zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 {\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)]. 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].
53A05Surfaces in Euclidean space
33E05Elliptic functions and integrals
49Q10Optimization of shapes other than minimal surfaces
49Q20Variational problems in a geometric measure-theoretic setting