CS284 Surface Topology

CS 284: CAGD

Genus = an integer that indicates the number of handles (or tunnels) on a body bounded by a closed surface.
Examples:

Integrating Gaussian curvature over a closed surface will yield a result that is equal to (1-genus)*720 degrees.
In particular, if we consider a polyhedral surface and we sum the angle deficits/excesses at each of its vertices, we will get the same formula.

Angle deficit at a vertex is equal to: (360 degrees - sum of all angles of all the facets that come together at that vertex).

polyhedra of genus 0 will have an angle deficit of 720 degrees
(e.g., a cube has 8 corners with angle sums of only 270 degrees);
polyhedra of genus 1 will have an angle deficit of 0 degrees;
polyhedra of genus 2 will have an angle excess of 720 degrees;
polyhedra of genus 3 will have an angle excess of 1440 degrees;

The topology of a control mesh will have to match the topology of the desired surface.
Consequences for the topology of control meshes:

Quadrilateral control meshes of genus 0 will have a "valence deficit" of 8 links;
- i.e., they may have 8 vertices with valence 3 (like a cube).
Quadrilateral control meshes of genus 1 will have a "valence deficit" of 0;
- i.e., they can be formed with regular recangular B-spline control meshes.
Quadrilateral control meshes of genus 2 will have a "valence excess" of 8 links;
- i.e., they may have 8 vertices with valence 5,
- - or perhaps, 4 vertices of valence 6,
- - or any combination of extraordinary points that produce an extra 8 links.

for genus 0: V + F = E + 2
generalization to objects of arbitrary dimension and arbitrary genus
it tells us how many and what kind of extraordinary vertices our control mesh should have;
this allows us to make "better", more regular, more symmetrical meshes that converge more quickly and more smoothly;
and if one wants to do a really fine job modeling a surface, one should put vertices with a valence deficit in areas of positive curvature,
and vertices with excess valences near saddle points.