Monday, June 3, 2013

Corrugated weaving of CP meshes

If a triangulation of a surface has the property that the incircles of its triangles form a circle packing (that is, the two incircles adjacent to an edge are tangent to that edge at the same point T) designing a corrugated weaving of the surface is greatly simplified. That is because the problem inside each triangle can be solved independently because the weavers cross the triangle boundaries in a fixed and symmetrical way. A. Schiftner, M. Hobinger, J. Wallner, and H. Pottmann in Packing circles and spheres on surfaces, have described triangulations of this type and their generation, naming them circle packing or CP meshes.

In such a mesh, each edge is crossed by two weavers (i.e., crossed by their centerlines) at the point T, and these weavers are angle symmetrically about the perpendicular to the edge at that point. The perpendicular connects the incircle centers (incenters) on both sides of the edge, which become identified with the 3-way crossings in the weaving.

Moreover, the weavers all enter a given triangle at the same distance from its incenter. Though the incircles vary somewhat in diameter, this variation can be accommodated by raising the actual three-way crossing above the plane of the triangle.

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