Monthly Archives: May 2019

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

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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

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Plateau problem

Let $\Sigma = (V,E,F)$ be a triangulated surface (with boundary). A realization of the surface in $\mathbb{R}^3$ is given by a map $p:V \rightarrow \mathbb{R}^3$ such that $p_i,p_j,p_k$ form a non degenerated triangle in $\mathbb{R}^3$ for all $\{i,j,k\} \in \Sigma$, … Continue reading

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Let $M= \left(V, E, F \right)$ be an oriented triangulated surface without boundary and $p:V \rightarrow \mathbb{R}^3$ a realization. In earlier lectures we considered the space of piecewise linear functions on $M$: $W_{PL}:=\left\{ \tilde{f} :M\rightarrow \mathbb{R} \, \big \vert \,\left. … Continue reading Posted in Lecture | Leave a comment Dirichlet Energy Let M be a triangulated domain with the functionspace: \[W_{PL}:=\bigl\{f:M\rightarrow\mathbb{R}\,\bigl\vert\bigr.\,\,\,\left. f\right|_{T_{\sigma}} \mbox{ is affine for all } \sigma \in \Sigma_2 \bigr\}.$ On the interior of each triangle $T_{\sigma}$ in $M$ the gradient of a function $g \in W$ is well … Continue reading

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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

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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

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Smooth Dirichlet Energy and Laplace Operator

Let $M \subset \mathbb{R}^2$ be a domain with smooth boundary $\partial M$ and outpointing normal vector field $N$. For a smooth function $f \in C^{\infty}(M,\mathbb{R})$ the gradient vector field $\mbox{grad} \, f :M \rightarrow \mathbb{R}^2$ is defined as : \[ … Continue reading

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