The Game of Poncelet

Play the game of Poncelet! Place two circles, red and blue, set a starting point on the blue circle, and watch what happens. For more explanations see below.

What happens?

Starting with a point on the blue circle, and a tangent to the red circle containing this point, there are two elementary operations we may perform.

On the one hand, if a line meets the blue circle, it does so with with multiplicity 2, meaning that either there are two intersection points, or if there is only one intersection point, the line is tangent to the circle. So if already one intersection point is given, we can replace it with the other (or do nothing if the line is tangent).

On the other hand, if for a given point there is a tangent to the red circle, then either the point lies already on the red circle, or there is also a second tangent. The second operation then switches the tangents (or does nothing if the point is on the red circle).

If we compose these two operations, we obtain the Game of Poncelet, which you can play above. The question is, can you find a configuration where the game returns to the starting point?

How to use the web app

  1. You can place the circles by clicking three times on the canvas. Unless your three points lie on a line, there exists a unique circle containing these three points.
  2. Once you have placed the blue circle, click anywhere on the canvas to choose a starting point, which will automatically be projected on the blue circle, if you avoid the centre.
  3. You do not need to choose the tangent to the red circle, it will be computed automatically. However, this is only possible if the starting point lies outside the red circle, so keep this in mind when setting up the configuration of the circles.
  4. Now you can play, and also apply the operation 10 times at once.
  5. If you wish to try a different configuration, click the reset button, or start selecting a circle again.