Steve's Hyperbolic Tools Page
There are many models of hyperbolic geometry. Three common models are the Poincaré disk model, the the Poincaré half-plane model, and the Beltrami-Klein model. Within those models, we often want to perform standard constructions, such as bisecting angles, dropping perpendiculars, or drawing circles. However, in the non-Euclidean models, the constructions are not the same constructions as in Euclidean geometry. For example, in the Poincaré models, "lines" are defined to be arcs of certain circles.
On this and other linked pages can be found dynamic geometry tools that automate 10 "standard" non-Euclidean constructions in the three models:
|1.||Construct a non-Euclidean line, given two points on the line.|| |
|2.||Construct a non-Euclidean line segment, given the endpoints of the segment.|| |
|3.||Measure the length of a non-Euclidean line segment.|| |
|4.||Calculate the measure of an angle.|| |
|5.||Construct the bisector of a given angle.|| |
|6.||Construct a perpendicular to a given line through a given point on the line.|| |
|7.||Construct a perpendicular to a given line through a given point not on the line.|| |
|8.||Construct the perpendicular bisector of a non-Euclidean line segment.|| |
|9.||Construct a circle, given its center and a point on the circle.|| |
|10.||Construct a circle, given its center and two points determining the radius of the circle.|
The two most popular dynamic geometry software packages are the commercial software Geometer's Sketchpad and the open-source software GeoGebra.Tools for both packages are available here. Unfortunately, I do not have access to the most recent versions of Geometer's Sketchpad, and the tools are only updated through version 4 of that software.
- Downloadable Geogebra sketches that illustrate each of the 10 "standard" constructions (as well as supporting constructions) in each of the three hyperbolic models. If you want to understand how the constructions are done, this is a great place to start!
- scripts.pdf: An older (2001) description of the hyperbolic construction tools.
This file is in .pdf format and requires a reader
such as Adobe Acrobat, which can be found
- How to install and use the tools:
Links to others' work.
- Teaching Ideas: Some very brief ideas on ways to use the tools in the classroom. For more teaching ideas, you might want to look at the article "The Hyperbolic Toolbox" that I wrote for the Journal of Online Mathematics and its Applications.
Many others have created tools for hyperbolic constructions using dynamic geometry software. Here are links (last checked in April 2009) to some of their work.
- Alexander and Finzer's Geometer's Sketchpad scripts for constructions in the Poincaré disk.
- Bochaca's more recent Geometer's Sketchpad scripts for constructions in the Poincaré half-plane. The page includes a link to Bennett's earlier half-plane tools.
- Tim Peil's Geometer's Sketchpad and Geogebra scripts for constructions in the Poincaré half-plane.
- David Flesner's Cabri Geometry scripts for constructions in the Klein Disk Model.
As GeoGebra has grown in popularity, the number and diversity of tools for hyperbolic geometry has increased dramatically. A simple search of the web or the GeoGebra Tube will yield many other examples of hyperbolic geometry construction tools and sketches.