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

If you experiment with 2z=(x²/p)-(y²/q) ,,, p*q>0

on graph paper you will find hyperbolic paraboloid

let p=q=1. when x=0, -y²=2z,,,,when y=0, x²=2z, parabolas, one open upward one downward

when z=0 (the xy plane) the result is two lines

when z=h where h>0 then two hyperbola curves are generated which intersect x-axis

when z=h where h<0 then two hyperbola curves are generated which intersect y-axis

x=l plane is parabola, y=l plane is parabola

note h and l are constants that can be experimented with to see how surface grows

Note: the surface equation z=xy is a hyperbolic parabolooid.

rotate xy plane by 45 degrees x=.707x'-.707y' and y=.707x'+.707y'

now equation is in identifable form,

2z'=x'²-y'²