To give
a brief forecast of what follows this section, let us say that this
plane intersects the light cone in a curve in standard (
) coordinates. Which curve it will be will depend on how
is moving relative to the standard observer .
However, when
uses certain coordinates for that plane that are adapted to the motion of ,
he will discover that it is a Euclidean plane, in the sense that all of its
vectors are space-like, so it inherits a Euclidean metric, and will be able
to conclude that the curve is an ellipse in the usual Euclidean sense. And something
new will emerge. When he writes the equation of the projected ellipse to standard
coordinates, his adapted system of coordinates will pass to a Euclidean basis
for the
plane in which he will find an ellipse with one focus at the origin, and for
which the center, the other focus, the major and minor axes and the directrix
will all have dynamical interpretations in terms of the motion of .
I will return to this topic when I analyze the focus-locus construction beginning
with **A Thought Experiment**.