Again we consider curves in in the real plane. Last time we regarded the real plane as a subset of the real projective plane. This time we want to regard it as a complex line, i.e. a \(1\)-dimensional complex vector space, and hence as a part of the Riemann sphere \(\overline{\mathbb{C}}=\mathbb{C}\cup\{\infty\}\). The reason is that we are interested in Darboux transforms of plane curves.
In the smooth world two arc length parametrized curves \(\gamma\) and \(\eta\) are Darboux tranforms of each other if the curve \(T := \gamma-\eta\)
- lies on a circle of radius \(l\),
- is not constant.
The second condition simply prevents that one curve is just a translation of the other. The smooth definition directly translates to the discrete world. In the following we will call \(l\) the rodlength.
Definition (Discrete Euclidean Darboux Transform). Let \(\gamma\) and \(\eta\) be two discrete plane curves with the same number of vertices and such that corresponding edges have equal length. Then \(\gamma\) and \(\eta\) are Darboux transforms of each other if its difference \(T=\gamma-\eta\) lies on a circle of radius \(l\) and is not constant.
Let \(\gamma, \gamma_+, \eta\) be given. The subscript \(+\) shall be thought of as a shift by \(1\) later. We want to construct \(\eta_+\). If we look at the possible configurations we see that there exist two distinct points, that satisfy the first condition. Exactly one of both satisfies also the second condition. The construction of a so called Darboux butterfly is shown in the following picture.
Hence if we start with a discrete curve \(\gamma\) and an arbitrary point \(\eta_0\) we can successively construct a Darboux transform that starts at \(\eta_0\).
Now, if we treat the real plane as complex line, then both conditions can be parsed in just one. Therefore we must first introduce the cross ratio, which is not defined on \(\mathbb C\), but on the Riemann sphere \(\overline{\mathbb{C}}\). This will become clear when we look at its
Definition (Cross Ratio). Let \(z,u,v,w\in\overline{\mathbb{C}}\) be distinct points. The cross ratio \(\{z,u,v,w\}\) of the four points is defined as follows\[\{z,u,v,w\}:=\frac{(u-v)(w-z)}{(z-u)(v-w)}.\]
It has some properties that are quiet interesting for our purpose, but first the following
Definition (Moebius transformation). A Moebius transformation is a map \(f\colon \overline{\mathbb{C}}\to \overline{\mathbb{C}}\) of the form\[ z \mapsto \frac{az+b}{cz+d},\quad ad-bc\not=0, \, a,b,c,d\in \mathbb{C}.\]Here we set\[f(\infty)=\frac{a}{c},\quad f(-\frac{d}{c})=\infty.\]
As Euclidean geometry is geometry that is invariant under Euclidean transformations, i.e. essentially translations and rotations, Moebius geometry is geometry that is invariant under Moebius transformations, i.e. essentially translations, stretch rotations and inversions in circles.
We shall list some properties here: Moebius transformations
- build a group (composition of functions as multiplication)
- are orientation preserving
- can be decomposed into translations, stretch rotations and inversions
- map circles to lines or circles
- are uniquely determined by its values at three different points
- different from the identity have exactly one or two fixed points
Note: With Moebius geometric eyes one cannot distinguish between straight lines and circles anymore. They can be mapped on each other by a Moebius transformation. Hence, we will simply speak only of (generalized) circles from now on – in this context a straight line is seen as a circle of infinite radius.
Here is a video to get an impression what a Moebius transformation can do. It shows quiet well what is going on.
Let’s come back to the cross ratio. The cross ratio is a Moebius geometric invariant, i.e. if \(f\) denotes a Moebius transformation, then \[\{z,u,v,w\}=\{f(z),f(u),f(v),f(w)\},\quad \forall z,u,v,w \in \overline{\mathbb{C}}.\]
We immediately obtain two more properties of the cross ratio:
- \(\{z,u,v,w\}\in\mathbb{R}\) if and only if the four points lie on a circle (or a line)
- \(\{z,u,v,w\}<0\) if and only if the four points are in successive order
Now let us consider the following map. For three given points \(u,v,w \in \mathbb{C}\) we define a map as follows\[z\mapsto \{u,v,z,w\}.\] This is a Moebius transformation and maps the circumcircle of the three points to the real line. In particular:\[ u \to 1,\, v \to 0,\, w \to \infty.\]
As a Moebius transformation it is in particular bijective and hence for every given \(\delta\in\overline{\mathbb{C}}\) we can find a unique \(z_\delta \in\overline{\mathbb{C}}\) such that\[\{u,v,z_\delta,w\}=\delta.\]
This is interesting for us since we have to solve exactly such a problem in the Darboux transform construction: We know the cross ratio of the points \(\gamma,\gamma_+,\eta_+,\eta\). You can easily convince yourself that the absolute value of the cross ratio has to be equal to \(\frac{l^2}{s^2}\), where \(s\) denotes the length of the edge between \(\gamma_+-\gamma\). Further we know by construction that the points \(\gamma,\gamma_+,\eta_+,\eta\) lie on a common circle and hence the cross ratio has to be real – that specifies two points on the circle. Finally, we know that the points are not in cyclic order on this circle, i.e. the cross ratio has to be positive. Altogether we see that \(\eta\) and \(\gamma\) are Darboux transforms of each other with rodlength \(l\) if and only if\[ \{\gamma,\gamma_+,\eta_+,\eta\}=\frac{l^2}{s^2},\quad s=|\gamma_+-\gamma|.\]
The notion of a Darboux transform can be generalized by allowing complex rodlengths.
