# Monthly Archives: April 2019

## Differential Forms

Let $M\subset \mathbb{R}^n$ be an open set and $f:M\to \mathbb{R}^k$ a smooth map. Then for each $p\in M$ the Jacobian $f'(p)$ is a $k\times n$ matrix. The derivative $d_pf$ of $f$ at $p$ then is the linear map given by … Continue reading

## Tutorial 3: Lawson’s Minimal Surfaces and the Sudanese Möbius Band

In the last tutorial we constructed certain minimal surfaces in hyperbolic space. These hyperbolic helicoids were generated by a 1-parameter family of geodesics: while moving on a geodesic – the axis of the helicoid – another geodesic perpendicular to the axis was … Continue reading

## 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 Maps III: Stereographic Projection

Stereographic projection \[\sigma: \mathbb{R}^n \to S^n \setminus \{\mathbf{n}\}\] where \[\mathbf{n}=(0,\ldots,0,1)\in\mathbb{R}^{n+1}\] is the northpole of $S^n$ is a special case of an inversion: Let us consider the hypersphere $S\subset\mathbb{R}^{n+1}$ with center $\mathbf{n}$ and radius $r=\sqrt{2}$ and look at the image of \[\mathbb{R}^n =\left\{\mathbf{x}\in\mathbb{R}^{n+1}\,{\large … Continue reading

## Conformal Maps II: Inversions

Let $M\subset \mathbb{R}^n$ be a domain. A smooth map $f:M \to \mathbb{R}^n$ is called conformal if there is a smooth function $\phi:M\to \mathbb{R}$ and a smooth map $A$ from $M$ into the group $O(n)$ of orthogonal $n\times n$-matrices such that … Continue reading

## Conformal Maps I: Holomorphic Functions

If $M\subset\mathbb{R}^2$ is a plane domain and the image of parametrized surface $f:M\to \mathbb{R}^3$ is contained in \[\mathbb{R}^2=\{(x,y,z)\in \mathbb{R}^3\,\,|\,\, z=0\}\] then the defining equations of a conformal map \begin{align*}\left|f_u\right|&=\left|f_v\right| \\\\ \langle f_u,f_v\rangle &=0\end{align*} imply that $\left|f_v\right|$ arises from $\left|f_v\right|$ by … 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