13.3: Circles, Horocycles, and Equidistants

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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.

Exercise $$\PageIndex{1}$$

Find the leg of an isosceles right h-triangle inscribed in a horocycle.

Hint

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 $$PQ_h$$.

This page titled 13.3: Circles, Horocycles, and Equidistants is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Anton Petrunin via source content that was edited to the style and standards of the LibreTexts platform.