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Special Relativity and Conic Sections - Introduction: Ellipses and Hyperbolic Geometry

James E. White
John Kepler observed that every ellipse that is not a circle has a pair of distinct foci. If one of these foci is distinguished, so that the pair is ordered, then this ordered pair determines an oriented line, and there is another line perpendicular to it called the directrix of the ellipse. (A circle has a single focus and no directrix.) Geometric similarities between non-circular ellipses correspond both to the foci and to the directrices. Here are the similarity class properties of ellipses to which I referred in the abstract.
  1. The focus-locus property: An ellipse is the locus of points the sum of whose distances from the two foci is constant, and that constant is the length of the major axis.

  2. The focus-directrix property: An ellipse (when it is not a circle) is the locus of points whose distance from a certain focus has constant ratio with the distance to the line called the directrix. That ratio (strictly between 0 and 1) is called the eccentricity of the ellipse.

In the Euclidean geometry of the plane, Apollonian ellipses may be represented in certain coordinates with origin at the center as


Then a straightforward bit of analysis shows that ellipses do have the focus-locus and focus-directrix properties. But that analysis does not usually offer an interpretation of the similarity properties (1) and (2). In order to find an interpretation, I cast the plane-intersecting-cone construction in the light of a certain hyperbolic geometry on .

We will discover that the "slicing" plane (the one that intersects the cone) inherits in both geometries a Euclidean metric structure, and we will find coordinates that reduce it to the standard form above. Also, we will see that each slice construction determines a new ellipse by orthogonal projection to the "base plane" perpendicular to the axis of the cone and passing through the origin. This projection is the same in both geometries. The latter plane also inherits a Euclidean metric structure, and the new projected ellipse is the one for which the similarity class properties will have interpretations.

These new projected ellipses were always available in a purely Euclidean context, but special relativity provides a clue that tells us how to restrict the plane-slicing-cone construction to a certain family of slices for which the projected ellipses (as well as the slice ellipses) give representatives from every similarity class of ellipses in the plane. Further, when we use the hyperbolic metric structure on , we find the dynamic interpretation of the similarity class properties of these ellipses, the focus-locus and focus-directrix properties. In particular, for the latter, each projected ellipse has associated with it a "directrix" that has a dynamic meaning. We will see, for example, that the eccentricity of an ellipse is simply a speed (strictly between 0 and 1, where 1 is the speed of light) that determines its shape.

In the language of 2+1 spacetime geometry, we will discover a number of interesting analogies with the familiar Euclidean constructions. For example, with 1 as the speed of light, it is not a coincidence that numbers between 0 and 1 can be the eccentricities of ellipses. Since the projected ellipses range through all similarity classes, we finish with the similarity class properties (1) and (2) for all ellipses in the plane. In that context, we will also see the "conic sections" as "conic intersections," the intersections of pairs of light cones.

James E. White, "Special Relativity and Conic Sections - Introduction: Ellipses and Hyperbolic Geometry," Convergence (October 2004)