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Math Help - Betweenness of a Sphere

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by geometrywiz View Post
    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.

    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.

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

    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.
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