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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.

geometrikal
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  • if you don't mind using calculus you can find the 4 axis points by optimizing the quantity $r^2(t) = (x(t))^2 + ( y(t))^2$ – WW1 May 25 '15 at 01:21

1 Answers1

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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 $|d_2|$.

orangeskid
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