# Area of the minimal surface of a non-planar quadrilateral in 3d

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 $x_1,x_2,x_3,x_4$?

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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 '18 at 23:17