Poincaré disk model

If P and Q are on a diameter of the boundary circle that diameter is the hyperbolic line.

The vertical bars indicate Euclidean length of the line segment connecting the points between them in the model (not along the circle arc), ln is the natural logarithm.

Another way to calculate the hyperbolic distance between two points is

Such a distance function is defined for any two vectors of norm less than one, and makes the set of such vectors into a metric space which is a model of hyperbolic space of constant curvature −1. The model has the conformal property that the angle between two intersecting curves in hyperbolic space is the same as the angle in the model.

An orthonormal frame with respect to this Riemannian metric is given by

Compare the formulas for stereographic projection between a sphere and a plane.

A basic construction of analytic geometry is to find a line through two given points. In the Poincaré disk model, lines in the plane are defined by portions of circles having equations of the form