# Betweenness of a Sphere

• Feb 7th 2010, 06:09 PM
geometrywiz
Betweenness of a Sphere
Let A and B be two points on the sphere S^2. Define D(A,B), the distance from A to B, to be the length of the shortest arc of a great circle containing A and B. More specifically, if A and B are antipodal points, define D(A,B)= pie. If A and B are not antipodal points, then they lie on a unique great circle and they are the endpoints of two subarcs of that great circle. The distance from A to B is the length of the shortest arc. As usual, define C to be between A and B if A, B, and C are collinear and D(A,C) + D(C,B)= D(A,B). Also define segment and ray in the usualy way.

a) Find all points that are between A and C in case A and C are nonantipodal points. Sketch the segment from A to C.

I don't understand what the nonantipodal points would look like.

b)Find all points that are between A and C in case A and C are antipodal points. Sketch the segment from A to C.

c) Does the plane separation postulate hold in this setting?

d) Let A and B be distinct, nonantipodal points on S^2. Find all points C such that A * B* C. Sketch the ray AB.
• Feb 8th 2010, 04:47 AM
HallsofIvy
Quote:

Originally Posted by geometrywiz
Let A and B be two points on the sphere S^2. Define D(A,B), the distance from A to B, to be the length of the shortest arc of a great circle containing A and B. More specifically, if A and B are antipodal points, define D(A,B)= pie. If A and B are not antipodal points, then they lie on a unique great circle and they are the endpoints of two subarcs of that great circle. The distance from A to B is the length of the shortest arc. As usual, define C to be between A and B if A, B, and C are collinear and D(A,C) + D(C,B)= D(A,B). Also define segment and ray in the usualy way.

a) Find all points that are between A and C in case A and C are nonantipodal points. Sketch the segment from A to C.

I don't understand what the nonantipodal points would look like.

"Antipodal points" are points on the sphere that are two ends of a diameter. Two poinst are "non-antipodal" as long as they are NOT the two ends of one diameter. For example, take any point, A, on the sphere. draw a diameter through the center of the sphere to the opposite side. That point, B, is anti-podal to A. If C is any point on the sphere [tex]except[/b] B, then C is "non-antipodal" to A.

Every pair of points on a sphere lie on some great circle. They cut that great circle into two arcs. If they are anti-podal, then those two arcs have exactly the same length. If they are non-anti-podal, then those two arcs are not of the same length, one is shorter than the other.

If A and B are non-antipodal points then the set of points between them is the set of points on the smaller of the two great circle arcs between them.

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b)Find all points that are between A and C in case A and C are antipodal points. Sketch the segment from A to C.
If A and C are anti-podal points then the two arcs are of the same length and every point on the great circle, except A and C themselves, is "between" A and C.

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c) Does the plane separation postulate hold in this setting?
What is the "plane separation postulate"?

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d) Let A and B be distinct, nonantipodal points on S^2. Find all points C such that A * B* C. Sketch the ray AB.