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{{About|a geometric figure|balloons made of [[mylar]]|Balloon}}
In [[geometry]], a '''mylar balloon''' is a [[surface of revolution]].  While a [[sphere]] is the surface that encloses a maximal [[volume]] for a given [[surface area]], the mylar balloon instead maximizes volume for a given generatrix [[arc length]].  It resembles a slightly flattened sphere.

The shape is approximately realized by inflating a physical balloon made of two circular sheets of flexible, inelastic material; for example, a popular type of toy balloon made of [[BoPET|aluminized plastic]].  Perhaps counterintuitively, the surface area of the inflated balloon is less than the surface area of the circular sheets.  This is due to physical crimping of the surface, which increases near the rim.

"Mylar balloon" is the name for the figure given by W. Paulson, who first investigated the shape.  The term was subsequently adopted by other writers.  "Mylar" is a trademark of [[DuPont]].

== Definition ==

The positive portion of the generatrix of the balloon is the function ''z''(''x'') where for a given generatrix length ''a'':
[[File:Mylar Balloon Profile Sketch Temp.svg|right|Profile of the mylar balloon in the ''xz'' plane]]

:<math>z(r)=0</math>

:<math>\int_0^r \!\sqrt{1+z'(x)^2}\,dx \, = a</math>  (i.e.: the generatrix length is given)

:<math>\int_0^r \! 4\pi x z(x) \, dx </math> is a maximum  (i.e.: the volume is maximum)

Here, the radius ''r'' is determined from the constraints.

== Parametric characterization ==
The parametric equations for the generatrix of a balloon of radius r are given by:

: <math>x(u) = r \cos u;\qquad  z(u) = r \sqrt{2} \left[ E(u,\frac{1}{\sqrt{2}})-\frac{1}{2}F(u, \frac{1}{\sqrt{2}})\right]\text{ for }u \in [0, \frac{\pi}{2}] \, </math>

(where ''E'' and ''F'' are [[elliptic integrals]] of the [[Elliptic_integrals#Incomplete_elliptic_integral_of_the_second_kind|second]] and [[Elliptic_integrals#Incomplete_elliptic_integral_of_the_first_kind|first]] kind)

== Measurement ==
The "thickness" τ of the balloon (that is, the distance across at the axis of rotation) can be determined by calculating <math>2 z({\frac{\pi}{2}} )</math> from the parametric equations above.  The thickness is approximately
 
: τ ≈ 0.599 &middot; 2''r''.  

Note that the ratio of ''τ'' to ''r'' is independent of the size of the balloon.

The ratio of the generatrix's arc length a to the radius of the balloon is approximately 

: ''a''/''r'' ≈ 1.3110.

The [[volume]] of the balloon is given by:

: <math>V =  \frac{2}{3} \pi a r^2, </math>

where ''a'' is the arc length of the generatrix).

or alternatively:

: <math>V = \frac{4}{3} \tau a^2, </math>

where τ is the thickness at the axis of rotation

== Surface geometry ==
The ratio of the [[principal curvatures]] at every point on the mylar balloon is exactly 2, making it an interesting case of a [[Weingarten surface]]. Moreover, this single property fully characterizes the balloon.  The balloon is evidently flatter at the axis of rotation; this point is actually has zero curvature in any direction.

== See also ==

* [[Paper bag problem]]

== References ==
* {{cite journal|
author=Mladenov, I. M.|
year=2001|
title=On the Geometry of the Mylar Balloon|
journal=[[Comptes Rendus de l'Académie Bulgare des Sciences|C. R. Acad. Bulg. Sci.]]|
volume=54|
pages=39–44}}
* {{cite journal|
author=Paulsen, W. H.|
year=1994|
title=What Is the Shape of a Mylar Balloon?|
journal=[[American Mathematical Monthly]]|
volume=101|
pages=953–958|
doi=10.2307/2975161|
jstor=2975161|
issue=10}}
* {{cite web|
last=Finch|
first=Steven|
title=Inflating an Inelastic Membrane|
date=13 August 2013|
url=http://www.people.fas.harvard.edu/~sfinch/csolve/mylr.pdf
}}
[[Category:Surfaces]]