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Skew Quadrilateral

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SkewQuadrilateral

A four-sided quadrilateral not contained in a plane. The lines connecting the midpoints of opposite sides of a skew quadrilateral intersect (and bisect) each other (Steinhaus 1999).

The problem of finding the minimum bounding surface of a skew quadrilateral was solved by Schwarz (Schwarz 1890, Wells 1991) 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

Hyperbolic Paraboloid, Quadrilateral, Skew Polygon

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References

Altshiller-Court, N. "The Skew Quadrilateral." Ch. 3 and §5.1 in Modern Pure Solid Geometry. New York: Chelsea, pp. 42-47 and 111-115, 1979.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., p. 53, 1967.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, 1972.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 242 and 244, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 186-187, 1991.

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Skew Quadrilateral

Cite this as:

Weisstein, Eric W. "Skew Quadrilateral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SkewQuadrilateral.html

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