. One such type of meshes results from a nonlinear optimization problem and the other type is obtained from a conformal subdivision scheme.