Hyperbolic Half-Plane Geometry Environment
Hyperbolic Geometry in the Beltrami-Poincare Half-Plane Model
This activity provides an environment for exploring Hyperbolic Geometry. Hyperbolic Geometry is a geometry with all of the postulates of Euclidean Geometry, except for the Parallel Postulate. In Hyperbolic Geometry there are infinitely many parallel lines to a given line through a given point.
The Half-plane model is one of several isomorphic models for Hyperbolic Geometry. The points in the geometry are the points in an open Euclidean half-plane (those with positive y-coordinates). The horizontal line (x-axis) is the edge of the Hyperbolic plane. These points are at infinity and are not part of the Hyperbolic plane. Similarly, the shaded region (y<0) is not included in the plane. Do not place points on the edge or in the shaded region. Work exclusively in the white region.
Be sure to open the activity in the app so that you have access to the custom toolbars. Hyperbolic Geometry counterparts to all of the standard Euclidean tools are included in the toolbar, grouped in a similar way to the standard tools.
There are two types of Hyperbolic lines: Euclidean vertical ascending rays with a non-included endpoint on the half-plane edge and Euclidean semi-circles with the non-included endpoints and center on the half-plane edge. Because of these two cases and the fact that intersecting circles gives two different intersection points, some of the tools actually produce different objects, depending on the position of the defining points. The tools have been designed to correctly show the constructed Hyperbolic figures, regardless of the positions of the input points. However, when using these tools to create further constructions, some care needs to be taken to insure that all possible figures will result.
Explore this geometry by experimenting with these tools and using them to create other Hyperbolic Geometry figures. Examine what standard Euclidean Geometry results are still true in Hyperbolic Geometry and which are not.
Although they are not Hyperbolic tools the standard tools are still available under the last group of tools on the right, in case you need them. These typically create Euclidean figures and measurements that may be meaningless in Hyperbolic Geometry.