what data am I looking for:

major/minor axis/radius

what data I have

semi-major/minor axis/radius. axes can be non-perpendicular and that's when I have trouble getting parametric form from data.

transformation applied is transformation from

origin (0,0)

x axis (1,0)

y axis (0,1)

into

origin ellipse center

x axis semi-major * major radius

y axis semi-minor * minor radius

ellipse is generated by drawing radius 1. circle at origin in source coordinate system and transforming result into destination.

I need to get parametric form (or major/minor axis/radius) in destination coordinate system from the above if possible.

sorry if I'm not clear, but engrish speaker here after few hours of thinking and googling for solution