Suppose I have an equation for a conic section, with the usual meaning: $$Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0$$
I know the current values of the conic $A,B,C, D, E, F$ and the angle of rotation $\Theta$. I'd like to now rotate the conic by some amount $\phi$, the new rotation angle therefore being (${\Theta} + {\phi}$).
I know that the values of the coefficients can be calculated as follows (with the usual meaning of $a,b$ and $(x0,y0)$, all of which are known):
\begin{aligned}A&=a^{2}(\sin \Theta )^{2}+b^{2}(\cos \Theta )^{2}\\B&=2\left(b^{2}-a^{2}\right)\sin \Theta \cos \Theta \\C&=a^{2}(\cos \Theta )^{2}+b^{2}(\sin \Theta )^{2}\\D&=-2Ax_{\circ }-By_{\circ }\\E&=-Bx_{\circ }-2Cy_{\circ }\\F&=Ax_{\circ }^{2}+Bx_{\circ }y_{\circ }+Cy_{\circ }^{2}-a^{2}b^{2}.\end{aligned}
Now, I could simply plug my new value of ($\Theta + \phi$) in the above equations to obtain my new Conic coefficients. However, I am trying to find out if I can use the already known values of the coefficients at angle ($\Theta$) and simply add/multiply some amount based on my new angle ($\Theta + \phi$)? The idea behind it is to save time on computation, if possible.
Example:
Taking the example of A, let's add (or multiply) some value of X which will be equal to the new value of A at ($\Theta + \phi$):
$$A + X = a^{2}(\sin \Theta + \phi)^{2}+b^{2}(\cos \Theta + \phi)^{2}$$ We know A to be equal to: $$a^{2}(\sin \Theta)^{2}+b^{2}(\cos \Theta)^{2} + X = a^{2}(\sin \Theta + \phi)^{2}+b^{2}(\cos \Theta + \phi)^{2}$$
$$ X = a^{2}(\sin \Theta + \phi)^{2}+b^{2}(\cos \Theta + \phi)^{2} - a^{2}(\sin \Theta)^{2}+b^{2}(\cos \Theta)^{2}$$
Is this mathematically possible? If so, can we find the value of X? Obviously, to have the whole exercise to be useful, X must also be easily computable/derivable, i.e. the above must be simplified somewhat, allowing us to find the coefficients at rotation ($\Theta + \phi$) with much ease: $$A' = A + X$$
... and similarly for other coefficients.