Special Relativity and Conic Sections

When an ellipse is presented as a "conic section" in the context of the Euclidean geometry of the construction is simple and visual. There is a standard cone at the origin, and one selects a plane whose normal vector is in the interior of the cone, and that does not contain the origin itself. The intersection of the cone and plane is the metric curve we call the ellipse.

This construction determines other metric invariants such as the lengths of major and minor axes, and the location of the foci, and it determines similarity invariants such as the ratio of the distance from the center to a focus, with the length of the semi-major axis, or another ratio that is known as the eccentricity. In addition, the following properties of similarity classes of ellipses are well known:

1. The ellipse is the locus of points, the sum of whose distances from the two foci is constant, the length of the major axis. We describe this as the focus-locus property.
2. 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. We describe this as the focus-directrix property.

Each of these properties of the similarity classes of ellipses actually characterizes them. But the usual construction of conic sections as the curves formed by planes intersecting cones in Euclidean does not make it evident why they should be true. In this story, we give a geometric and dynamic interpretation of the fact that the similarity classes of ellipses satisfy properties (1) and (2).

Of course one may simply settle for the observation that ellipses do have these properties, as a straightforward bit of analysis done in Euclidean shows. But, as we said, that analysis does not usually offer an interpretation of the similarity properties (1) and (2). In order to find an interpretation, we cast the plane-intersecting-cone construction in the light of a certain hyperbolic geometry on . We will discover that the "slice" plane (that intersects the cone) inherits in both geometries a Euclidean metric structure and gives in both cases the same ellipses. And we see that each slice construction determines a new ellipse by orthogonal projection to the 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 property, and the focus-directrix property. In particular, for the latter, each projected ellipse has associated with it a "directrix" that has a dynamic meaning. We will see, among other things for example, that the eccentricity of an ellipse is simply a speed (strictly between 0 and 1 -- the speed of light) that determines its shape.

It is convenient to use the language of 2+1 spacetime Geometry to develop these ideas. In particular, we describe a thought experiment in Special Relativity that gives the physical and dynamic interpretation for the fact that the sum of distances from a point to the two foci is constant. There is also a dynamic interpretation for the fact that the ratio of distance from a point to a focus, to the distance from the point to the directrix should be constant, as well of course 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 famous similarity class properties listed above have straightforward geometric interpretations. 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.

This story has two purposes. The first is to stimulate cross-disciplinary thinking. It makes connections between Euclidean and hyperbolic geometry for which the lexicon is the idiom of special relativity. And while the facts of hyperbolic geometry and linear algebra that we need are fairly elementary, the physical intuition required to apply them here depends strongly on the experience and the imagination of the reader. The section titled: Light Rays, Clocks, and Rulers: A Visual Primer in Special Relativity is a very modest attempt to reinforce and develop that style of thinking.

The second purpose of this story is to recruit 3 dimensional Graphics as a heuristic method that can illustrate the hyperbolic geometry of , and to support visual intuition of some elementary facts of special relativity. For that, we offer a large number of experiments in the body of the Microworld. But since we are experimenting with a new medium for the communication of mathematics, we understand that some readers may not have access to the technology required. So we will present the story here in two forms, an active form and a static form. In both forms, you may print the story, page by page, so that you have a hardcopy record to which you can refer.

The first form is a Mathwright Microworld, in which you have the opportunity to ask your own questions and experiment as you read along in your browser. If you are on a Windows platform (Windows 98/ME/NT/2000 or XP) and you have an ActiveX-enabled browser such as Internet Explorer 5.0 or later, we strongly recommend that you read and interact in the Microworld. The Microworld contains within it the entire text of the story. To read the Microworld, please click the following link to get the Player, and then start reading the Microworld. If you have already downloaded the Player, then simply visit the Microworld.

The second form is Static hypertext. If you prefer to read this story in the traditional way, you may read and print the full text in your browser. If you wish to do that, please proceed directly to the Contents. In the static form, you will not have access to the experiments. That text is supplemented with many illustrations of the outcomes of experiments, in any case.