12.1: Conformal Disc Model
 Page ID
 23654
In this section, we give new names for some objects in the Euclidean plane which will represent lines, angle measures, and distances in the hyperbolic plane. Let us fix a circle on the Euclidean plane and call it absolute. The set of points inside the absolute will be called the hyperbolic plane (or \(h\)plane). Note that the points on the absolute do not belong to the hplane. The points in the hplane will be also called hpoints.
Often we will assume that the absolute is a unit circle.
Definition: Hyperbolic lines
The intersections of the hplane with circlines perpendicular to the absolute are called hyperbolic lines or hlines.
By Corollary 10.5.3, there is a unique hline that passes thru the given two distinct hpoints \(P\) and \(Q\). This hline will be denoted by \((PQ)_h\).
The arcs of hyperbolic lines will be called hyperbolic segments or hsegments. An hsegment with endpoints \(P\) and \(Q\) will be denoted by \([PQ]_h\).
The subset of an hline on one side from a point will be called a hyperbolic halfline (or hhalfline). More precisely, an hhalfline is an intersection of the hplane with arc perpendicular to the absolute that has exactly one of its endpoints in the hplane. An hhalfline starting at \(P\) and passing thru \(Q\) will be denoted by \([PQ)_h\).
If \(\Gamma\) is the circline containing the hline \((PQ)_h\), then the points of intersection of \(\Gamma\) with the absolute are called ideal points of \((PQ)_h\). (Note that the ideal points of an hline do not belong to the hline.)
An ordered triple of hpoints, say \((P,Q,R)\) will be called htriangle \(PQR\) and denoted by \(\triangle_h P Q R\).
Let us point out, that so far an hline \((PQ)_h\) is just a subset of the hplane; below we will introduce hdistance and later we will show that \((PQ)_h\) is a line for the hdistance in the sense of the Definition 1.5.1.
Exercise \(\PageIndex{1}\)
Show that an hline is uniquely determined by its ideal points.
 Hint

Let \(A\) and \(B\) be the ideal points of the hline \(\ell\). Note that the center of the Euclidean circle containing \(\ell\) lies at the intersection of the lines tangent to the absolute at the ideal points of \(\ell\).
Exercise \(\PageIndex{2}\)
Show that an hline is uniquely determined by one of its ideal points and one hpoint on it.
 Hint

Assume \(A\) is an ideal point of the hline \(\ell\) and \(P \in \ell\). Suppose that \(P'\) denotes the inverse of \(P\) in the absolute. By Corollary 10.5.1, \(\ell\) lies in the intersection of the hplane and the (necessarily unique) circline passing thru \(P, A\), and \(P'\)
Exercise \(\PageIndex{3}\)
Show that the hsegment \([PQ]_h\) coincides with the Euclidean segment \([PQ]\) if and only if the line \((PQ)\) pass thru the center of the absolute.
 Hint

Let \(\Omega\) and \(O\) denote the absolute and its center.
Let \(\Gamma\) be the circline containing \([PQ]_h\). Note that \([PQ]_h = [PQ]\) if and only if \(\Gamma\) is a line.
Suppose that \(P'\) denotes the inverse of \(P\) in \(\Omega\). Note that \(O, P\), and \(P'\) lie on one line.
By the definition of hline, \(\Omega \perp \Gamma\). By Corollary 10.5.1, \(\Gamma\) passes thru \(P\) and \(P'\). Therefore, \(\Gamma\) is a line if and only if it pass thru \(O\).
Hyperbolic distance
Let \(P\) and \(Q\) be distinct hpoints; let \(A\) and \(B\) denote the ideal points of \((PQ)_h\). Without loss of generality, we may assume that on the Euclidean circline containing the hline \((PQ)_h\), the points \(A,P,Q,B\) appear in the same order.
Consider the function
\(\delta(P,Q):= \dfrac{AQ \cdot PB}{AP \cdot QB}.\)
Note that the right hand side is a crossratio; by Theorem 10.2.1 it is invariant under inversion. Set \(\delta(P,P)=1\) for any hpoint \(P\). Let us define hdistance as the logarithm of \(\delta\); that is,
\(PQ_h := \ln[\delta(P,Q)].\)
The proof that \(PQ_h\) is a metric on the hplane will be given later. For now it is just a function that returns a real value \(PQ_h\) for any pair of hpoints \(P\) and \(Q\).
Exercise \(\PageIndex{4}\)
Let \(O\) be the center of the absolute and the hpoints \(O\), \(X\), and \(Y\) lie on one hline in the same order. Assume \(OX=XY\). Prove that \(OX_h<XY_h\).
 Hint

Assume that the absolute is a unit circle.
Set \(a = OX = OY\). Note that \(0 < a < \dfrac{1}{2}\), \(OX_h = \ln \dfrac{1+ a}{1 a}\), and \(XY_h = \ln \dfrac{(1 + 2 \cdot a) \cdot (1  a)}{(1  2 \cdot a)\cdot (1 + a)}\). It remains to check that the inequalities
\(1 < \dfrac{1 + a}{1  a} < \dfrac{(1 + 2 \cdot a) \cdot (1  a)}{(1  2 \cdot a)\cdot (1 + a)}\)
hold if \(0 < a < \dfrac{1}{2}\).
Hyperbolic angles
Consider three hpoints \(P\), \(Q\), and \(R\) such that \(P\ne Q\) and \(R\ne Q\). The hyperbolic angle \(PQR\) (briefly \(\angle_h PQR\)) is an ordered pair of hhalflines \([QP)_h\) and \([QR)_h\).
Let \([QX)\) and \([QY)\) be (Euclidean) halflines that are tangent to \([QP]_h\) and \([QR]_h\) at \(Q\). Then the hyperbolic angle measure (or hangle measure) of \(\angle_h PQR\) denoted by \(\measuredangle_h PQR\) and defined as \(\measuredangle XQY\).
Exercise \(\PageIndex{5}\)
Let \(\ell\) be an hline and \(P\) be an hpoint that does not lie on \(\ell\). Show that there is a unique hline passing thru \(P\) and perpendicular to \(\ell\).
 Hint

Spell the meaning of terms "perpendicular" and "hline" and then apply Exercise 10.5.4.