I have an ellipsoid (That can be rotated), described by its conic equation :
$$E : a_1x² + a_2y² + a_3z² + 2a_4xy + 2a_5xz + 2a_6yz + a_7x + a_8y + a_9z + \gamma = 0$$
This ellipsoid can be intersected by 3 different planes, each parallel to one of the 'coordinate plane', and with a third known coordinate. For example, I have $P_1$ parallel to $P_{(x,y)} : z = 0$, and with a known Z, so its equation would be $P_1 : z = Z$. I hope I'm not wrong so early on
I would like to find the equation for the ellipse determined by the intersection of E and $P_1$ (for example).
Maths isn't my specialty so I could be wrong, even on the most basic stuff, but I injected my plane's equation in E, giving :
$$e : a_3Z² + a_9Z + \gamma = 0$$
Where $e$ would be my ellipse's equation. That looks pretty easy (too easy, actually), so I'm wondering if I did anything wrong. More specifically, I'm afraid that, since my original ellipsoid can be rotated, I'm "losing" this rotation in my calculation. Am I being paranoid or is there something I missed ?
Thanks
You're doing fine. You missed a couple of $z$s in the equation, so it's really
$$ a_1 x^2 + a_2y^2 + a_3Z^2 + 2a_4xy + 2a_5xZ + 2a_6yZ + a_7x + a_8y + a_9Z + \gamma = 0 $$ which we'd typically restructure a bit by writing it $$ a_1 x^2 + a_2y^2 + 2a_4xy + 2(a_5Z+ a_7)x + (2a_6Z + a_8)y + (a_3Z^2 + a_9Z + \gamma) = 0 $$ where $x$ and $y$ are the "variables" and the things in parentheses are all just numbers, since $Z$ is a known value rather than a variable. This latter expression has the form of an ellipse in the $xy$-plane.