Ellipsoidal Paths

Given an , and a smaller strictly inside it, start at a point on the outer , and in a counterclockwise fashion (say), follow a line tangent to the inner until you hit the outer 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 . You can learn more about Poncelet's theorem in any classical text on

How to Cite this Page: 
Su, Francis E., et al. “Ellipsoidal Paths.” Math Fun Facts. .

Fun Fact suggested by
Jorge Aarao, Johannes Huisman

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