Skew quadrilateral

Note that all three possible skew quadrilaterals (ABCD = ADCB, ABDC = ACDB, ACBD = ADBC) are equivalent and thus any one of them can be selected for the canonical nomenclature. 

Fig. 4.10 The minimum, catenoid, surface contained by two coaxial funnels. Fig. 4.11 The minimum surface contained by a skew quadrilateral.

At the end of the last century H. A. Schwarz s solved analytically another problem in which the minimum surface is continuous and has continuous derivatives. This problem concerns the minimum surface area contained by a skew quadrilateral. That is a quadrilateral which is not confined to a plane. The solution can be expressed in terms of hyperelliptic integrals and has the shape of a saddle (Fig. 4.11). The analogue surface can be obtained by forming a soap film with a boundary formed by a wire frame which has the shape of the skew quadrilateral. 

The Laplace-Young equation was used, in section 4.4, to discuss the solution to such problems as the minimum surface area contained by two rings and that contained by a skew quadrilateral. In these problems the soap film produces a double surface. The pressure difference across the soap film is zero as the pressure is the same on both sides of the film. The Laplace-Young equation, (5.3), reduces to... 

The solution of this equation, with appropriate boundary conditions, will give the minimum area contained by a closed boundary. Such problems as the minimum area contained by two parallel coaxial rings and the minimum area contained by a skew quadrilateral can be solved using this equation. In principle all the minimum area problems discussed in Chapter 4 can be solved using (5.10). However most of them have not been found to be amenable to an analytic solution. 

This shape is, as one would expect, in the form of a saddle in addition to the sides of the skew quadrilateral, it admits two other straight hues, which would respectively join the midpoint of each of the sides to the midpoint on the opposite side. Finally Mr. Schwarz continues the surface beyond the contour indicated, and to deteimine the general shape of it the total surface is made of portions identical to that above, juxtaposed in a certain way, and it is very curious ... 

Finally, Mr. Schondorff treated, for his part, in a Memoire also prize-winning in 1867, but by the Gottingen Society, the question of the surface with minimum area which rests on a skew quadrilateral this quadrilateral satisfies the same conditions as that of Mr. Schwarz. 

Mr. Schwarz determination of surface supported on the sides of a skew quadrilateral, in a case more general than that of 141 the same of another surface deriving from the preceding one by inflection. - Symmetry in surfaces of minimum area on which one can trace a straight line, and in those which can be cut by a plane in such a way that, all along the section, their elements are perpendicular to this plane. Determination, by elliptic functions, of two particular surfaces. 143... 

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