*(English)*Zbl 1123.53006

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 $x(u,v)=(ucosv,usinv,z(u\left)\right)$ where $z\left(u\right)$ is expressed by hypergeometric functions using MAPLE.

For the second approach the authors use the parametrization $x(s,v)=\left(r\right(s)cosv,r(s)sinv,z(s\left)\right)$ (whith $s$ the arclength on the meridian curve $\left(r\right(s),z(s\left)\right)$ 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 $({k}_{\mu}=2{k}_{\pi})$. The paper also contains visualizations using MAPLE.

##### MSC:

53A05 | Surfaces in Euclidean space |

33E05 | Elliptic functions and integrals |

49Q10 | Optimization of shapes other than minimal surfaces |

49Q20 | Variational problems in a geometric measure-theoretic setting |