I have an equation describing an oblique ellipse in parametric form
$$ \left(\begin{matrix} x\\ y \end{matrix}\right) = \left(\begin{matrix} \cos(\alpha) & -\sin(\alpha);\\ \sin(\alpha) & \cos(\alpha) \end{matrix}\right)\cdot \left(\begin{matrix}a \cos(t) \\ b \sin(t) \end{matrix}\right) $$ where $a$ and $b$ are the semi axes, $\alpha$ is the angle the ellipse is tipped and $t \in [0,2\pi]$, and I would like to transform it into its normal form
$$ Ax^2+Bxy+Cy^2 = D $$
But I can't calculate the first into the other to get the connection of $\alpha, a, b$ and $A,B,C,D$.
I am grateful for any help.
How do you transform nromal parametric equation $(a\cos t, b\sin t)$ to algebraic form? You have $X = a\cos t, Y = b\sin t$ which easily gives you $\dfrac{X^2}{a^2} + \dfrac{Y^2}{b^2} = 1$ right? Now in your case you basically have $(x, y)' = A(X, Y)'$ for some invertible matrix $A$ and for $X$ and $Y$ as above. Find $A$ inverse and get $(X, Y)' = A^{-1}(x, y)'$. You know what equation $X$ and $Y$ satisfy so just substitute.