# 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 $$S\colon \mathbb R \to \mathbb S^2 \subset \mathbb R^3$$ such that $\eta=\gamma + \ell S.$Moreover, you proved that Darboux transformations preserve arc length. This led to the following definition of the discrete Darboux transformation: A discrete space curve $$\eta\colon \mathbb Z \to \mathbb R^3$$ is called a Darboux transformation of a discrete space curve $$\gamma\colon \mathbb Z \to \mathbb R^3$$ with rod length $$\ell$$ and twist $$r$$ if $\eta_j = \gamma_j + \ell S_j,$ where $$S\colon \mathbb Z \to \mathbb S^2$$ satisfies$S_{j+1} =(r+\ell S_j-v_jT_j)S_j(r+\ell S_j-v_jT_j)^{-1}.$Here $$v_jT_j = \gamma_{j+1}-\gamma_j$$ with $$v_j=|\gamma_{j+1}-\gamma_j|$$. Compare also with this post from 2013.

The discrete Darboux transformation is easily implemented – given a discrete space curve $$\gamma=(\gamma_0,\ldots,\gamma_{n-1})$$, a point $$p\in\mathbb R^3$$ and $$r\in \mathbb R$$, then we obtain a discrete Darboux transformation $$\eta$$ of rod length $$\ell := |\gamma_0 – p|$$ and twist $$r$$ starting at $$p$$ ($$\eta_0 = p$$) as follows:

1. Set $$S_0 := (\eta_0 – \gamma_0)/\ell$$,
2. iteratively compute $$S$$, and
3. define $$\eta_j := \gamma_j + \ell S_j$$.

As we want to play around with the Darboux transformation we want to have control over the initial data $$p$$ and $$r$$. The parameter $$r$$ can be just given as a node parameter. The initial point $$p$$ can just be given by a geometry containing just a single point which is then made draggable by an edit node. Here is how such a network could look like:

In general, a Darboux transformation of a closed discrete space curve $$\gamma \colon \mathbb Z/n\mathbb Z \to \mathbb R^3$$ is not closed. Though, for given $$\ell$$ and $$r$$ the map $$M_{\ell,r} \colon \mathbb R^3 \to \mathbb R^3$$ sending the initial rod direction $$S_0$$ to the end rod direction $$S_n$$ is a Moebius transformation, a generic discrete space curve will have exactly two closed Darboux transformations corresponding to the fixed points of $$M_{\ell,r}$$. These points could be computed directly from $$M_{\ell,r}$$. Unfortunately if we try to compute $$M_{\ell,r}$$ we run into numerical problems . Fortunately, we quickly converge to a fixed point when running around $$\gamma$$ several times (always computing a new Darboux transformation starting from the end of the previous).

Affectively this amounts to applying $$M_{\ell,r}$$ several times but turns out to be numerically stable and quite fast. So we can compute a closed Darboux transformation of a discrete space curve $$\gamma\colon \mathbb Z/n\mathbb Z \to \mathbb R^3$$ with rod length $$\ell$$ and twist $$r$$ as follows:

1. Compute for sufficiently large $$m \in\mathbb N$$ the direction $$\tilde S_{mn}$$ starting from some initial rod direction $$S_0\in\mathbb S^2$$,
2. Compute the Darboux transformation $$\eta$$ starting at $$\gamma_0+\ell \tilde S_{mn}$$.

Similarly, a second fixed point is found when running $$\gamma$$ backwards (iterating $$M_{\ell,r}^{-1}$$).

Homework (due 19 June). Write a digital asset which computes for a closed discrete space curve a closed Darboux transformation with rod length $$\ell$$ and twist $$r$$. Use a checkbox to specify whether to iterate forward of backward.

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