## 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].

### MSC:

 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

### Keywords:

Mylar balloon; Weingarten surface; equilibrium

### Citations:

Zbl 0847.49030; Zbl 1072.74046; Zbl 1044.33008

Maple