For an ellipse defined by
$$x = a \cos(t + \alpha)$$ $$y = b \cos(t + \beta)$$
What are the angles and lengths of each axis?
I've tried to work backwards from the expression for a rotated ellipse but I can't seem to equate the trigonometric expressions.
You can write the coordinates at time $t$
\begin{eqnarray} \left( \begin{array}{c} x(t) \\ y(t) \end{array} \right) = \left( \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right)\cdot \left( \begin{array}{c} \cos t \\ \sin t \end{array} \right) \end{eqnarray}
Consider a singular value decomposition $$\left( \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right ) = \left( \begin{array}{cc} \cos u & -\sin u \\ \sin u & \cos u \end{array} \right ) \cdot \left( \begin{array}{cc} d_1 & 0 \\ 0 & d_2 \end{array} \right ) \cdot \left( \begin{array}{cc} \cos v & -\sin v \\ \sin v & \cos v \end{array} \right ) $$ So we get $$\left( \begin{array}{c} x(t) \\ y(t) \end{array} \right) = \left( \begin{array}{cc} \cos u & -\sin u \\ \sin u & \cos u \end{array} \right ) \cdot \left( \begin{array}{c} d_1 \cos (t+v) \\ d_2 \sin (t+v) \end{array} \right) $$
So this is a rotation by angle $u$ of a common ellipse with semi-axes $|d_1|$ and $|d_2|$.