Triangles in the Half-Plane Model of the Hyperbolic Plane

There are several models of the hyperbolic plane, one of which is the upper half-plane model denoted by H. H is the set of all complex numbers whose imaginary part is positive. In the hyperbolic half-plane, the shortest path between two points is represented by ab arc of a circle whose center lies on the line Im = 0, that is, the lower boundary of the half-plane.

A hyperbolic triangle is a set of three points and the "lines" that join them in the hyperbolic plane. In the applet below, you can move the points around the plane to see different examples of hyperbolic triangles. When two points have the same horizontal coordinate, the line that joins them is a vertical line (essentially, an arc of a circle with infinite radius).

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Applet created with GeoGebra.

Area of a Hyperbolic Triangle

In the hyperbolic half-plane, the area of a triangle is bounded and cannot exceed π. The formula for area is simply π minus the sum of the angles. Since the half-plane model is conformal (preserves angles), you can find each angle of the triangle by computing the intersection angle between circular arcs in radians.

Perimeter of a Hyperbolic Triangle

Distance in the hyperbolic half-plane is defined differently than in Euclidean geometry. If a + bi and c + di represent two points in H, then the distance between a + bi and c + di is given by

cosh^-1[(b² + d² + (a-c)²)/(2bd)]

When a = c, the distance formula simplifies to

|ln(b/d)|

An equivalent formulation of distance can be described in terms of the chord lengths shown in the image below:

To find the perimeter of a hyperbolic triangle you simply sum the hyperbolic distances between points. Unlike area, perimeter and distance are unbounded in the hyperbolic half-plane model.