Math 487 Lab 7

Spherical Geometry

The idea of most of the lab today is to do constructions both with the plastic Lenart spheres and also with the stereographic image of a sphere on the plane via Sketchpad.

On the Lenart sphere you will have the sphere and the great circle tool which can be used to measure angles and to draw right angles.

In Sketchpad, you have scripts that, given the equatorial circle in the plane, will construct the antipodal point of a point A. Also we still have circle inversion--inversion in a great circle is the same as line reflection on sphere.

You also have a handout with "Adventure" on the sphere.

Adventures 2.1, 2.2, 2.3

Carry out these first on the Lenart sphere and then use the script and Sketchpad.

• The constructions of perpendiculars leads to construction of the Pole of a great circle.
• The extra reading in section 2 can be done later.

Adventures 3.1, 3.2, 3.3, 3.4, 3.5

It is not hard to do 3.1-3.3 with both methods. In 3.4, measuring angles may be a bit tricky with Sketchpad than on the plastic spheres. Don't get hung up on this point but return to it later if necessary. Both media should be OK for 3.5 (Hint: If circles a and b are two great circles that don't meet at right angles, what can you say about the great circle that passes through the poles A and B of a and b?)

INTERLUDE. MAJOR QUESTION about AREA and TRIANGLES

• A lune is a region enclosed by two great circles. If the surface areas of a sphere is S, what is the area of a lune with vertex angle a at each vertex?
• Draw a triangle ABC on the sphere (be sure to draw the entire great circles, not just arcs). Let the area of this triangle be T. Notice that the sphere is covered by overlapping lunes. Figure out how the lunes (whose areas you know) and the triangle area T can be combined to get the area S. Figure out this relationship and solve for T.