#### Note 4. Directrix and eccentricity of the projected ellipse

We observed earlier that the graph of is the tangent plane to the hyperboloid at the point .

This plane intersects the plane ( ) in the line in the plane, ,

since . If . Now suppose that is an arbitrary point in the plane. Letting our unit vector ,

it is clearly perpendicular to the line, and there is a unique number and point such that .

The number is the distance from to the line . Now it is not difficult to see that if is the dihedral angle between the planes and the graph of (that is, the angle formed by intersecting these planes with a plane perpendicular to ), then for each , .

Note that . I will interpret the line as the directrix of the ellipse, and the number as the eccentricity in the relativistic context in the section Interpretation of the Experiment.

For now, I will simply determine the number in terms of . First of all, it is easy to see that the major axis of the ellipse is generated by and that the intersection of and the line generated by is the point .

This tells us that the distance from the directrix to the origin, a focus of our ellipse, is .

To determine the number , it will be enough to calculate for a single point . Since the origin , we calculate ,

and so we conclude that .

When we interpret the ellipse dynamically using Special Relativity, we will be able to conclude that is the eccentricity of the ellipse that was characterized above.