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 Wave and heatequation on surfaces
 Partial differential equations involving time
 Tutorial 11 – Electric fields on surfaces
 Laplace operator 2
 Triangulated surfaces with metric and the Plateau problem
 Dirichlet energy 2
 Gradient and Dirichlet energy on triangulated domains.
 Triangulated surfaces and domains
 Laplace operator 1
 Tutorial 10 – Discrete minimal surfaces
 Tutorial 9 – The Dirichlet problem
 Tutorial 8 – Flows on functions
 Tutorial 7 – Visualization of gradient fields
 Random Fourier polynomials
 Tutorial 6: Closetoconformal parametrizations of Hopf tori
 Tutorial 5: Lawson’s minimal surfaces and the Sudanese Möbius band
 The 3Sphere
 Tutorial 4: Hyperbolic helicoids
 Tutorial 3: Framed Closed Curves
 Conformal maps III: Stereographic Projection
 Conformal Maps II: Inversions
 Quaternions
 Tutorial 2: Framed Discrete Curves
 Mandelbrot Set
 Conformal Maps I: Holomorphic Functions
 Conformal Parametrizations of Surfaces
 Parallel Frame for Curves
 ArclengthParametrized Curves
 Sampled Parametrized Curves
 Tutorial 1: Implicit Surfaces with Houdini
 Creating Geometry From Scratch
 Combinatorial Geometry in Houdini
 Combinatorial Geometry: Simplicial Complexes
 Combinatorial Geometry: Cell Complexes
 Scenes with White Background
 Simple Ambient Scenes
 Visualizing Discrete Geometry with Houdini II
 Rendering and Working with Cameras
 Visualizing Discrete Geometry with Houdini I
 Using Houdini on MacBooks
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Author Archives: Felix
Tutorial 11 – 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
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Tutorial 10 – Discrete minimal surfaces
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 9 – The Dirichlet 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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Tutorial 8 – Flows on functions
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 realvalued Fourier polynomials \(\mathcal F_N^{\mathbb R} \subset … Continue reading
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Tutorial 7 – Visualization of gradient fields
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
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Random Fourier polynomials
What is a typical function? The answer of this question certainly depends on the branch of mathematics you are into – while functions in differential geometry are usually smooth, the wiggling graphs appearing at the stock market are far from … Continue reading
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Tutorial 6: Closetoconformal parametrizations of Hopf tori
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 3space by stereographic projection. For tori we will encounter a problem similar to … Continue reading
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Tutorial 5: 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 1parameter family of geodesics: while moving on a geodesic – the axis of the helicoid – another geodesic perpendicular to the axis was … Continue reading
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Tutorial 4: Hyperbolic helicoids
A ruled surface is a surface in \(\mathbb R^3\) that arises from a 1parameter 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
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Tutorial 3: Framed Closed Curves
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
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