Find if two points are on the same sheets of a quadric surface

Question

I have the equation of a hyperboloid cylinder (having two sheets), and I want to test if two points are on the same sheet or different sheet. How can I do this?

Answer 1

One possibility is to find a plane separating the sheets of the surface, expressed in the form $F(x, y, z) = 0$ for some affine function $F$. Two points $p_{1}$ and $p_{2}$ are on the same sheet if and only if $F(p_{1}) F(p_{2}) > 0$ (i.e., the values of $F$ at the two points have the same sign).

For example, if your surface has equation $x^{2} - y^{2} = 1$, any of the following $F$ would be suitable: $$ F(x, y, z) = x;\qquad F(x, y, z) = x - y;\qquad F(x, y, z) = x + y. $$

Without knowing the details (e.g., the equation of your surface) it's hard to be more precise.