What does it imply when two lines cross each other in a level curve

Question

I'm having trouble to articulate how the actual graph looks like when z = 0, meaning in the middle of the level curve where intersection occurs.

What shape would the graph look like and what curvature is it supposed to have?

Answer 1

This hyperbolic paraboloid has has a pair of horizontal lines crossing at the central saddle point. That's a level set, but not a level curve.

A level set is the intersection of a surface with a plane. If the surface is smooth that will be a one dimensional object in the plane. Sometimes it will be a single curve in the usual sense - then you would call it a level curve. But it might be an intersection of lines or curves, or come in several pieces. Then calling it a level curve is a mild abuse of language, but will not confuse anyone.

https://commons.wikimedia.org/wiki/File:Hyperbolic-paraboloid.svg

Answer 2

This is a saddle point, like a Pringle. If you move in certain direction away from the origin (positive or negative $x$-directions) the function will increase, while if you move away from the origin in other directions (positive or negative $y$-directions) the function will decrease.