Math 487 Lab 7 - Stereograpic Tools and Maps

The goal of this lab, as in the last one, is to explore spherical geometry using the stereographic projection to do spherical geometry on a "map" of the sphere laid out on the plane.

Basic tools (on the Lenart sphere)

• Opposite (antipodal) point
• Great circle through 2 points.
• Reflection in great circle
• Great circle through P orthogonal to given great circle c (the diameter of c through P).

Establish scripts for the same basic tools in Sketchpad or straightedge and compass

In both cases, use the basic tools to get new tools and constructions

• Given a great circle c, construct the two poles. (Hint: intersect two great circles orthogonal to c.)
• Given two points A and B, construct the great circle that reflects A to B. (This is the perpendicular bisector.)
• Given a point A, construct the polar circle (the "equator" with A as north pole).

Important constructions related to polyhedra and spherical tessellations

• Construct the Vertices of an inscribed octahedron.
• Construct the Vertices of an inscribed cube.
• Construct the Vertices of an inscribed tetrahedron.
• Construct a Wulff net with circles at angles of 15 degrees apart.

General Triangles

• Construct a general triangle ABC and its (spherical) perpendicular bisectors. Are they concurrent?
• Construct a general triangle ABC and its (spherical) angle bisectors. Are they concurrent?