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© 1999-2010 by Francis Edward Su

From the Fun Fact files, here is a Fun Fact at the Medium level:

# Ellipsoidal Paths

 Figure 1

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 shows 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.

Su, Francis E., et al. "Ellipsoidal Paths." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Keywords:    algebraic geometry
Subjects:    calculus, analysis
Level:    Medium
Fun Fact suggested by:   Jorge Aarao, Johannes Huisman
Suggestions? Use this form.
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