hyperbolic paraboloid "saddle"
Slicing this 3D surface with equation in planes perpendicular to either the x-axis or y-axis gives 2D curves that are parabolas. Slicing in planes perpendicular to the vertical z-axis (z≠0) gives 2D curves that are hyperbolas.
That's why this 3D surface is called a hyperbolic paraboloid.
(Hmmm, is there such a thing as "parabolic hyperboloid? If so, are these two types of surfaces synonymous?)
Notice that one set of parabolas has maxima, and the other set of parabolas has minima. The origin lies on this surface and is both a local minimum and local maximum, depending on the direction in which you slice the surface.
Drag the magenta point on the vertical z-axis to adjust the magenta cross section that is perpendicular to that z-axis.
Drag the red point and/or green point on the horizontal x- or y-axis to adjust the red cross section and/or green cross section that is perpendicular to those axes.
Click/drag in the 3D graphics view and/or use zoom buttons to change your viewing perspective.
A hyperbolic paraboloid is an example of a "ruled surface" for which every point on the surface lies on a perfectly straight line that also lies entirely on the surface. In fact, every point on the the hyperbolic paraboloid has two lines in the surface that pass through it, making it a "doubly-ruled" surface.
Click the "ruled surface" button to cycle through some options for viewing the straight lines that generate this surface.
Resource that gives equations used to model this: https://mathcurve.com/surfaces.gb/paraboloidhyperbolic/paraboloidhyperbolic.shtml