TITLE: Area of the minimal surface of a non-planar quadrilateral in 3d
QUESTION [14 upvotes]: Consider a non-planar quadrilateral in three dimensions, i.e. four points $x_1,\dots,x_4$ in $\mathbb{R}^3$ that do not lie on a plane and connected by straight lines. Then, by general theory of minimal surfaces and the Plateau problem there exists a surface of minimal area with this lines as boundary. The situation looks like this:

(Picture from http://mathworld.wolfram.com/SkewQuadrilateral.html) but note that the obvious bilinear interpolation is not the minimal surface.
There are formulae for the minimal surfaces such as the Weierstraß-Enneper formula but I haven't come across a formula for this particular case of a quadrilateral.
In fact, I am not interested in a formula for the surface but only look for an answer to the question:

What is area of the minimal surface of the quadrilateral in terms of the four corner points $x_1,x_2,x_3,x_4$?

REPLY [5 votes]: This paper seems to give a partial answer to the posed question, for
skew quadrilaterals that project to rectangles:

Furui, Sadataka, and Bilal Masud. "Numerical calculation of a minimal surface using bilinear interpolations and Chebyshev polynomials." 
  arXive: math-ph/0608043 (2006).
  
            
  


Abstract.
"We calculate the minimal surface bounded by four-sided figures whose projection on a plane is a rectangle, starting with the bilinear interpolation and using, for smoothness, the Chebyshev polynomial expansion in our discretized numerical algorithm to get closer to satisfying the zero mean curvature condition. We report values for both the bilinear and improved areas, suggesting a quantitative evaluation of the bilinear interpolation. An analytical expression of the Schwarz minimal surface with polygonal boundaries and its 3-dimensional plot is also given."