Combinatorial Surfaces

Everything should be made as simple as possible, but no simpler.

Assumptions

• shapes are manifolds idea: looks like Rn up close
• around any point you can find neighbourhood that is a topological disk

Definitions:

• Convex set: for every p,q points in S the line segment between p and q in S.

• Convex set hull (s): intersection of all convex sets containing S

• Simplex: k-simplex is the convex hull of k+1 affinely-independent points (vertices)

• Face: vertices are a subset of vertices of simplex

• Simplicial Complex: “ bunch of simplices”key concept: recursive data structure

  • Definition: collection of simplicies where:
    • the intersection of any 2 simplicies is a simplex
    • every face of every simplex in the complex is also in the complex

• Abstract Simplicial Complex: idea: specifies how vertices are connected, but not where they are in space.

• K-simplicies: set of k-1 vertices connected

• Face: subset of simplicial

• Proper face: strict inclusion

• Oriented k-simplex: ordered tuple, up to even permutation. (even +, odd -)

• Relative orientation: 2 maximal faces in their intersection have same/diff orientation