
Recent Posts
 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
Recent Comments
Archives
Categories
Meta
Monthly Archives: May 2016
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
Posted in Tutorial
Comments Off on Tutorial 5: Lawson’s minimal surfaces and the Sudanese Möbius band
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
Posted in Lecture
Comments Off on The 3Sphere
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
Posted in Tutorial
Comments Off on Tutorial 4: Hyperbolic helicoids
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
Posted in Tutorial
Comments Off on Tutorial 3: Framed Closed Curves
Conformal maps III: Stereographic Projection
Stereographic projection \[\sigma: \mathbb{R}^n \to S^n \setminus \{\mathbf{n}\}\] where \[\mathbf{n}=(0,\ldots,0,1)\in\mathbb{R}^{n+1}\] is the northpole of $S^n$ is a special case of an inversion: Let us consider the hypersphere $S\subset\mathbb{R}^{n+1}$ with center $\mathbf{n}$ and radius $r=\sqrt{2}$ and look at the image of \[\mathbb{R}^n =\left\{\mathbf{x}\in\mathbb{R}^{n+1}\,{\large … Continue reading
Posted in Lecture
Comments Off on Conformal maps III: Stereographic Projection
Conformal Maps II: Inversions
Let $M\subset \mathbb{R}^n$ be a domain. A smooth map $f:M \to \mathbb{R}^n$ is called conformal if there is a smooth function $\phi:M\to \mathbb{R}$ and a smooth map $A$ from $M$ into the group $O(n)$ of orthogonal $n\times n$matrices such that … Continue reading
Posted in Lecture
Comments Off on Conformal Maps II: Inversions
Quaternions
Quaternions, $\mathbb{H}$ are a number system like real or complex numbers but with 4 dimensions. In particular, $\mathbb{H}$ is nothing but $\mathbb{R}^4$ together with a multiplication law. The identification of $\mathbb{H}$ and $\mathbb{R}^4$ is given by: $$ \mathbb{H} = \lbrace … Continue reading
Posted in Lecture
Comments Off on Quaternions
Tutorial 2: Framed Discrete Curves
A discrete curve \(\gamma\) in a space \(\mathrm M\) is just a finite sequence of points in \(\mathrm M\), \(\gamma = (\gamma_0,…,\gamma_{n1})\). In case we have discrete curve \(\gamma\) in \(\mathbb R^3\) we can simply draw \(\gamma\) by joining the points by straight … Continue reading
Posted in Tutorial
Comments Off on Tutorial 2: Framed Discrete Curves
Mandelbrot Set
If you google “mathematics visualization”, the resulting wikipedia page shows the Mandelbrot set as a famous example. The Mandelbrot set \(M\subset{\Bbb C}\) is defined as the following. For each \(c\in {\Bbb C}\), consider the iteration \(z_{n+1}(c) = ({z_n}(c))^2+c\), \(n=0,1,2,\ldots\) and \(z_0(c) = … Continue reading
Posted in Tutorial
Comments Off on Mandelbrot Set
Conformal Maps I: Holomorphic Functions
If $M\subset\mathbb{R}^2$ is a plane domain and the image of parametrized surface $f:M\to \mathbb{R}^3$ is contained in \[\mathbb{R}^2=\{(x,y,z)\in \mathbb{R}^3\,\,\,\, z=0\}\] then the defining equations of a conformal map \begin{align*}\leftf_u\right&=\leftf_v\right \\\\ \langle f_u,f_v\rangle &=0\end{align*} imply that $\leftf_v\right$ arises from $\leftf_v\right$ by … Continue reading
Posted in Lecture
Comments Off on Conformal Maps I: Holomorphic Functions