9846

Two Models of Projective Geometry

In the first model of 2D elliptic geometry, "points" and "lines" are the lines and planes through the origin in 3D, respectively. These axioms are satisfied: two "points" determine a "line" (because the two ordinary lines determine an ordinary plane), and two "lines" determine a "point" (intersect the two ordinary planes to get an ordinary line). There are no parallel "lines", because all ordinary planes intersect.
Another definition uses the points and great circles on a sphere with opposite points identified. This definition seems more natural than the first because the points are more point-like and the lines are one-dimensional, but the identification of opposite points is somewhat disorienting, especially for the great circles.
The two models are clearly related: a line through the center of a sphere intersects the sphere in two opposite points and a plane through the center intersects the sphere in a great circle.
For the choice of a pair of planes, the sliders govern the planes' normals.

THINGS TO TRY

SNAPSHOTS

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