Lab 7

Part 1. Inverting Circles to and from Concentric Circles

Spend 10-15 minutes contemplating the methods of this GSP file that shows how to invert two circles to concentric. The work is all done in this case, but you can learn some methods and also explore the possible ways these figures can appear.

Part 2. The Sphere

This part of the lab will sort of bounce back between hands-on and computer visualization using Stereographic projection. The definition and the theorems (not yet proved) that we will use about Stereographic projection are these:

• The Stereo projection will project from a point called N to the plane tangent to the sphere at the opposite point S. The points equidistant from N and S form a great circle e (the equator).
• The circle e' will be given in the plane. The center of e' is S.
• Circles are mapped to circles or lines. Angles are preserved.
• Vertical Cross-Section of Stereographic projection shows how to transform a point P in the plane to a point P* on the sphere.
• Antipodal Point Construction. Then if the opposite (antipodal) point of P* is Q*, then the stereo image Q of Q* can be constructed by the method shown in this figure (where the labels are different, alas. There P1 and P2 are on the sphere and Q1 and Q2 are in the plane -- sorry).

Sphere Basics

On the physical sphere, explore these ideas or answer these questions:

1. There is a unique circle through any 3 points (why? how do you see this?)
2. Two points determine a unique great circle (with one exception).
3. Investigate the nature of parallel great circles.
4. How are the points of intersection of two great circles related?
5. Why does any circle on the sphere have two centers on the sphere? What does this mean? What are the centers of two great circles?
6. True or false? For any great circle c, all the great circles orthogonal to c pass through the centers of c. Is this still true if c is a circle that is not a great circle?
7. Spherical triangles. What is a spherical triangle? Can you find a triangle with angle sum not equal to 180? (How do you measure angle?)

On the GSP plane, where the sphere is mapped by stereographic projection, the circle e' is given.

1. For any point A, construct the (stereo image) of the antipodal point of A*.
2. For any two points A and B (not antipodal), construct the great circle through A and B.
3. Show in a figure an example that two great circles intersect in antipodal points.
4. Given the great circle through A and B and give a point C, construct the great circle through C orthogonal to the great circle AB.
5. Construct the (stereo image) of the spherical centers of the spherical great circle through A and B. (Hint: Construct two orthogonal circles.)