In practice do these extraordinary points pose any problems? They seem very evident in untextured meshes like this one, and it seems like these points might also affect mesh traversals. They also seem to form at extrema of the surface, most notably maximums or saddle points. Is this a coincidence cause by this specific mesh or do extraordinary points generally appear at these extrema?
wangcynthia
What does it mean for a mesh to be topologically equivalent to a sphere?
Carpetfizz
@wangcynthia I think it means that if you start with a sphere and just change the positions of the vertices without changing any half edges, you can end up at a new shape with different geometry but same topology.
dtseng
Is there a proof somewhere that describes why all convex closed objects cannot have all vertices with degree 6? I know a icosahedron has vertices with degree 5, it's just interesting how 6 is somehow the magic number at which point it no longer works.
Edit, nevermind, didn't realize that it would be explained in the next slide. Still very fascinating though.
go-lauren
@wangcynthia Someone once explained topologically equivalent to a sphere to me as:
Imagine the polygon in 3D space. If you poked a hole in one of the faces (as of a balloon), and stretched out the rest of the rubber and can lay the surface flat, then it is topologically equivalent to a sphere. In other words, you can draw the mesh on a balloon (all edges between vertices are preserved, relative length doesn't matter).
If I'm not mistaken, this also is the same as being a planar graph (graph that can be drawn without intersecting edges), or having an Euler characteristic of 2.
In practice do these extraordinary points pose any problems? They seem very evident in untextured meshes like this one, and it seems like these points might also affect mesh traversals. They also seem to form at extrema of the surface, most notably maximums or saddle points. Is this a coincidence cause by this specific mesh or do extraordinary points generally appear at these extrema?
What does it mean for a mesh to be topologically equivalent to a sphere?
@wangcynthia I think it means that if you start with a sphere and just change the positions of the vertices without changing any half edges, you can end up at a new shape with different geometry but same topology.
Is there a proof somewhere that describes why all convex closed objects cannot have all vertices with degree 6? I know a icosahedron has vertices with degree 5, it's just interesting how 6 is somehow the magic number at which point it no longer works. Edit, nevermind, didn't realize that it would be explained in the next slide. Still very fascinating though.
@wangcynthia Someone once explained topologically equivalent to a sphere to me as:
Imagine the polygon in 3D space. If you poked a hole in one of the faces (as of a balloon), and stretched out the rest of the rubber and can lay the surface flat, then it is topologically equivalent to a sphere. In other words, you can draw the mesh on a balloon (all edges between vertices are preserved, relative length doesn't matter).
If I'm not mistaken, this also is the same as being a planar graph (graph that can be drawn without intersecting edges), or having an Euler characteristic of 2.