# Monthly Archives: April 2016

## Parallel Frame for Curves

If you have a polygonal curve, say arc-length parameterized, you might want to visualize the curve as the following rendering This can be done in Houdini by the “copy” SOP node. The “copy” node has two input and one output.  The … 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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## Tutorial 1: Implicit Surfaces with Houdini

It is amazingly simple to create high-quality renderings with Houdini. Several small tricks to achieve nice renderings of geometries in a mathematical context are already described in the first posts of this blog. If you not have looked at it yet you should … 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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## Combinatorial Geometry: Cell Complexes

Roughly speaking, Combinatorial complexes play a similar role in the discrete world as differentiable manifolds in the smooth world. They are able to capture the “intrinsic” properties of a geometric object, i.e. those properties that are independent of any embedding … Continue reading