## 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

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 Posted in Tutorial | Leave a comment ## 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

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