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 Triangulated surfaces and domains
 Laplace operator 1
 Tutorial 10 – Discrete minimal surfaces
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 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
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Author Archives: pinkall
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
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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
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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
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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
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Conformal Parametrizations of Surfaces
In the context of surfaces the strict analog of arclength parametrized curves is an isometric immersion \[f: M\to \mathbb{R}^3\] of a standard surface into $\mathbb{R}^3$. Here “isometric” means that lengths of curves and intersection angles of curves on the surface … Continue reading
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ArclengthParametrized Curves
A parametrized curve $\gamma_[0,L]\to\mathbb{R}^3$ is called parametrized by arclength provided that $\gamma(t)$ moves with unit speed if we interpret $t$ as time: $\left\gamma'(t)\right=1$ for all $t\in [0,L]$. Sampling an arclengthparametrization $\gamma$ at evenly spaced points $t=\frac{mL}{n}$ for integer values of $m$ … Continue reading
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Sampled Parametrized Curves
In the last post we created geometric objects from scratch, including the underlying combinatorics. In most situations it is much more convenient to start with already existing geometry and transform it. This approach is similar to the standard way of … Continue reading
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Creating Geometry From Scratch
Here we explain how to generate geometry procedurally. There are two node types that allow tho do this: Nodes of type Attribute Wrangle allow to specify geometry using the programming language VEX. Nodes of type Python allow to do the same in the … Continue reading
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Combinatorial Geometry in Houdini
As primitive geometric objects (objects that do not arise as combinations of other objects) Houdini supports also round spheres, cylinders (called tubes in Houdini), volumes and other things. We will focus here on Houdini’s implementation of the combinatorial complexes described … Continue reading
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Combinatorial Geometry: Simplicial Complexes
While (for good reasons) we have restricted our treatment of combinatorial cell complexes to the twodimensional case, the theory of $n$dimensional simplicial complexes is rather straightforward: Definition: A simplicial complex is a finite set $P$ together with a set $\mathcal{S}$ … Continue reading
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