To describe the Euclidean plane in terms of complex numbers, the basic idea is to represent each point with cartesian coordinates (x, y) by a single complex number: . A single fictional point, called the point at infinity, is adjoined to this plane and denoted by the “number” ∞. This point is visualized to be infinitely far away from all other points.

Most configurations in this extended plane can then be described in terms of cross ratios. The cross ratio of any four distinct points a, b, c, d, in the plane is the complex number given by

If one of the points, say a, is the point at infinity, then the cross ratio is defined to be the limit of the above expression as a becomes infinite, i.e. since

then

.

 

You can use cross ratios to determine if points lie on a circle or line: four points lie on a circle whenever their cross ratio is real; three points lie on a line whenever their cross ratio with ∞ is real. Angles and ratios of distances can also be expressed in terms of cross rations. (The discussion for the mathematical underpinnings of the Lester Circle contains more details and references.)

The Poincaré universe can be modeled as the interior of the unit disk |z| < 1 in this extended plane. Lines in the Poincaré universe are sections of this disc with euclidean circles or lines perpendicular to the boundary of this disk. Distances along Poincaré lines and angles between Poincaré lines can be defined in terms of these euclidean circles and lines, and can be measured by formulas involving cross ratios. Isometries of the Poincaré universe (transformations which preserve its metrical structures) can be described as a certain class of linear fractional transformations of the euclidean plane which preserve cross ratios. And so on - most things of interest in the Poincaré universe can be expressed in terms of cross ratios of points in the extended euclidean plane.

Many objects and configurations in the Poincaré universe resemble their counterparts in the euclidean plane – a Poincaré circles is a euclidean circle, for example (though its centre is not its euclidean centre). Other properties of the Poincaré universe are different from their euclidean counterparts – the angle sum of a Poincaré triangle is always less than 180°, for example. What is most interesting – and impressive to students – is that the Poincaré universe is “locally euclidean” – in a small enough region, it is impossible to perform measurements accurate enough to distinguish Poincaré geometry from euclidean geometry. Small enough portions of Poincaré lines appear straight, for example, and small enough triangles have angle sums indistinguishable from 180°.