When an ellipse is presented as a "conic section" in the context of 3-dimensional Euclidean geometry, the construction is simple and visual: You're given a standard cone with vertex at the origin, and you select a plane not through the origin whose normal vector at the origin is in the interior of the cone. The intersection of the cone and plane is the metric curve we call an ellipse. This is essentially the definition given by the Greek geometer Apollonius of Perga, about 200 BCE.
The similarity classes of these curves have certain well known properties that we call the focus-locus and focus-directrix properties. Each of these properties of ellipses actually characterizes their similarity class. But the usual construction of conic sections as the curves formed by planes intersecting cones does not make it evident why the properties should be true. In this story, I give a geometric and dynamic interpretation of the fact that similarity classes of ellipses have these properties.
It is convenient to use the language of 2+1 spacetime geometry to develop the ideas. In particular, I describe a thought experiment in special relativity that gives a physical and dynamic interpretation of the fact that the sum of distances from a point to the two foci is constant. I also give a dynamic interpretation of the constant ratio of distances from a point to a focus and from the point to the directrix, as well as an interpretation of the directrix itself. In this view, the "plane-slicing-cone" description of an ellipse is the first step in the description of a second ellipse for which the similarity class properties have straightforward geometric interpretations.
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Publication date: October, 2004
© Copyright 2004 by James E. White
James E. White, "Special Relativity and Conic Sections," Loci (October 2004)
Journal of Online Mathematics and its Applications