Equation for a sphere after a projective transformation? I was wondering if there is a general equation for a sphere that has undergone a projective transformation (i.e, the sphere has been multiplied by a 4x4 homography matrix)?
I attempted to multiple the elements of the homography through the sphere equation, but it became really messy rather quickly and didn't look like something I could use. 
My goal is to do a least square fit of such an object to some points, but I can't seem to find an equation (or even the terminology) for such an object.
 A: I believe that a sphere that's undergone a projective transformation is called a conic (or the equivalent in 3D...right... it's a "quadric surface"). Suppose that you have the sphere defined by 
$$
\newcommand{\xvec}{\mathbf x}\newcommand{\Imat}{\mathbf I}\newcommand{\Mmat}{\mathbf M}\newcommand{\yvec}{\mathbf y}\newcommand{\Amat}{\mathbf A}
\xvec^t \Imat \xvec = 1
$$
where $\xvec$ is the vector $(x, y, z, 1)^t$, and $\Imat$ is the 4x4 identity. 
Now suppose that $\yvec = \Mmat \xvec$, where $\Mmat$ represents a nice invertible projective map. Let's say that $\Amat$ is the inverse matrix. Then you have 
\begin{align}
\xvec^t \Imat \xvec & = 1 \\
(\Amat\yvec)^t \Imat (\Amat\yvec) & = 1 \\
\yvec^t \Amat^t \Imat \Amat\yvec & = 1 \\
\yvec^t (\Amat^t \Amat) \yvec & = 1 
\end{align}
In other words, $\yvec$ satisfies an equation defined by the symmetric matrix $\Amat^t \Amat$. That's basically ends up being a quadratic expression in the entries of $\yvec$ being set equal to 1. So you get an ellipsoid, hyperboloid of one or two sheets, a sphere, or a paraboloid. I think that's the whole list. 
