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       #Post#: 119--------------------------------------------------
       ELLIPSOIDAL PATHS
       By: eba95 Date: July 30, 2010, 6:53 am
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       Ellipsoidal Paths
       Given an ellipse, and a smaller ellipse
       strictly inside it, start at a point on
       the outer ellipse, and in a
       counterclockwise fashion (say),
       follow a line tangent to the inner
       ellipse until you hit the outer ellipse
       again. Repeat. Figure 1 gives an
       example.
       Now it is quite possible that this path
       will never hit the same points on the
       outer ellipse twice. But if it does
       "close up" in a certain number of
       steps, then something amazing is
       true: all such paths, starting at any
       point on the outer ellipse, close up in
       the same number of steps!
       This fact is known as Poncelet's
       Theorem.
       Presentation Suggestions:
       Intuition may be gained by
       presenting special cases, such as
       where the ellipses are concentric
       circles.
       The Math Behind the Fact:
       This process that produces this path
       may be thought of as a dynamical
       system on the outer ellipse, and is
       related to the study of circle maps
       and rotation numbers in dynamical
       systems. You can learn more about
       Poncelet's theorem in any classical
       text on algebraic geometry.
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