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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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Category Archives: Lecture
Wave and heatequation on surfaces
Wave equation Let $M$ be a surface and $f : \mathbb{R} \times M \rightarrow \mathbb{R}$ a function. The wave equation is given by: \[\ddot{f} = \Delta f.\] After the space discretization we have $f:\mathbb{R} \times V \rightarrow \mathbb{R},$ $p:= \left(\begin{matrix} … Continue reading
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Partial differential equations involving time
Let $\Omega \subset \mathbb{R}^n$ be a domain, we will consider $\Omega$ to be the “space” and functions: \begin{align} f : \mathbb{R} \times \Omega \rightarrow \mathbb{R}, \\ (t,x) \mapsto f(t,x),\end{align} will be view as time dependent functions \begin{align} & f_t : … Continue reading
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Laplace operator 2
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
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Triangulated surfaces with metric and the 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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Dirichlet energy 2
Let $M = (V,E,F)$ be a triangulated domain in the plane where $V$ denotes the set of vertices, $E$ the set of edges and $F$ the set of triangles. We consider the set of functions on the vertices : \begin{align*} … Continue reading
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Gradient and Dirichlet energy on triangulated domains.
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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Triangulated surfaces and domains
We want to derive a discrete version of the laplace operator defined on triangulated surfaces. At first we will define what a triangulated surface with and without boundary is, and consider triangulated domains of $\mathbb{R}^2$ as an important example. Let … Continue reading
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Laplace operator 1
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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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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The 3Sphere
So far we have seen geometries in 2D and 3D, which are the dimensions we are familiar with. But mathematician have found interesting geometries in higher dimensions and it would be great if we could visualize them. In particular a huge … Continue reading
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