<s>[INST] Axis angle and length of ellipse 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.
 [/INST] 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 $|d_2|$. 
</s>