The equation of an ellipsoid is
$$ax^2+by^2+cz^2+2fyz+2gxz+2hxy+2px+2qy+2rz+d=0$$
The ellipsoid is arbitrary rotated and the orientation angle are given as θ, β and Ѱ and the center is at (x',y',z'). The radius of the ellipsoid are $r_a$,$r_b$,$r_c$. $r_a$>$r_b$ and $r_a$>$r_c$ and $r_b=r_c$.
I want the projection of the ellipsoid into the XY plane. i do not know where to start. Any help will be appreciated. Thanks in advance!
Some representations of the ellipsoid:
1. $$ (u - u_c)^\top A (u - u_c) = 1 $$ where $u = (x,y,z)^\top$ and $u_c = (x_c, y_c, z_c)^\top$ and $A$ is a definite matrix with eigen values $r_a^{-2}$, $r_b^{-2}$ and $r_c^{-2}$
2.
$$ u = u_c + u_x x + u_y y + u_z z $$ where $A = (u_x, u_y, u_z)$ is a regular $3\times 3$ matrix.
Where one can choose $u_x$, $u_y$, $u_z$ as orthogonal system with $\lVert u_x \rVert = r_a$, $\lVert u_y \rVert = r_b$, $\lVert u_z \rVert = r_c$
Which using a parametrization of the unit sphere gives $$ u_c + u_x \cos \theta \cos \varphi + u_y \cos \theta \sin \varphi + u_z \sin \theta \quad (-\pi/2 \le \theta \le \pi/2, \ 0 \le \varphi < 2 \pi) $$
Projection:
The projection seems just taking a vector $u=(x,y,z)$ from the ellipsoid and then force the $z$ coordinate to $0$: $P u = (x,y,0)$.