#### Note 11. Interpretation of the eccentricity

Clearly, is perpendicular to the line generated by , the major axis of the ellipse. If we let be an arbitrary point of the plane, there is a unique number and point such that .

The number is the distance from to the line .

Now we saw in the Euclidean context 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 , .

I noted that .

For a point in the ellipse, is, by the interpretation of the experiment above, , the time of arrival as measured by of a light ray emitted by . This number in turn is the length of the segment as measured by connecting the focus at the origin to the point on the ellipse. Therefore the ratio of {the length of the segment connecting the focus to the point in the ellipse} to {the distance from to the line } is the constant . We usually call the eccentricity of the ellipse.

The calculation in Planes Intersecting Cones actually gave a value for . The intersection of the directrix with the extension of the major axis is the point = .

And the value of the eccentricity is the speed of as measured by , = .