NonEuclid is Interactive Java Software
for Creating Ruler and Compass Constructions
in both the Poincaré Disk and the Upper Half-Plane Models
of Hyperbolic Geometry.

Copyright©: Joel Castellanos, 1994-2009


NonEuclid allows the curious explorer to gain experience in Hyperbolic Geometry and to empirically investigate questions such as: "In Hyperbolic Geometry, are the base angles of an isosceles triangle congruent?"

The NonEuclid software and documentation are accessible to anyone with high school level geometry.

Aside from being interesting in itself, a study of Hyperbolic geometry can, through its novelty, enable a deeper understanding of a formal proof.

Hyperbolic Geometry also has practical aspects such as orbit prediction of objects within intense gradational fields. Hyperbolic Geometry is used in Einstein's General Theory of Relativity and Curved Hyperspace.

Joel Castellanos, Dept. of Computer Science, University of New Mexico
Joe Dan Austin, Dept. of Education, Rice University
Ervan Darnell, Dept. of Computer Science, Rice University

Italian Translation by Andrea Centomo, Scuola Media "F. Maffei", Vicenza

Funding for NonEuclid has been provided by:
The Center for High Performance Software Research (HiPerSoft), Rice University, and
The Institute for Advanced Study / Park City Mathematics Institute

Run NonEuclid using Java Web Start WITHOUT Save and Print Permissions

Some firewalls prevent downloading of jar files. This will result in the error message:
Unable to load resource:

Download NonEuclid.jar

If you have the Java Runtime Enviroment 1.5 or newer installed, then clicking on this link in most browsers will give you the option of running NonEuclid or of saving it to a file. If you choose to save it, then remember where you saved it and from the command line enter:
java" -jar NonEuclid.jar

Running NonEuclid in this way does not require an Internet connection. Additionally, the Java security manager will not prevent NonEuclid from saving or printing files.


1) What is Non-Euclidean Geometry: - Euclidean Geometry, Spherical Geometry, Hyperbolic Geometry, and others.
2) Using NonEuclid - My First Triangle:
3) Activities: - Exploring properties in Hyperbolic Geometry of Adjacent Angles, General Triangles, Isosceles Triangles, Equilateral Triangle, Right Triangles, Congruent Triangles, Rectangles, Squares, Parallelograms, Rhombuses, Polygons, Circles, and Tessellations of the Plane.
4) The Shape of Space: - Curved Space, Flatland, Ourland, and Mercury's Orbit.
5) The Pseudosphere: - A description of the space of which NonEuclid is a model.
6) Parallel Lines: - In Hyperbolic Geometry, a pair of intersecting lines can both be parallel to a third line.
7) Axioms and Theorems: - Euclid's Postulates, Hyperbolic Parallel Postulate, SAS Postulate, Hyperbolic Geometry Proofs.
8) Area: - Exaimation of A=½bh and A=s² in Hyperbolic Geometry, Properties Necessary for an Area Function, Altitudes of a Hyperbolic Triangle, Defect of a Triangle, Defect of a Polygon, and an Upper Bound to Area.
9) X-Y Coordinate System: - A description of how an x-y coordinate system can be set up in Hyperbolic Geometry.
10) Disk and Upper Half-Plane Models: - An informal development of these two models of Hyperbolic Geometry.

References, Appendices, and Supporting Information:

For The Teacher: Why is it Important for Students to Study Hyperbolic Geometry?
Conceptual Mechanics of Expression in Non-Euclidean Fields by Artist/Mathematician, Clifford Singer.
Palm OS Application for Exploring Non-Euclidean Geometry. The package includes two files: MathLib.prc and HypGeom.prc. MathLib is a library that contains mathematical functions missing on the standard palm libraries. HypGeom is the application. This package was written by Felipe Grajales, Faculty, Universidad de los Andes, Colombia.

References & Further Reading.

Change History.

For more information, questions, bug reports, or comments send e-mail to Joel Castellanos

Copyright©: Joel Castellanos, 1994-2009
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Updated April, 03, 2009