9860

Triangles in the Poincaré Disk

This Demonstration gives a representation of hyperbolic geometry showing a hyperbolic triangle. You can drag the vertices.

THINGS TO TRY

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

Poincaré used the points of the unit disk without its boundary as a model of hyperbolic geometry. A hyperbolic straight line is represented as an arc of a circle that is perpendicular to the disk edge. If a hyperbolic line passes through the center, it is straight. Angles are preserved so they can be measured directly from the figure. The angles of a hyperbolic triangle do not sum to 180°; the sum is zero for the limiting case of all vertices on the edge and close to 180° for vertices close to center or for small triangles, whose sides are almost straight.
Poincaré rediscovered this model in 1882; the conformal disk realization of hyperbolic geometry was found by Eugenio Beltrami and published in 1868 but does not carry his name.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+