CS 284: CAGD
Remarks on the Topology of Surfaces

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

Examples:
 sphere has genus 0;
 torus (doughnut) has genus 1;
 a "figure8 shaped pretzel" has genus 2.
 Exercise: Determine the genus of various objects ...

Integrating Gaussian curvature over a closed surface will yield a result
that is equal to (1genus)*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).

Examples:
 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 Bspline 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.

Euler's rule concerning polyhedral (or meshed) objects.
 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.
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