I've been trying to learn more about envelope of lines and came over this proof for the cardioid inside a cup. I have arrived at the two parametric equations which I've written in matrix form.
$$\begin{pmatrix}cos3\theta & -sin3\theta\\\ sin3\theta & 1+cos3\theta\end{pmatrix} \begin{pmatrix}x \\ y \end{pmatrix} = \begin{pmatrix}\frac{1}{3} cos\theta-\frac{2}{3}cos2\theta \\ sin\theta\ sin2\theta \end{pmatrix}$$
I'm using the Cramer's Rule approach to get a general solution for x and y in terms of $\theta$, and so far I've gotten the determinant. The final answer in the linked proof has written the final solution in the cartesian form for a cardioid, $(\frac{1}{3}(2−2),\frac{1}{3}(2−2))$. But, I've been struggling to get $D_x$ and $D_y$, I don't know which trigonometric identities to use and I seem to be going in circles!
I would appreciate if some direction to solve this can be given!