"Cylindrical" membrane Vesicles

 From: bemfarmer 24 Mar 2015  (1 of 2)
 Here are some closed 2d curves which may be of interest. Aspire 8 is out, and is now 64 bit. Sorry, no script for this, due to programming complexity of finding the roots nu and qu. Source papers by V M Vassilev, P A Djondjorov and I M Mladenov: Cylindrical equilibrium shapes of fluid membranes: http://arxiv.org/pdf/0803.0843 Analytic description and explicit parametrization of the equilibrium shapes of elastic rings and tubes under uniform hydrostatic pressure: http://arxiv.org/pdf/1008.0533 (This paper uses the Mathematica notebooks.) The Mathematica notebook files: http://www.bio21.bas.bg/ibf/dpb_files/mfiles/ The first paper helps in understanding the second paper. The second paper derives parametric solutions to the "membrane shape equation." Sigma, an o with a tail, is the "pressure." Sigma > n*n - 1. n is the number of lobes, e.g. 2, 3, 4, 5... The Mathematica notebooks are used to solve all of the equations, and plot the curves. It is necessary to solve two trancendental equations to find the roots nu and qu. (How to script this?) The complete and incomplete elliptic integrals of the third kind are used. The complete elliptic integral of the first kind is used. Several other parameters are calculated, and then used in the parametric equations to plot the curves. The downloaded Mathematic workbooks include several closed curves. The files called "Lines" have portions of the membranes collapsed. Right click in the area of the closed curve, select "Save Graphic As," select pdf, and save. Load the pdf into Moi. Other curves can be generated. It takes a few seconds. After selecting ""appropriate" sigma and n, evaluate the Mathematica notebook. Then save the closed curve as a pdf. I've found it necessary to have the cursor located on a line between formulas, and the notebook must be reloaded between evaluations. The lobes of the pdf curves do not seem to be precisely identical. Moi can locate center(s), and one of the lobes can be used to circularly array the other lobes, for symmetry. At some point, Rebuild is definitely in order. If there is any interest, more of these 2d shapes could be uploaded. -Brian