13.3: Circles, Horocycles, and Equidistants
Note that according to Lemma 12.3.4 , any h-circle is a Euclidean circle that lies completely in the h-plane. Further, any h-line is an intersection of the h-plane with the circle perpendicular to the absolute. In this section we will describe the h-geometric meaning of the intersections of the other circles with the h-plane.
You will see that all these intersections have a perfectly round shape in the h-plane. One may think of these curves as trajectories of a car with a fixed position of the steering wheel. In the Euclidean plane, this way you either run along a circle or along a line. In the hyperbolic plane, the picture is different. If you turn the steering wheel to the far right, you will run along a circle. If you turn it less, at a certain position of the wheel, you will never come back to the same point, but the path will be different from the line. If you turn the wheel further a bit, you start to run along a path that stays at some fixed distance from an h-line.
Equidistants of h-lines. Consider the h-plane with the absolute \(\Omega\) . Assume a circle \(\Gamma\) intersects \(\Omega\) in two distinct points, \(A\) and \(B\) . Suppose that \(g\) denotes the intersection of \(\Gamma\) with the h-plane.
Let us draw an h-line \(m\) with the ideal points \(A\) and \(B\) . According to Exercise 12.1.1 , \(m\) is uniquely defined.
Consider any h-line \(\ell\) perpendicular to \(m\) ; let \(\Delta\) be the circle containing \(\ell\) .
Note that \(\Delta\perp \Gamma\) . Indeed, according to Corollary 10.5.1 , \(m\) and \(\Omega\) invert to themselves in \(\Delta\) . It follows that \(A\) is the inverse of \(B\) in \(\Delta\) . Finally, by Corollary 10.5.2 , we get that \(\Delta\perp \Gamma\) .
Therefore, inversion in \(\Delta\) sends both \(m\) and \(g\) to themselves. For any two points \(P',P\in g\) there is a choice of \(\ell\) and \(\Delta\) as above such that \(P'\) is the inverse of \(P\) in \(\Delta\) . By the main observation ( Theorem 12.3.1 ) the inversion in \(\Delta\) is a motion of the h-plane. Therefore, all points of \(g\) lie on the same distance from \(m\) .
In other words, \(g\) is the set of points that lie on a fixed h-distance and on the same side of \(m\) .
Such a curve \(g\) is called equidistant to h-line \(m\) . In Euclidean geometry, the equidistant from a line is a line; apparently in hyperbolic geometry the picture is different.
Horocycles. If the circle \(\Gamma\) touches the absolute from inside at one point \(A\) , then the complement \(h=\Gamma\backslash\{A\}\) lies in the h-plane. This set is called a horocycle . It also has a perfectly round shape in the sense described above.
The shape of a horocycle is between shapes of circles and equidistants to h-lines. A horocycle might be considered as a limit of circles thru a fixed point with the centers running to infinity along a line. The same horocycle is a limit of equidistants thru a fixed point to sequence of h-lines that runs to infinity.
Since any three points lie on a circline, we have that any nondegenerate h-triangle is inscribed in an h-circle, horocycle or an equidistant.
Find the leg of an isosceles right h-triangle inscribed in a horocycle.
- Hint
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As usual, we assume that the absolute is a unit circle.
Let \(PQR\) be a hyperbolic triangle with a right angle at \(Q\), such that \(PQ_h = QR_h\) and the vertices \(P, Q\), and \(R\) lie on a horocycle.
Without loss of generality, we may assume that \(Q\) is the center of the absolute. In this case \(\measuredangle_h PQR = \measuredangle PQR = \pm \dfrac{\pi}{2}\) and \(PQ = QR\).
Note that Euclidean circle passing thru \(P, Q\), and \(R\) is tangent to the absolute. Conclude that \(PQ = \dfrac{1}{\sqrt{2}}\). Apply Lemma 12.3.2 to find \(PQ_h\).