Category Archives: Tutorial

A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. A simple pendulum is an idealization of a real pendulum—a point of mass \(m\) moving on circle … Continue reading →

In previous tutorials we compute immersed minimal surfaces. These immersions were the critical points of Dirichlet energy. You have seen in the lecture that, if we additionally include a volume constraint, then the critical points have no longer mean curvature … Continue reading →

Let \(M\) be a Riemannian surface and let \(\Delta\colon C^\infty M \to C^\infty M\) denote the corresponding Laplace operator. The second order linear partial differential equation \[\ddot{u} = \Delta u\]is called the wave equation and—as the name suggests—describes the motion of … Continue reading →

We are back at minimal surfaces in \(\mathbb S^3\). This time the goal is to visualize minimal surfaces in \(\mathbb S^3\) first described by Lawson, which are constructed by reflections and rotations of a fundamental piece which is a minimal … Continue reading →

As described in the lecture a charge distribution \(\rho\colon \mathrm M \to \mathbb R\) in a uniformly conducting surface \(M\) induces an electric field \(E\), which satisfies Gauss’s and Faraday’s law\[\mathrm{div}\,E = \rho, \quad \mathrm{curl}\, E = 0.\]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 →

Let \(M\subset \mathbb C\) be an open set and \(V\) a complex vector space with a non-degenerated complex-valued symmetric complex-bilinear form \(\langle.,.\rangle\). A null-curve is a map \(\gamma \colon M \to V\) such that\[\gamma^\ast\langle.,.\rangle = \langle d\gamma,d\gamma\rangle = 0\,.\]Unless explicitly stated … Continue reading →

The real projective plane \(\mathbb R\mathrm P^2\) is obtained by identifying the antipodal points of the a \(2\)-sphere \(\mathbb S^2\), i.e. \(\mathbb R\mathrm P^2 = \mathbb S^2/_\sim\) with equivalence relation given by \[ x\sim y \Leftrightarrow y = \pm x.\] The … 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 →