How would I explain the maths leading to the equation of a hyperbolic paraboloid?

Question

How could I explain, in a clear and relatively easy manner how one ends up the function of an hyperbolic paraboloid? Like how do we know that its formula is what it is and how would one explain how to get there?

I hope my questions make sense...

Answer 1

For every given point x, the cross section in y direction is a parabola with branches up

For every given point y, the cross section in x direction is a parabola with branches down

z = y^2/b^2 - x^2/a^a