Geometry :: Notes :: Optional Topics
1 Optional Topics
1.1 Hyperbolic Geometry
1.1.1 Introduction
Let \(z_0,z_1,z_2\) and \(z_3\) be four distinct point is \(\mathbb{C}\), and \(T\) a Mobius transformation. Then \((z_0,z_1,z_2,z_3) = (Tz_0,Tz_1,Tz_2,Tz_3)\).
The cross ratio \((z_0,z_1,z_2,z_3)\) is real if and only if \(z_0,z_1,z_2\), and \(z_3\) all line on a cline.
Let \(C\) be a cline containing three distinct points \(z_1,z_2\), and \(z_3\). Two points \(z\) and \(z^\star\) are symmetric with respect to \(C\) if \((z^\star,z_1,z_2,z_3) = \overline{(z,z_1,z_2,z_3)}\).
If \(T\) is a Mobius transformation and \(T\) maps the unit disk \(D\) onto itself, then \(T\) is of the form
\[ Tz = e^{i\theta}\frac{z-z_0}{1-\overline{z_0}z} \]
for some \(\theta \in \mathbb{R}\) and some \(z_0 \in D\).
Let \(\mathcal{H}\) be the set of Mobius transformations mapping the unit disk \(D\) onto itself, that is
\[ \mathcal{H} = \left\{z \mapsto e^{i\theta}\frac{z-z_0}{1-\overline{z_0}z} : \theta \in \mathbb{R}, z_0 \in D\right\}. \]
The pair \((D, \mathcal{H})\) is a model for hyperbolic geometry. When talking of hyperbolic geometry, \(D\) is called the hyperbolic plane, and \(\mathcal{H}\) is called the hyperbolic transformation group.
A hyperbolic strait line is a Euclidean circle of line in \(\mathbb{C}\) that intersects the unit circle at right angles.
In hyperbolic geometry, all hyperbolic strait lines are congruent. Two points in the hyperbolic plane determine a unique hyperbolic start line.
Point on the unit circle are called ideal points.
Two hyperbolic lines are parallel if the don't intersect in \(D\), but share and ideal point.
Two hyperbolc lines are hyperparallel if they don't intersect in \(D\), and don't have an ideal point in common.
1.2 More Mobius Geometry
1.2.1 Cline Types
Let \(A\) and \(B\) be points in \(\mathbb{C}^+\).
A type I cline of \(A\) and \(B\) is a cline containing both \(A\) and \(B\).
A cline \(c\) is a type II cline of \(A\) and \(B\) if \(A\) and \(B\) are symmetric with respect to \(c\).
The standard cartesian form of \((x,y)\)-pairs is useful for graphing many objects. For other objects, the polar form of \((r,\theta)\)-pairs is more useful. The usual polar form has two special points, 0 and \(\infty\), and gives one a coordinate system based on these two points.. In this example, we will use the usual polar form to generate a coordinate system about the two points \(P = -4 -2i\) and \(Q = 5+3i\) by applying a Mobius transformation which takes 0 to \(P\) and \(\infty\) to \(Q\).
First we show the usual polar coordinate system. The angular lines differ by \(\pi/12\), start at \(\theta = 0\), and are shown in blue. The circles start at \(r=1\), differ by 1, and are shown in red. Also plotted in black here are the points \(P\) and \(Q\). We can think of the blue lines as circles through 0 and infinity, and hence are tye I clines for the points 0 and infinity. We can think of the red circles as circles about 0, but also as circles about infinity; they are type II clines for the points 0 and infinity.
The result of applying the Mobius transformation taking 0 to \(P\) and \(\infty\) to \(Q\) is shown below. Again the blue are type I clines, and the red are type II clines, this time of the points \(P\) and \(Q\). There are actually more red clines about the point \(P\) but they are not shown (since there aren't many clines around \(\infty\) in the above graphic, the red line corresponds to a circle of radius 1 above).
1.2.2 Spiral Examples
We will examine the transformation \(T: z \mapsto (11/10)e^{\pi/10i}z\). This transformation scales by 1.1, and rotates by \(\pi/10\). It is interesting to look at the iterated application of \(T\) to a starting fixed number. What we mean by this is to look at \(\{T^0z_0, T^1z_0, T^2z_0 \ldots T^{100}z_0\}\), where \(T^0 z_0 = z_0\) and \(T^nz_0 = T(T^{n-1}z_0)\) for \(n\geq 1\). The graph for \(z_0 = 1\) is shown below.
The second plot shows \(\{S^0z_1, S^1z_1, S^2z_1 \ldots S^{100}z_1\}\), where \(S: z \mapsto RTR^{-1}z\) and \(R: z \mapsto (z-1)/(z+1)\), with \(z_1 = -96/100\). Notice that \(R(0) = -1\) and \(R(\infty) = 1\). Thus \(S\) takes maps -1 to 0, applies the transformation \(T\) and then maps everything back to -1. The result is a spiral with source \(z=-1\) and sink \(z=1\).