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:

enter image description here

(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 $x_1,x_2,x_3,x_4$?

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  • What sort of formula are you expecting for your answer? Given that (as you note) the surface is likely to be very complicated, your best-case scenario is probably going to be getting the result in terms of ratios of elliptic functions of cross-ratios or somesuch, but all of the examples I've seen solved explicitly (see the 'four lines' surface at people.fas.harvard.edu/~sfinch/csolve/ge.pdf , for instance) seem to take essential advantage of some of the symmetries of the example. – Steven Stadnicki Nov 9 '14 at 17:11
  • Also, a caveat: you use the phrase 'a minimal surface' but it's not clear that there's a single minimal surface spanning the polygon and picking out the one of least area may not be trivial/formulaic. – Steven Stadnicki Nov 9 '14 at 17:13
  • A formula in elliptic functions or another non-elementary integral would be OK. In the end I would be happy with a simple numerical method to calculate the value. – Dirk Nov 9 '14 at 18:54
  • Regarding your other comment: I thought 'minimal surface' would mean 'surface of minimal area' but probably there are some local minima (whatever this means in this context)? – Dirk Nov 9 '14 at 19:00
  • This was asked at math.stackexchange.com on Nov 7 2014.<br> This was migrated to mathoverflow.net (here) on Nov 20 2014.<br> A similar question was asked at math.stackexchange.com on Nov 30 2015.<br> That one generates the curve by simple linear interpolation between the 4 points, and doesn't say whether the result is "minimal".<br> Something like an answer was added on Nov 30 2015. (It is not an expression in terms of a, b, c, d.)<br> **Compute the area defined by four non-planar points**<br> math.stackexchange.com/questions/1552551/… – A876 Nov 8 at 23:17

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).
          FigB2


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."

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