## Surface area of a hyperbolic paraboloid patch bounded by a skew quadrilateral

Given four non-planar 3D points, these points define a skew quadrilateral ABCD.

Example:

A = (0,0,0); B = (0,3,0); C = (3,3,2); D = (3,0,-1);

The bilinear interpolation of these vertices define a patch for a doubly-ruled surface that is an hyperbolic paraboloid. My question is:

Is there an analytical formula to compute the surface area of this patch?

(Note: this is not the minimal surface solved by Schwarz using abelian integrals.)

In a trivial case, with the surface given by:

z = x y

Parametric equations: x(u,v) = u; y(u,v) = v; z(u,v) = u v

And points given by:

A = (-1, -1, 1)

B = ( 1, -1, -1)

C = ( 1, 1, 1)

D = (-1, 1, -1)

The surface area can be found by integrating dS over this domain:

$dS = \sqrt{1 + u^2 + v^2}$

I've found the surface area for this trivial surface patch to be:

$S = 2/9 (-\pi + 6 (\sqrt{3} + \log{(7 + 4 \sqrt{3})})) \approx 5.12316$

It can be extended to other simple cases, such as A = (0, 0, 0), B = (0, b, 0), C = (a, b, a b), D = (a, 0, 0), but I could not extend it to the generic case (i.e., any A, B, C, D points). Note that the generic case is just an affine transformation of the trivial case.

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