"The circle and parabola are unique among conic sections in that they have a universal constant. The analogous ratios for ellipses and hyperbolas depend on their eccentricities. This means that all circles are similar and all parabolas are similar, whereas ellipses and hyperbolas are not." I think this is not correct... As circle is a special case of ellipsis, so is unit hyperbola a special case of hyperbola. So a universal constant of circle is somehow on same level as "universal constant of a unit hyperbola" would be. In this sense, universal parabolic constant is somehow superior to those two, because it works for all parabolas, not just special cases. (See further for example affine geometry, where we are able to recognize just general ellipsis, parabola and hyperbola. A circle is much more subtle notion.) But I let it for discussion, so I've not made any edits yet.