Instructor:Spherical Geometry Exploration Solutions

Spherical Geometry Exploration

Straight Lines
1. Straight line
2. Straight line
3. Straight line
Geodesics
4. Straight lines on the sphere are great circles, which cut the sphere into two equal hemispheres. The shortest way to get from one point to another is along a great circle. If you walk straight, you'll follow a great circle. If you pull a piece of string tight between two points, it will lie on a great circle.
5. Straight lines are great circles
Between
6. Not really. For example, three equally spaced points on the equator. Now it's not clear if any one of these is between the other two.
7. There's two real options here: If three points are on a great circle then each point is between the other two. Or, you could say that a point is between two others if it is on the short great circle segment joining the two (there's always two segments joining two points, and unless they are antipodal one of these two is shorter.)
Circles
8. The circle grows until it becomes a great circle. Then it begins to shrink again. Then it becomes a single point, antipodal to the center.
9. Both sides of a circle are finite, so it's not clear which is the inside and which is the outside. For most circles, one could say the small part is the inside, but even this fails for great circles.
Squares
10. Sketch a spherical rhombus
11. Sketch a regular 4-gon
12. A regular 4-gon is not a square if you want a square to have 90° angles.
Postulates
• I. There is exactly one geodesic through any two points, unless those points are antipodal. In that case, there are infinitely many geodesics through the two.
• II. Any geodesic can be extended until it meets itself and forms a great circle.
• III. There is a circle with any given center and radius, until the radius is 1/2 the circumference of the sphere, at which point the circle becomes a single point.
• IV. The sphere looks the same at every point. (Compare with, say, a football, which has different sorts of points).
• V. There are no parallel geodesics on the sphere. Any pair of geodesics must meet at two antipodal points.