# Author Archives: Felix

## Tutorial 7 – Obstacles

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

## Tutorial 6 – Biot-Savart field of polygons

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

## Tutorial 5 – Solid angle of space curves

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

## Tutorial 4 – Darboux transformations

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

## Tutorial 3 – Möbius tangent and vortex filament flow

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

## Tutorial 2 – Frames and tubes

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

## Tutorial 1 – Solving differential equations with RK4

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