As we have learned, one can visualize hyperbolic geometries as a surface in Euclidean Three-Space and manipulate lines, points, and other objects on those surfaces. But, as someone who has interacted with geometry in a purely euclidean way, hyperbolic geometry seems next to impossible to easily explore.[br]This leads to the question: Is it possible to model hyperbolic geometry on a Euclidean plane?[br]This question was answered in the 19th century, with both the Klein Model and Poincaré Model of hyperbolic geometry.

Both the Klein and Poincare models use a disk [math]\gamma[/math] (a unit circle) to model hyperbolic geometry. All points are contained within the disk, and the boundaries of the disk are ideal endpoints of every line.[br][br]On the Klein model, lines are defined as open chords of the circle, which is the line segment between two points on boundary of the circle without the endpoints being included.[br][br]On the Poincare model, lines are defined as either: 1) open diameters of the disk (open chords that go through the center of the disk) or 2) Open circular arcs orthogonal to the disk.[br][br]On the section below, play around with lines on the Poincare and Klein models.

As the Klein and Poincare Models are modeling hyperbolic geometry, it should be possible to violate Euclid's 5th postulate. [br][br]Remember that a statement equivalent to Euclid's 5th postulate is[br]"Given a line and a point not on the line, at most one line parallel to the given line can be drawn through the point." (Playfair's posulate)[br][br]Use the figures below to show that Euclid's 5th postulate does not hold on the Klein or Poincare Models

Now, let's take a look at one of the largest problems with the Klein model. Angles.[br][br]Perpendicular lines on the Klein model are constructed using the Pole of a chord C. Any chord that, when extended into a line, passes through the pole of C is perpendicular to C.[br][br]Play around with the model below. Both open BC and AD are perpendicular to AB.

Angle measure is complicated on the Klein Model. Angles that look congruent aren't always congruent, and angles that look incongruent could be congruent.[br][br]This is a problem that the Poincare Model does not have. [b]On the Poincare disk, angles are measured in the same way as in Euclidean geometry.[br][/b][br]Because of this, the easiest way to measure angles on the Klein Model is to translate the points to the Poincare Model, and then measure the angle in that model. To do this, we need a way to translate the Klein Model to the Poincare Model (and vice versa).

It is the case that the two models can be related via an isomorphism (a one to one and onto function which maps points and lines from one model to the other while preserving betweenness and congruence). One such isomorphism is given as follows.[br][br]Let S be the unit sphere in [math]\mathbb{R}^3[/math] [br]Let [math]\gamma[/math] be the unit circle on the xy plane. [math]\gamma[/math] represents both the Klein and Poincare model.[br][br]To start, project every point in the Poincare Model onto the southern hemisphere of S stereographically from the north pole of S (i.e. construct a line from the north pole through a point P on the Poincare model. The intersection point P' of line NP and the southern hemisphere of S is the steriographic projection of P from the north pole of S.)[br][br]Now, project orthogonal upward to the disk . (i.e. construct a line through P' orthogonal to the disk. The point P'' where the line intersects the disk is the orthogonal projection of P'. Note that because the disk is contained in the xy-plane, if P' = (a, b, c), then P'' = (a, b, 0.)[br][br]This process maps every point on the Poincare Model to every point on the Klein model (while preserving lines, congruence, and betweenness).[br][br]Take a look at the model below to see how points are mapped from the Poincare to Klein model

This process can be represented by a function F where

This allows us to easily map from the Poincare Disk to the Klein Disk (or vice versa). [br]Explore the model below to see how lines are related between the two models.

Let's use the Poincare Model to try and discover facts about triangles in hyperbolic geometry.

Finally, let's take a look at quadrilaterals on a Poincare Disk.[br]Take some time to explore and discover as much as you can.

Poincare and Klein Models are great ways to develop ideas about hyperbolic geometry from within a Euclidean plane. It makes it easier to understand for our Euclidean-trained brains (it's also very fun to perform compass and straight edge constructions in the Poincare disk. Highly recommended).[br][br]I hope you learned more about hyperbolic geometry and the ways it can be represented.