Poncelet Porism in Ellipses

Poncelet's porism is an assertion that provided there is a polygon circumscribed around one conic and inscribed into another, there are infinitely many such polygons. For two circles, one nested in the other, a demonstration is available elsewhere.

The applet below illustrates Poncelet's porism in a more general situation of two ellipses, one inside the other.

Denote the bigger ellipse Γ and the smaller one γ. Choose point A on Γ and find a tangent from A to γ. Extend this tangent through the intersection with Γ at, say, B. For B, repeated the process of finding the tangent to γ and extending it to the intersection with Γ. Let the process go on. It is quite possible that one of the tangents will pass through the starting point A, thus closing a polygon inscribed in Γ and circumscribed around γ. Such a polygon is sometimes referred to as a Poncelet's traverse. The essence of Poncelet's porism is an assertion that if a traverse exists for one point A, a traverse will exist for any starting point A.

In the applet, ellipse Γ is defined by the 4 corners of the enclosing rectangle. These can be dragged. If relative to the center of the applet viewing area the equation of Γ is

( Γ )

x²/a² + y²/b² = 1,

then the smaller ellipse γ is defined by the equation

( γ )

x²/(a - λ)² + y²/(b - λ)² = 1,

where λ can be changed with the scrollbar at the bottom of the applet. I found that these controls allow finding Poncelet's traverses with relative ease. The starting point can be dragged along ellipse Γ.

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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Conic Sections > Ellipse

Poncelet Porism

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