Category Archives: Lecture

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

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

Posted in Lecture | Comments Off on Conformal Maps I: Holomorphic Functions

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

Posted in Lecture | Comments Off on Conformal Parametrizations of Surfaces

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

Posted in Lecture | Comments Off on Arclength-Parametrized Curves

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

Posted in Lecture | Comments Off on Sampled Parametrized Curves

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

Posted in Lecture | Comments Off on Creating Geometry From Scratch

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

Posted in Lecture | Comments Off on Combinatorial Geometry in Houdini

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

Posted in Lecture | Comments Off on Combinatorial Geometry: Simplicial Complexes