Poincaré Disk Model: Definition of the Model

The Poincaré disk model is one model for hyperbolic non-Euclidean geometry. This geometry can be studied in a manner like Euclidean geometry, with axioms about points and lines, but as for Euclidean geometry, it is helpful to have physical models and models on which to draw. We will call this model the P-model for short.

The building blocks for the P-model are objects in the Euclidean plane, or more properly the inversive plane (since we will consider Euclidean lines and circles to be the same kind of objects.). We will write "circle" when we mean a circle in the sense of inversive geometry (it is either a Euclidean circle or a Euclidean line).

To construct a P-model, start with a Euclidean circle h with center O.

• The points in the P-model are the points inside circle h.
• The lines in the P-model are "arcs" m consisting of the intersection of a "circle" m_ with the interior of h. The "circle" m_ is required to be orthogonal to h! From the Euclidean point of view there are two cases: (1) m_ is a Euclidean circle orthogonal to h and the P-line is the arc of m_ interior to h or (2) m_ is a Euclidean line through the center O, and m is the segment interior to h.

We call the points and lines in the Poincaré model (when it is not clear from the context) P-points and P-lines. Every P-line m is an arc of a Euclidean circle m_ or a segment of a Euclidean line m_. Then m_ is called the support or the supporting circle of m. The supporting circle is not really part of the model, but it will be used in constructions.

The points on the circle h (i.e., on the circle itself, not the interior) are called ideal points. They are not true points of the model. We will see that they represent directions at infinity. They also can be useful in making some constructions in the model even though they are not P-points.

• P-line reflection of a P-point A in a P-line m is the inversion A' of A in the supporting circle of m (this means line reflection of m_ is a Euclidean line). It can be shown that if A is a P-point (i.e., interior to h), then so is A'.
• The angle between P-lines is measured as the usual angle measure between Euclidean circles.

GENERAL CONSTRUCTION NOTES:

(1) For each construction in non-Euclidean geometry, interpret the statement as a construction in the P-model using "circles" and then carry out the construction. Begin by drawing the circle h whose interior is the "universe" that you will be operating in.

(2) For P-lines, you can either draw the whole support circle orthogonal to h and just ignore the part outside or else you can also construct the arc interior to h. The latter is a bit more complicated with Sketchpad, but it looks better (and prints better, since everything is inside h). In practice you may want to keep the center of the support showing in your figure as a construction aid. You can use Hide/Show buttons to hide the supports when they are not needed.