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 ## 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*}\left|f_u\right|&=\left|f_v\right| \\\\ \langle f_u,f_v\rangle &=0\end{align*} imply that $\left|f_v\right|$ arises from $\left|f_v\right|$ 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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## Arclength-Parametrized 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 arclength-parametrization $\gamma$ 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 two-dimensional 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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