Here is a simple little script, which creates a physics mathematics based Mylar Balloon.
The "oval" Mylar Balloon starts off as two circular disks of Mylar, fused together at their perimeters, which is then inflated.
Wrinkles or crimps or crinkles or creases are not included.


The (Mylar) Balloon script is a 2d curve generator, which creates a curve in one quadrant of the xz plane.
Mirror and Revolve to complete the balloon, are left up to the user. (Or Revolve and Mirror)
Mylar is a Dupont trademark.

A general link is:
http://en.wikipedia.org/wiki/Mylar_balloon_(geometry) add on the trailing )

Pretty pictures:
http://obzor.bio21.bas.bg/conference/Conference_files/sa14/Pulov%202014.pdf

There are several similar papers that can be downloaded from the internet:
The most detailed is viewable on jstor. (Sign up and use one of three "shelves".):
Mladenov I. and Oprea J., The Mylar Balloon Revisited. November 2003.

The script uses Carlson symmetric form elliptic integrals, and the Jacobi functions.
Additional papers, remarks and information are in the htm file.

The default modulus of .707106... (sqrt(2) /2) is correct for the mylar balloon.
Other values of the modulus between 0 and 1, exclusive, make a different 2d curve.

- Brian





Corrected script is in _Balloon2.zip, Oct31, 2014

Using the MylarBalloon curve as an isotensoid, here is a fake "pumpkin" balloon.
It is just a swept arbitrary arc.

- Brian

ps The MylarBalloon script does not seem to have quite the correct height. (?)

pps Should have styled it orange.

Timely - and impressive!

The mylar balloon profile script is incorrect.
The aspect ratio is too low. The height is too low.
Using an alternate parameterization a curve was generated which created the correct aspect ratio (height over width), (about 0.599).

The error is yet to be found.

So I will delete the script until corrected.

- Brian

Just uploaded, in message 1, the corrected _Balloon2 script, which is using the simple formula, without cn or dn.

Added a modulus k slider to create other, non-mylar shapes, which are likely not physics based.
For a modulus k = -0.908908557548541+, the origin is (very nearly) crossed, and a revolve results in a solid.

The value was found by using a wide slider with very narrow range -0.9089056 to -0.9089055, then sliding until the z axis is crossed just above or below
the origin. More decimals can be manually added.

For the mylar balloon curve, the aspect ration (Thickness/Diameter) = 0.599+ approximately, is now correct.
Also, geodesics could be calculated for the mylar balloon curve.

- Brian