Category Archives: Tutorial

The Biot-Savart formula provides for a given set of curves \(\gamma = (\gamma_1,\ldots,\gamma_m)\) a divergence-free velocity field whose vorticity is concentrated along these curves. This led to a simple algorithm to model fluids on entire \(\mathbb R^3\). Here we present … Continue reading →

The velocity field of a fluid in \(\mathbb R^3\) (at rest at infinity) can be reconstructed from the vorticity by the Biot-Savart formula. Let us assume a fluid with vorticity concentrated along a filaments: Given a curve \(\gamma\colon \mathbb S^1 … Continue reading →

Let \(\gamma\colon \mathbb S^1 \to \mathbb R^3\) be a closed space curve and \(m\in\mathbb R^3\setminus \gamma(\mathbb S^1)\). The solid angle \(\Omega_{\gamma}(m)\) of \(\gamma\) subtended to the point \(m\) is defined as the (signed) spherical area enclosed by the projection \(\hat\gamma\) … Continue reading →

In class you defined the Darboux transformation \(\eta\colon \mathbb R\to \mathbb R^3\) of a smooth space curve \(\gamma\colon \mathbb R\to \mathbb R^3\). By construction, the distance between corresponding points is constant. Or phrased differently, there is \(S\colon \mathbb R \to … Continue reading →

The goal of this post is to implement certain time-continuous flows on discrete space curves. By a time-continuous flow of a closed discrete space curve \(\gamma\colon \mathbb Z/n\mathbb Z \to \mathbb R^3\) we mean the continuous solution of an equation\[\dot\gamma … Continue reading →

Under a discrete space curve \(\gamma\) we understand a (finite or periodic) sequence \(\gamma_i\) of points in \(\mathbb R^3\). The \(i\)-th edge vector is then denoted by \(e_i = \gamma_{i+1} – \gamma_i\) and has length \(\ell_i = |e_i|\). If \(\ell_i\neq … Continue reading →

In this very first tutorial we will describe how to solve initial value problems using Houdini. We will first play this through for an first order ODE – the Lorenz system – before you apply the solver to compute the … Continue reading →