# Author Archives: Felix

## Tutorial 12: The Pendulum

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

## Tutorial 11: CMC Snails, Tires and Soap Bubbles

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

## Tutorial 10: Wave Equation and Verlet Method

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

## Tutorial 9: Minimal Surfaces and Plateau Problem in \(\mathbb S^3\)

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

## Tutorial 8: Electric Fields on Surfaces

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

## Tutorial 7: The Discrete Plateau Problem

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

## Tutorial 6: The Dirichet Problem

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

## Tutorial 5: Holomorphic Null-Curves—Associated Family and Goursat transforms

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

## Tutorial 4: Boy’s Surface

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

## 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