Poincaré disc model

The Poincaré disc model for $\mathbb{H}^{2}$ is the disc $\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}<1\}$ in which a point is similar to the Euclidean point and a line must be one of the following:

• a diameter (excluding its endpoints) of the unit circle;

• an arc (excluding its endpoints) of a circle such that it intersects the unit circle at two distinct points and the two circles are perpendicular at both intersection points.

The Poincaré disc model has the drawback that lines in the model do not resemble Euclidean lines; however, it has the advantage that it is angle preserving. That is, the Euclidean of an angle within the model is the angle measure in hyperbolic geometry. For this reason, this model is also referred to as the conformal disc model. (See the entry conformal for more details.)

Some points outside of the Poincaré disc model are important for constructions within the model. The following is an example of such:

Let $\ell$ be a line in the Poincaré disc model that is not a diameter of the circle. The pole of $\ell$ is the intersection of the Euclidean lines that are tangent (http://planetmath.org/TangentLine) to the circle at the endpoints of $\ell$.

Note that this matches the definition of pole for the Beltrami-Klein model. Also, poles are important for the same reason that they are important in the Beltrami-Klein model: Given a line $\ell$ that is not a diameter of the Poincaré disc model, one constructs a line perpendicular to $\ell$ by considering Euclidean lines passing through $P(\ell)$. Thus, two disjointly parallel lines $\ell$ and $m$ that are not diameters of the Poincaré disc model, one constructs their common perpendicular by connecting their poles. It is actually much easier to do this construction by finding the poles of the two lines, finding the common perpendicular with respect to the Beltrami-Klein model, then converting the common perpendicular to the Poincaré disc model. See the entry on converting between the Beltrami-Klein model and the Poincaré disc model for more details.

In all pictures in this entry from this point on, blue segments are lines in the Beltrami-Klein model, and red arcs are lines in the Poincaré disc model.

Below is a picture of two disjointly parallel lines $\ell$ and $m$ in the Poincaré disc model, neither of which is a diamter of the unit circle:

Their poles can be found:

The common perpendicular with respect to the Beltrami-Klein model can be found:

From this, the common perpendicular $n$ with respect to the Poincaré disc model can be found:

Title Poincaré disc model PoincareDiscModel 2013-03-22 17:06:56 2013-03-22 17:06:56 Wkbj79 (1863) Wkbj79 (1863) 25 Wkbj79 (1863) Definition msc 51M10 msc 51-00 conformal disc model ConvertingBetweenTheBeltramiKleinModelAndThePoincareDiscModel ConvertingBetweenThePoincareDiscModelAndTheUpperHalfPlaneModel