I have a set of phasors (complex numbers) x, y & z.

using a 3d vector of these, q = [x, y, z]

then real(q.e^(i*theta)) traces an ellipse in 3d space as theta is swept over the range 0->2pi

is there a way to derive the equation of this ellipse in some nicer or more general ellipse representation

such as-

u = 3d vector representing the semi-major axis

v = 3d vector representing the semi-minor axis