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Zbl 1123.53006
Mladenov, Iva\"ilo M.; Oprea, John
The Mylar balloon: new viewpoints and generalizations. (English)
[A] Mladenov, Iva\"ilo (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. 246-263 (2007). ISBN 978-954-8495-37-0/pbk

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 {\it W. Paulsen} [Am. Math. Mon. 101, No. 10, 953--958 (1994; Zbl 0847.49030)], {\it F. Baginski} [SIAM J. Appl. Math. 65, No. 3, 838--857 (2005; Zbl 1072.74046)], {\it G. Gibbons} [DAMTP Preprint, Cambr. Univ. (2006)] and the authors [Am. Math. Mon. 110, No. 9, 761--784 (2003; Zbl 1044.33008)]. \par 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. \par 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.
[Friedrich Manhart (Wien)]

MSC 2000:: *53A05 Surfaces in Euclidean space
33E05 Elliptic functions and integrals
49Q10 Optimization of the shape other than minimal surfaces
49Q20 Variational problems in geometric measure-theoretic setting

Keywords: Mylar balloon; Weingarten surface; equilibrium

Citations: Zbl 0847.49030; Zbl 1072.74046; Zbl 1044.33008

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