Skew Quadrilateral

A four-sided Quadrilateral not contained in a plane. The problem of finding the minimum bounding surface of a skew quadrilateral was solved by Schwarz (1890) in terms of Abelian Integrals and has the shape of a Saddle. It is given by solving

$$(1+{f_y}^2)f_{xx}-2f_xf_yf_{xy}+(1+{f_x}^2)f_{yy}=0.$$

See also Quadrilateral

References

Isenberg, C. The Science of Soap Films and Soap Bubbles. New York: Dover, p. 81, 1992.

Forsyth, A. R. Calculus of Variations. New York: Dover, p.  503, 1960.

Schwarz, H. A. Gesammelte Mathematische Abhandlungen, 2nd ed. New York: Chelsea.