The traditional parametric equations of an ellipse centered at the origin are given by: $$ \{x(t)= a\cos(t)\cos(B) - b\sin(t)\sin(B); y(t)= a\cos(t)\sin(B) + b\sin(t)\cos(B)\} $$ where B is the angle of rotation of the major axis from the x-axis. Here, the major and minor axes lengths are given as a and b. However I have a different parametric form of an ellipse given by: $$ \{x(t)=A\cos(t) + B\sin(t); y(t)=C\cos(t) + D\sin(t)\}. $$ Each equation represents sinusoids along the the x and y axes with independent magnitudes and phases, thus forming an ellipse as the locus $(x(t),y(t))$.
The major and minor axes lengths are not apparent in this form. My approach has been to set the derivative $dx/dy = -(x/y)$ since the slope of the tangent is perpendicular to the radius only at the intersections with the major and minor axes. My result is $t=.5(\tan^{-1}((AB+CD)/2(A^2+C^2-B^2-D^2))$. This value of t does not quite match my numerical simulation (although it is close) and I am perplexed. Any help is greatly appreciated.
Thanks, Bob T.