Assume I have a surface $S$, defined as $Ax + By + Cz + D = 0$, and a point $p$ defined as $(x, y, z)$ that is not on that plane. Assuming the $S$ is a plane, I understand that the set of all points on $S$ that are at distance $d$ from $p$ form an ellipse but I don't know how to calculate the equation of that ellipse.
When I relax the restriction to include all points on $S$ at distance $\leq d$ from $p$, how do I calculate the equation of that ellipse?
If $S$ is a plane of $\mathbb{R}^3$ and $p \in \mathbb{R}^3$ is a point not in $S$, notice that the set of points at a distance $d$ from $p$ is a sphere with center $p$ and radius $d$, call it $S_{p,d}$. If you want to find the points in $S_{p,d}$ that are also in $S$, you just have to find the intersection between a sphere and a plane. I leave it to you to find this intersection, but you can visually see that it is not an ellipse (by symmetry, it could not be, right?).