What would be called as the shape of $xy=10$ in 3-dimensional space?

Question

As title says, what would be called as the shape of $xy=10$ in 3-dimensional space? It doesn't seem to be paraboloid nor hyperboloid...

Answer 1

Such a shape is often called a hyperbolic cylinder, which as you say is neither a paraboloid nor a hyperboloid. As a general rule a quadric in 3 dimensions whose equation does not involve one variable (like $z$) will be called a cylinder. For example $x^2 + 2y^2 = 1$ forms an elliptic cylinder, as the equation defines an ellipse in 2 dimensional space.

Addition to respond to comment on question: To see the equivalence with the standard form for a hyperbolic cylinder, make the change of variables $x \mapsto X + Y$, and $y\mapsto X-Y$, then $xy=10$ becomes $(X+Y)(X-Y)=10$ or $X^2 - Y^2 =10$.