Are all conics completely defined (up to rotation, translation, and dilation) by a single parameter, their eccentricity?
I believe this to be true but am surprised that this fact is not more widely identified. This means, for examples, that all of the various equations in Cartesian coordinates, with all of their complexity, are simply artifacts of the rotation, translation, and dilation, wrapped around a very simple single parameter, eccentricity.
What about the degenerate conics? It seems that a point is simply the limit of dilating any bound figure (i.e. $e < 1$) around both axes, a pair of lines is the limit as $e \to \infty$, and a single line is the limit of dilating a pair of lines across a single axis. The one omission I'm aware of is parallel lines, which don't seem producible under this scheme.
