Mylar balloon (geometry)

Definition

The positive portion of the generatrix of the balloon is the function z(x) where for a given generatrix length a:

:z(r)=0

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

:\int_0^r ! 4\pi x z(x) , dx 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:

: 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}] ,

(where E and F are elliptic integrals of the second and first kind)

Measurement

The "thickness" of the balloon (that is, the distance across at the axis of rotation) can be determined by calculating 2 z({\frac{\pi}{2}} ) from the parametric equations above. The thickness τ is given by

: {\tau} = 2Br,

while the generatrix length a is given by

: a = Ar

where r is the radius; A ≈ 1.3110287771 and B ≈ 0.5990701173 are the first and second lemniscate constants.

Volume

The volume of the balloon is given by:

: V = \frac{2}{3} \pi a r^2,

where a is the arc length of the generatrix).

or alternatively:

: V = \frac{4}{3} \tau a^2,

where τ is the thickness at the axis of rotation.

Surface area

The surface area S of the balloon is given by

: S = \pi^2r^2

where r is the radius of the balloon.

Derivation

Substituting u = \arccos(x/r) into the parametric equation for z(u) given in yields the following equation for z in terms of x:

z(x) = r \sqrt{2} \left[ E(\arccos(x/r),\frac{1}{\sqrt{2}})-\frac{1}{2}F(\arccos(x/r), \frac{1}{\sqrt{2}}) \right]

The above equation has the following derivative:

\frac{dz}{dx} = -\frac{x^2}{\sqrt{r^4 - x^4}}

Thus, the surface area is given by the following:

S = \int_0^r ! 4\pi x \sqrt{1 + \left(\frac{dz}{dx}\right)^2}, dx

Solving the above integral yields S = \pi^2r^2.

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.

References

• {{cite journal| author=Mladenov, I. M.| year=2001| title=On the Geometry of the Mylar Balloon| journal=C. R. Acad. Bulg. Sci.| volume=54| pages=39–44| bibcode=2001CRABS..54i..39M}}
• {{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