Hyperbolic Geometry: Poincaré Disc

One of the most common models used to visualize hyperbolic geometry is the Poincaré disc.  The Poincaré disc maps the point at infinity of a hyperbolic space to a circle where hyperbolic lines are represented as arcs of circles intersecting the Poincaré disc at 90 degrees.  As we move away from the origin of a hyperbolic space, the space itself expands due to negative curvature, so as we reach the perimeter of the Poincaré disc, the scale of the space changes dramatically, subdividing into an infinite number of pieces.

Typically, the Poincaré disc is given a radius of 1, but any radius will work.  There are quite a number of other models for hyperbolic space, including the Klein disc model and the Weierstrass model depicted below.  The Weierstrass model places the hyperbolic plane on a single sheet of a hyperboloid.

The biggest difference between Euclidean space and hyperbolic space is the change in curvature.  This is expressed in the Poincaré model by the mapping of hyperbolic lines to circular arcs.  These arcs must be interior to the Poincaré disc and can only be on circles that have their center outside of or on the disc perimeter and that intersect the disc at 90 degree angles.  Lines passing through the center of the Poincaré disc are on circles of infinite radius and this look like straight lines.  Below are some examples with the supporting constructive geometry visualized:

In these images, the hyperbolic line is black, the Poincaré disc has a thick grey border, and the circle on which the arc lies is light grey and thin.  Notice the dramatic change in scale of the Poincaré disc relative to the circle constructing the hyperbolic line.  Also notice how each line, if continued along its circle, would intersect infinity from 2 directions.  In hyperbolic space, each line is parallel to exactly 2 other sets of lines and is “ultraparallel” to many other lines that do not intersect it either directly or asymptotically at infinity like parallel lines.

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