Suppose I have an arbitrarily oriented ellipsoid, centered at $r_C$, and whose equation is given by:
$$ (r - r_C)^T Q (r - r_C) = 1$$
where the $3 \times 3 $ symmetric matrix $Q$ is known, and the $3 \times 1 $ vector $r_C$ is known, and the position vector $r = [x, y, z]^T$. I want to find the equation of the projection (i.e. shadow outline) of this ellipsoid onto the $xy$ plane. The projection is clearly an elliptical region, centered at $p_1$, and whose boundary is given by:
$$ (p - p_1)^T Q_1 (p - p_1) = 1 $$
where $p = [x, y]^T $
So I'd like to find the steps to identify the $2 \times 2 $ symmetric matrix $Q_1$.
A detailed answer or hints on this is greatly appreciated.
