The Poincaré Disk
Hyperbolic Geometry is any geometrical system which satisfies the axioms of Neutral Geometry (Ruler Postulate, Protractor Postulate, SAS, etc) but in which the Euclidean Parallel Postulate does not hold, and instead the Hyperbolic Parallel Postulate holds:
Given a line L and a point A not on L, there are infinitely many lines through A that are parallel to L.The Poincaré Disk is a model of Hyperbolic Geometry. Here are a few definitions:
- The "plane" is all points inside the Euclidean unit circle.
- A "line" through two points is the unique Euclidean circle (or line) through the points which is perpendicular to the unit circle.
- The "distance" between two points is calculated by a technical formula.
- The "angle" between two lines (or rays or segments) is the Euclidean angle measure of the tangents at the vertex.
In the next few slides we'll go over a few facts in Hyperbolic Geometry (mostly without proof) and then learn about a process for constructing hyperbolic tessellations.