MaximalRegInEllipse

Maximal Polygon in Ellipse

Given an ellipse with axes a and b (a >b) and a number N, construct a maximal in area N-gon inscribed in the ellipse. Show that there are infinitely many solutions and each point of the ellipse is the vertex of exactly one such maximal N-gon. All of them have the same area equal to (N/2)*a*b*sin(2*pi/N).

A quick solution can be found using the affinity f, which maps the unit circle to the given ellipse. This can be realized with a diagonal matrix diag(a, b). f maps the square to the maximal inscribed rectangle of the ellipse (see MaximalRectangle.html ) with sides sqrt(2)*a, sqrt(2)*b. Since affinities preserve area-ratios the proof follows at once. The maximal N-gons are images of the corresponding maximal in area inscribed N-gons in the unit circle. These are the regular N-gons and the formula gives their area if we replace a=b=r, with the radius of the circle.
Note that all these maximal N-gons are also circumscribed about a second ellipse which is the image under f of the corresponding inscribed circle of the regular N-gons. Rewriting the area E(N) in a convenient form and taking the limit we find the area of the ellipse:

The exercise gives an example for Poncelet's "Great" theorem (see Poncelet.html ) on polygons simultaneously circum/in-scribed on two ellipses (conic sections more general).

The exercise generalizes also the examples of maximal triangles/quadrangles inscribed in ellipse, whose discussion starts in the file ParaInscribedEllipse.html .

Switch to the selection on contour-tool (press CTRL+2), catch and move points P and watch the corresponding pentagons inscribed in the ellipse and their counterparts inscribed in the circle.

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