Take the following ellipsoid in implicit form as an example:
$$x^2 + 2 y^2 + 3 z^2 + x y + y z - 2 xz = 5$$
The parallel projection of the ellipsoid onto $xoy$ coordinate plane can be obtained as:
$$ 8 x^2 + 16 x y+23 y^2=60$$
Is it possible to prove:
- The parallel projection of an ellipsoid is always an ellipse and how?
I guess this should be able to be generalized into:
- the perspective projection of an ellipsoid is a conic curve.
How to prove it?
In prjective geometry, the quadratic form of conics is useful in such proof. This one seems a little more difficult.