Greedy Computation of a Homotopy Basis for a Genus 2 Surface

Several tools from topology are useful for mesh processing and computer graphics. These tools often operate on the 1-skeleton of a surface, i.e., the graph of edges embedded in the surface. A common task is to find a collection of edges called a cut graph - cutting along these paths turns the surface into a shape which can be flattened into the plane. This kind of flattening is necessary for texture mapping, remeshing, etc.

One way to find a cut graph is to find a set of loops, no two of which are homologous, which cut the surface into a disk when removed. Intuitively, two loops on a surface are homologous if one can be deformed into the other while always keeping it entirely on the surface. For a closed orientable surface with genus g (i.e., a torus with g handles), there are 2g classes of homologically independent loops. A homology basis consists of one loop from each class. Not every homology basis is a cut graph: some homology bases either disconnect the surface or cut it into a punctured sphere. However, a homotopy basis will cut the surface into a disk.

This video shows the greedy homotopy basis for each vertex of the mesh (the magenta square is the current basepoint). The end of the video illustrates the total length of the homotopy basis at each point: the brightness of a vertex corresponds to the total length of the corresponding basis.

For more information see http://users.cms.caltech.edu/~keenan/project_topology.html

Category:

Science & Technology

Tags:

• topology
• geometry
• homology
• homotopy
• cut
• graph
• computer
• graphics
• mesh
• processing
• torus

[1czelaya]

How about making a video for fiber bundles applied to the object presented in R3

[SheaDesign]

Looks like a baked occlusion pass! Advanced unwrapping tools are always welcome... -Shea

[Leonida1986]

what's this?