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The Mylar balloon: new viewpoints and generalizations. (English) Zbl 1123.53006
Mladenov, Ivaïlo (ed.) et al., Proceedings of the 8th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 9–14, 2006. Sofia: Bulgarian Academy of Sciences (ISBN 978-954-8495-37-0/pbk). 246-263 (2007).
The Mylar balloon is physically determined: Sew together two disks of Mylar and inflate it for instance by air, the resulting balloon is called a Mylar balloon. This object has rotational symmetry (but is different from a sphere). Investigations are due to W. Paulsen [Am. Math. Mon. 101, No. 10, 953–958 (1994; Zbl 0847.49030)], F. Baginski [SIAM J. Appl. Math. 65, No. 3, 838–857 (2005; Zbl 1072.74046)], G. Gibbons [DAMTP Preprint, Cambr. Univ. (2006)] and the authors [Am. Math. Mon. 110, No. 9, 761–784 (2003; Zbl 1044.33008)].
In the present paper the Mylar balloon is first modelled as a linear Weingarten surface of revolution. A parametrization of such a surface is given as \(x(u,v)=(u \cos v, u \sin v, z(u))\) where \(z(u)\) is expressed by hypergeometric functions using MAPLE.
For the second approach the authors use the parametrization \(x(s,v)=(r(s) \cos v, r(s) \sin v, z(s))\) (whith \(s\) the arclength on the meridian curve \((r(s),z(s))\) and figure out the equilibrium conditions. Special solutions of these non linear equations are presented. Let denote \(H=(1/2)(k_{\mu}+k_{\pi})\) with \(k_{\mu}\) and \(k_{\pi}\) the main curvatures related to the meridian and to the parallel curves, respectively. Then one class of examples are the Delaunay surfaces \((H=\text{const.})\). The second example is the Mylar balloon \((k_{\mu}=2k_{\pi})\). The paper also contains visualizations using MAPLE.
For the entire collection see [Zbl 1108.53003].

53A05 Surfaces in Euclidean and related spaces
33E05 Elliptic functions and integrals
49Q10 Optimization of shapes other than minimal surfaces
49Q20 Variational problems in a geometric measure-theoretic setting