Poncelet's porism on non-circular conic sections

Oct 14, 2024
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Introduction

Poncelet's porism or Poncelet's closure theorem is one of the most remarkable results in plane geometry. It is most easily described in terms of circles: suppose we have two circles \(C\) and \(D\), with \(D\) lying entirely inside \(C\). Pick a point \(p_0\) on \(C\), and find the tangent from \(p_0\) to \(D\). Let \(p_1\) be the other intersection of the tangent line at \(C\). So the line \(p_0 - p_1\) is a chord of \(C\) which is tangential to \(D\). Continue creating \(p_2\), \(p_3\) and so on. The porism claims that: If at some stage these tangents "join up"; that is, if there is a point \(p_k\) equal to \(p_0\), then the tangents will join up for any initial choice of \(p_0\) on \(C\).

The polygon so created from the vertices \(p_0,\, p_1,\,\cdots,p_k\) is called a bicentric polygon: all its vertices lie on one circle (\(C\)), and all its edges are tangential to another circle \(D\).

If \(r\) and \(R\) are the vertices of \(D\) and \(C\) respectively, and \(d\) is the distance between their centres, much effort has been expended over the past two centuries determining conditions on these three values for an \(n\) sided bicentric polygon to exist. Euler established that for triangles:

\[ \frac{1}{R+d}+\frac{1}{R-d}=\frac{1}{r} \]

or that

\[ R^2-2Rr-d^2=0. \]

Euler's amanuensis, Nicholas Fuss (who would marry one of Euler's granddaughters) determined that for bicentric quadrilaterals:

\[ \frac{1}{(R+d)^2}+\frac{1}{(R-d)^2}=\frac{1}{r^2} \]

or that

\[ (R^2-d^2)^2=2r^2(R^2+d^2). \]

Looking at the first expressions, you might hope that further polynomials might have similarly nice expressions. Unfortunately, the expressions get considerably more complicated as \(n\) increases, and the only way to write them succinctly is with a sequence of substitutions.

There is a good demonstration and explanation at Wolfram Mathworld which has examples of some further expressions.

An example with two circles

Here's an example with a quadrilateral. To use it, move the point \(A\) along the \(x\) axis. You'll see that the inner circle changes size according to Fuss' formula. Then you can drag the circled point around the outer circle to demonstrate the porism.

An example with non-circular conic sections

Poncelet's porism ia in fact a result for conic sections, not just circles. However, circles are easy to work with and define - as seen above, just three parameters are needed to define two circles. This means that nobody has tried to develop similar formulas to Euler and Fuss for general conic sections: the complexity is simply too great. In the most general form, five points are needed to fix a conic section. that is: given any five points in general position, there will be a unique conic section passing through all of them.

Here's how this figure works:

  • The green dots define the interior ellipse (two foci and a point on the ellipse). They can be moved any way you like.

  • The red points on the ellipse: \(p_0\), \(p_1\), \(p_2\), \(p_3\) and \(p_4\) can be slid around the ellipse.

  • The tangents to these points and their intersections define a pentagon, whose vertices define a larger ellipse.

  • When you have a nice shape that you like, use the button "Hide initial pentagon". All labels will vanish, and you'll have one circled point which can be dragged around the outer ellipse to demonstrate the porism.

What happens if you allow two of the points \(p_i\) to "cross over"?

A note on the diagrams

These were created with the amazing JavaScript library JSXGraph which is a very powerful tool for creating interactive diagrams. I am indebted to the many answers I've received to questions on its Google group, and in particular to its lead developer, Professor Dr Albert Wassermann from the University of Bayreuth, who has been unfailingly generous with his time and detail in answering my many queries.