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# 12.1: Conformal Disc Model

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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 h-plane. The points in the h-plane will be also called h-points.

Often we will assume that the absolute is a unit circle.

Definition: Hyperbolic lines

The intersections of the h-plane with circlines perpendicular to the absolute are called hyperbolic lines or h-lines.

By Corollary 10.5.3, there is a unique h-line that passes thru the given two distinct h-points $$P$$ and $$Q$$. This h-line will be denoted by $$(PQ)_h$$.

The arcs of hyperbolic lines will be called hyperbolic segments or h-segments. An h-segment with endpoints $$P$$ and $$Q$$ will be denoted by $$[PQ]_h$$.

The subset of an h-line on one side from a point will be called a hyperbolic half-line (or h-half-line). More precisely, an h-half-line is an intersection of the h-plane with arc perpendicular to the absolute that has exactly one of its endpoints in the h-plane. An h-half-line starting at $$P$$ and passing thru $$Q$$ will be denoted by $$[PQ)_h$$.

If $$\Gamma$$ is the circline containing the h-line $$(PQ)_h$$, then the points of intersection of $$\Gamma$$ with the absolute are called ideal points of $$(PQ)_h$$. (Note that the ideal points of an h-line do not belong to the h-line.)

An ordered triple of h-points, say $$(P,Q,R)$$ will be called h-triangle $$PQR$$ and denoted by $$\triangle_h P Q R$$.

Let us point out, that so far an h-line $$(PQ)_h$$ is just a subset of the h-plane; below we will introduce h-distance and later we will show that $$(PQ)_h$$ is a line for the h-distance in the sense of the Definition 1.5.1.

Exercise $$\PageIndex{1}$$

Show that an h-line is uniquely determined by its ideal points.

Hint

Let $$A$$ and $$B$$ be the ideal points of the h-line $$\ell$$. 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 h-line is uniquely determined by one of its ideal points and one h-point on it.

Hint

Assume $$A$$ is an ideal point of the h-line $$\ell$$ 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 h-plane and the (necessarily unique) circline passing thru $$P, A$$, and $$P'$$

Exercise $$\PageIndex{3}$$

Show that the h-segment $$[PQ]_h$$ 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 h-line, $$\Omega \perp \Gamma$$. 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 h-points; 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 h-line $$(PQ)_h$$, 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 cross-ratio; by Theorem 10.2.1 it is invariant under inversion. Set $$\delta(P,P)=1$$ for any h-point $$P$$. Let us define h-distance as the logarithm of $$\delta$$; that is,

$$PQ_h := \ln[\delta(P,Q)].$$

The proof that $$PQ_h$$ is a metric on the h-plane will be given later. For now it is just a function that returns a real value $$PQ_h$$ for any pair of h-points $$P$$ and $$Q$$.

Exercise $$\PageIndex{4}$$

Let $$O$$ be the center of the absolute and the h-points $$O$$, $$X$$, and $$Y$$ lie on one h-line 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 h-points $$P$$, $$Q$$, and $$R$$ such that $$P\ne Q$$ and $$R\ne Q$$. The hyperbolic angle $$PQR$$ (briefly $$\angle_h PQR$$) is an ordered pair of h-half-lines $$[QP)_h$$ and $$[QR)_h$$.

Let $$[QX)$$ and $$[QY)$$ be (Euclidean) half-lines that are tangent to $$[QP]_h$$ and $$[QR]_h$$ at $$Q$$. Then the hyperbolic angle measure (or h-angle measure) of $$\angle_h PQR$$ denoted by $$\measuredangle_h PQR$$ and defined as $$\measuredangle XQY$$.

Exercise $$\PageIndex{5}$$

Let $$\ell$$ be an h-line and $$P$$ be an h-point that does not lie on $$\ell$$. Show that there is a unique h-line passing thru $$P$$ and perpendicular to $$\ell$$.

Hint

Spell the meaning of terms "perpendicular" and "h-line" and then apply Exercise 10.5.4.