About a month ago I wrote my first serious post in this blog. It was about circle packings. Well, that’s nothing special because pretty much everybody knows what a circle packing is and many have used it in the field of generative art to produce stunning images and patterns. But what most people don’t know is how many uses such a packing has in the context of discrete differential geometry. Quite interesting for instance is the so called CP mesh. Originally developed to rationalize freeform surfaces in architecture and design, a CP mesh is a triangle mesh whose incircles form a packing.
To compute the CP mesh we first need to set up an optimization problem based on the original mesh. This is done by formulating an energy function measuring the incircle packing property, the distance to the original mesh and the distance to the boundary. After solving the nonlinear minimization problem in the least squares sense we finally obtain new point positions forming the required CP mesh. However, it is essential to bear in mind that it’s not possible to reconstruct a CP mesh from an arbitrary surface. In other words, there is always a deviation to the original mesh no matter what energy function we use. This is clearly visible on the images below. The picture in the middle shows a CP mesh for which the surface proximity is used in the energy function whereas for the picture on the right the boundary proximity is used.
CP meshes have some useful properties which are quite relevant to various algorithms and applications. One of these properties is that they form an ideal primal/dual structure which guarantees orthogonality, non-intersection and convexity. This is important in case we are using discrete differential geometry operators: Orthogonality ensures good numerical qualities of the Laplacian and furthermore it ensures that we can safely compute primal values based on the dual values and dual values based on primal values respectively.
The combinatorial relation between primal and dual mesh is quite important in the context of discrete differential geometry, especially when we are using DEC methods. For tight error bounds it’s therefore essential to work with a good primal triangulation which ensures the above mentioned properties such as non-intersection, orthogonality and convexity of the dual. To achieve orthogonality, we usually place the dual vertices in the circumcenter of its corresponding primal complex. This however could lead to intersections in case our primal mesh contains obtuse triangles. To avoid this problem we could use the barycentric instead of the circumcentre dual. But this in turn would fail to satisfy orthogonality and convexity. We could circumvent this new problem partially by generally using the circumcenter but switching to the barycenter in case the dual vertex is outside of its corresponding primal triangle. What we could do as well, is to project the vertex onto the triangles boundary. However, to cut a long story short, the only way to avoid all these problems is to use a proper triangle primal mesh. And a CP mesh is such a mesh.
(a) Triangle mesh. (b1, b2) Dual mesh based on the circumcenter of its primal triangulation. Obtuse triangles for which the circumcenter is outside of the triangles boundary are leading to intersections of the dual mesh (c) Dual mesh based on the barycenter of its primal triangulation. In this case there is no intersection but it fails to satisfy orthogonality. (d) Dual vertex is projected onto the edge of its corresponding triangle. (e) Optimized CP mesh (f) Incircle packing. (g) Dual mesh based on the incircle packing of the CP mesh
Another important property of a CP mesh is that if we extrude the edges of the dual polygon we get a torsion free structure. In other words, its physical realization could be build out of beams which intersect in a common axis passing through the incircles center. And last but not least, CP meshes can be used to approximate circle packings on a surface. This could be done by either fitting a circle into each vertex gap between the incircle packing or by using the contact points of the sphere packing. Both methods generally produce a fairly good packing. A real circle packing however, is not possible on an arbitrary surface.
A. Schiftner, M. Höbinger, J. Wallner, H. Pottmann: Packing Circles and Spheres on Surfaces