Quaternions

Quaternions, $\mathbb{H}$ are a number system like real or complex numbers but with 4 dimensions. In particular, $\mathbb{H}$ is nothing but $\mathbb{R}^4$ together with a multiplication law. The identification of $\mathbb{H}$ and $\mathbb{R}^4$ is given by:  \mathbb{H} = \lbrace … Continue reading

Tutorial 2: Hyperbolic Helicoids

A ruled surface is a surface in $$\mathbb R^3$$ that arises from a 1-parameter family of straight lines, i.e. these surfaces are obtained by moving a straight line though the Euclidean space. E.g. a normal vector field of a curve defines such … Continue reading

Tutorial 1: Möbius Transformations

In this first tutorial we build a simple network to visualize Möbius transformations of a given geometry—below a picture of a cube together with its Möbius transform. The Möbius transformations are the group of transformations of \(\mathbb R^n\cup\{\infty\} \cong \mathbb … Continue reading

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

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
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