Nonlinear Tessellations

Background

We shall be using a model of hyperbolic geometry developed by Henri Poincare.

The space of this model is an open disc, the interior region bounded by a circle.

Points in this model are the (Euclidean) points of the disc, but in the model straight lines are not necessarily straight Euclidean lines.

The phrase "straight line" has such a strong visual meaning for most of us that we will use a different terminology to avoid confusion.

In any space where distance can be measured, a geodesic between two points is the shortest-distance path between those points.

For the Euclidean plane, geodesics are the straight line segments.

For the Poincare disc, geodesics are of two types:

(i) segments of Euclidean straight lines through the center of the disc, and

(ii) arcs of circles that intersect the boundary of the disc in right angles.

Poincare disc with geodesics through points A and B.

Return to background.