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From the Dance of the Foci to a Strophoid
The College Mathematics Journal
Vol. 42, No. 4 (September 2011), pp. 289-298
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/10.4169/college.math.j.42.4.289
Page Count: 10
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Summary The intersection of a plane and a cone is a conic section and rotating the plane leads to a family of conics. What happens to the foci of these conics as the plane rotates? A classical result gives the locus of the foci as an oblique strophoid when the plane rotates about a tangent to the cone. The analogous curve when the plane intersects a cylinder, in which case all the sections are ellipses, is a right strophoid. This article discusses both results and provides elementary geometric proofs. Rotation about a different axis, such as one meeting the axis of the cone or cylinder, gives a very different curve. We consider how the resulting curve relates to the classical one by analyzing the family of curves obtained as the axis of rotation moves.
Copyright the Mathematical Association of America 2011