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.

**How to Cite this Page:**

Su, Francis E., et al. “Ellipsoidal Paths.” *Math Fun Facts*. <https://www.math.hmc.edu/funfacts>.

**Fun Fact suggested by**:

Jorge Aarao, Johannes Huisman