Definition (Discrete generalized Darboux Transform). Let \(\gamma\) and \(\eta\) be two discrete plane curves with the same number of vertices such that corresponding edges have equal lengths \[s=|\gamma_+-\gamma|=|\eta_+-\eta|.\] Then \(\gamma\) and \(\eta\) are Darboux transforms of each other with parameter \(l\in\mathbb{C}\) if\[\{\gamma,\gamma_+,\eta_+,\eta\}=\frac{l^2}{s^2}.\]
One can check that the map \(f\) defined by \(\{u,v,f(z),z\}=\delta\) is a Moebius transformation. Hence we can construct Darboux transform to a given discrete curve \(\gamma\) by successively applying Moebius transformations that sit on the edges of \(\gamma\). To construct the transformations explicitly we will slightly change our point of view once more. The point is that a Moebius transformation can be regarded as a complex \(2\times2\)-matrix, what will simplify the implementation. How does this work?
The Riemann sphere can be identified with the complex projective line \(\mathbb{CP}^1\) by the following map\[ \mathbb{CP}^1 \ni \begin{bmatrix}z_1\\z_2\end{bmatrix} \longleftrightarrow \frac{z_1}{z_2} \in \overline{\mathbb{C}}.\] The coordinates one gets under this identification are refered to as homogeneous coordinates, which we will denote by capital letters from now on. Under this identification, a Moebius transformations is then given by an invertible complex \(2\times 2\) matrix:\[\left(z\mapsto\frac{az+b}{cz+d}\right) \longleftrightarrow \left( \begin{bmatrix} z_1 \\ z_2\end{bmatrix} \mapsto \begin{bmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix}z_1\\ z_2\end{pmatrix}\end{bmatrix}\right).\] In fact, this is a group isomorphism between the group of Moebius transformations and the projective special linear group \(PSL(2,\mathbb C)=SL(2,\mathbb C)/_{\{\pm Id\}}\).
For given \(\delta\in\mathbb{C}\) we search for the unique Moebius transformation \(f\) that satisfies the equation\[\{\gamma,\gamma_+,f(\eta),\eta\}=\delta.\] The corresponding matrix is easily given in terms of the homogeneous coordinates \[\gamma=[\Gamma],\quad \gamma_+=[\Gamma_+],\quad\eta=[\text{H}].\]
Claim: The Moebius transformation \(f\) is given by the matrix \(A\) determined by the following equations:\[ A\Gamma=\Gamma, \quad A\Gamma_+= \lambda\Gamma_+,\]where\[\delta=\frac{1}{1-\lambda}.\]
To see this, first let us check how the cross ratio looks in homogeneous coordinates: \begin{align*}\left\{Z,U,V,W\right\}&=\left\{\frac{z_1}{z_2},\frac{u_1}{u_2},\frac{v_1}{v_2},\frac{w_1}{w_2}\right\}\\&=\frac{(\frac{u_1}{u_2}-\frac{v_1}{v_2})(\frac{w_1}{w_2}-\frac{z_1}{z_2})}{(\frac{z_1}{z_2}-\frac{u_1}{u_2})(\frac{v_1}{v_2}-\frac{w_1}{w_2})}\\&=\frac{(u_1v_2-v_1u_2)(w_1z_2-z_1w_2)}{(z_1u_2-u_1z_2)(v_1w_2-w_1v_2)}\\&=\frac{\det(U,V)\det(W,Z)}{\det(Z,U)\det(V,W)}.\end{align*}
Since \(\Gamma\) and \(\Gamma_+\) are linearly independent they build a basis of \(\mathbb{C}^2\)and \(A\) can be written as follows:\[ A= \tfrac{1}{\det(\Gamma,\Gamma_+)}\left(\det(.,\Gamma_+)\Gamma+\lambda\det(\Gamma,.)\Gamma_+\right).\]Now let us plug this into the equation to get the right \(\lambda\). We find \[\delta= \{\Gamma,\Gamma_+,A\text{H}, \text{H}\}=\frac{\det(\Gamma_+,A\text{H})\det(\text{H},\Gamma)}{\det(\Gamma,\Gamma_+)\det(A\text{H},\text{H})}=\frac{1}{1-\lambda}.\]
Homework. Write a plugin DarbouxTransformationPlugin that displays the Darboux transform of a given discrete curve to a given initial value. The initial value shall be movable. Exclude all calculations to a class DarbouxTransformationUtility that provides static methods. It shall be able to calculate for a given \(\delta\in\mathbb{C}\) the Moebius transformation from one vertex to another and to return the Darboux transform for given curve and initial point.
Above a picture how this finally can look like. Usually a Darboux transform of a curve is not closed. This happens only at some points. To get this picture I had to play around a while. The thin lines connect the corresponding points of \(\gamma\) and \(\eta\). Even if not drawn one can see their envelope – it’s a tractrix.