Math 445A Winter 2002

Week 10

Monday 3/11 - Friday 3/15

    Course Home Page

  1. Week 1 - Jan 7-11

  2. Week 2 - Jan 14-18

  3. Week 3 - Jan 21-25

  4. Week 4 - Jan 28 - Feb 1

  5. Week 5 - Feb 4 - 8

  6. Week 6 - Feb 11 - 15

  7. Week 7 - Feb 18 - 22

  8. Week 8 - Feb 25 - Mar 1

  9. Week 9 - Mar 4 - 8

  10. Week 10 - Mar 11 - 15

  11. Exam Week Mar 18 - 22

    12.

Monday 3/11

Poincare Model of Hyperbolic Geometry

  • Some basic constructions
  • Comparison with stereographic sphere and DWEG.

Assignment for Week 10:

This is partly online and partly a set of constructions that may be done in lab or at other times. Follow link for details.

Take home Assignment

This will be considered as part of the Final Exam - Due Monday 3/18 at the Final Exam.

  • Hint - 3 approaches to measuring angles on a sphere: An angle of a spherical triangle is a dihedral angle between planes. Such a dihedral angle has been measured in this course in two ways at least: (1) by finding a suitable triangle in 3-space where the dihedral angle can be measured as an angle in the triangle or (2) by using coordinates and the equations of the planes, vectors, , dot products, etc. A third way to find angles in the sphere is to project the great circles onto the plane by stereographic projection and (3) find the angles between the image circles in the plane. This works since stereographic projection is conformal (preserves angles).

Wednesday 3/13

Class: Hyperbolic Geometry through constructions in the Poincare model.

  • The class does 8 constructions together.
  • Here are some interactive Java Sketchpad pages showing the perpendicular bisector in the Poincare model. (1) Sketch 1 shows: The Perpendicular Bisector of AB and the circle with diameter AB. (2) Sketch 2 shows the relationship among the center O of the Poincaredisk and the Euclidean center and the P-center of the circle with diameter AB.

Lab 10 - Hyperbolic Geometry Constructions

Friday 3/15

Class

Final Exam Prep