# Category Archives: Lecture

## Conformal Parametrizations of Surfaces

In the context of surfaces the strict analog of arclength parametrized curves is an isometric immersion \[f: M\to \mathbb{R}^3\] of a standard surface into $\mathbb{R}^3$. Here “isometric” means that lengths of curves and intersection angles of curves on the surface … Continue reading

## Structure derived from Halfedges

The halfedge description $M=(E,s,\rho)$ of an oriented surface contains all the information. It is nevertheless convenient to introduce some more derived stucture. For an edge $e\in E$ we define $\textrm{left}(e)=$ the face (cycle of $s$) containing $e$ $\textrm{right}(e)=$ the face … Continue reading

## The two most Simple Discrete Surfaces

We claim that any pair of permutations $s,\rho$ of a finite set $E$ (with $\rho$ involutive and without fixpoints) defines a unique cell decomposition of some compact surface without boundary. No further conditions are needed. To illustrate this, we look … Continue reading

## Halfedge Description of Oriented Surfaces

Imagine a cell decomposition $M$ of an oriented compact surface without boundary. Denote by $E$ the set of all oriented edges of $M$. Then for each $e \in E$ there is a unique face $\varphi$ on the left of $e$. … Continue reading

## Combinatorial Geometry: Cell Complexes

Roughly speaking, Combinatorial complexes play a similar role in the discrete world as differentiable manifolds in the smooth world. They are able to capture the “intrinsic” properties of a geometric object, i.e. those properties that are independent of any embedding … Continue reading

## Combinatorial Geometry: Simplicial Complexes

While (for good reasons) we have restricted our treatment of combinatorial cell complexes to the two-dimensional case, the theory of $n$-dimensional simplicial complexes is rather straightforward: Definition: A simplicial complex is a finite set $P$ together with a set $\mathcal{S}$ … Continue reading