Note 8. Digression

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.