Category Archives: Tutorial

As described in the lecture a charge distribution $\rho :\mathrm{M}\to \mathbb{R}$ in a uniformly conducting surface $M$ induces an electric field $E$, which satisfies Gauss’s and Faraday’s law$$\mathrm{div}E=\rho ,\mathrm{curl}E=0.$$In particular, on a simply … Continue reading →

In the lecture we have defined what we mean by a discrete minimal surface. The goal of this tutorial is to visualize such minimal surfaces. Let $\mathrm{M}$ be a discrete surface with boundary and let $V,E,F$ denote the set of vertices, … Continue reading →

In the lecture we saw that the Dirichlet energy has a unique minimizer among all functions with prescribed boundary values. In this tutorial we want to visualize these minimizers in the discrete setting. Let $\mathrm{M}\subset {\mathbb{R}}^2$ be a triangulated surface … Continue reading →

Last time we looked at the gradient flows of functions defined on the torus ${\mathrm{T}}^n$. This time we will look at flows on the space of Fourier polynomials ${\mathcal{F}}_N$. Let us first restrict ourselves to the real-valued Fourier polynomials \(\mathcal F_N^{\mathbb R} \subset … Continue reading →

In class we discussed how to generate smoothed random functions on the discrete torus and how to compute their discrete gradient and the symplectic gradient. In this tutorial we want to visualize the corresponding flow. As described in a previous … Continue reading →

In this tutorial we want to construct Hopf cylinders and Hopf tori. These are flat surfaces in ${\mathrm{S}}^3$ and allow for an easy conformal parametrization when mapped to Euclidean 3-space by stereographic projection. For tori we will encounter a problem similar to … Continue reading →

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 →

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 →

A closed discrete curve $\gamma$ is map from a discrete circle ${\mathfrak{S}}_n^1=\{z\in \mathbb{C}\mid z^n=1\}$, $n\in \mathbb{N}$, into some space $\mathrm{M}$. In some situations it is more convenient to consider the discrete circle just as $\mathbb{Z}/n\mathbb{Z}$,\[\mathbb … Continue reading →

A discrete curve $\gamma$ in a space $\mathrm{M}$ is just a finite sequence of points in $\mathrm{M}$, $\gamma =({\gamma }_0,\dots ,{\gamma }_{n-1})$. In case we have discrete curve $\gamma$ in ${\mathbb{R}}^3$ we can simply draw $\gamma$ by joining the points by straight … Continue reading →