Franz J. Vesely >   Notes > Spheroellipsoids
 


Figure 1: Definitions

Spheroellipsoids

Truncated ellipsoid capped by sphere segments; where the caps are fused to the body they have the same tangent as the ellipsoid.

Advantages:

  • (a) convex everywhere
  • (b) only one additional parameter ($z_{0}$, see below)
  • (c) shape may be tuned between spherocylinder and ellipsoid
  • (d) not homeomorphic to a sphere (like the ellipsoid); may therefore have a smectic phase
  • (e) physically more reasonable than spherocylinders, as many mesogens have a bulge around the waist

Equations:
Given long and short semiaxes $c$ and $a$ and a truncation half length $z_{0}$ we have
$z_{c}=z_{0}(1-a^{2}/c^{2})$ ... center of cap
$r_{c}^{2}=a^{2}[1-(z_{0}^{2}/c^{2})(1-a^{2}/c^{2})]$ ... radius of cap
$D=2 \, a$ ... particle width
$L=2 \, (z_{c}+r_{c})$ ... particle length



Figure 2: Hard spheroellipsoid with $c/a=6.0$, $z_{0}/a=3.06$

Here is an Applet to try out various shapes:


Obviously, the shape may be continuously tuned between an ellipse (simply set $z_{0}=c$) and a spherocylinder: with a large $c$ we achieve a very small curvature of the ellipsoid near the equator, and a small truncation height $z_{0} \ll c $ produces a near-spherocylinder. Try it out with the applet!

Contact Distance between Convex Bodies

This is a fascinating topic, and I will prepare a thorough treatment in a further communication. Here, for the time being, is an Applet for determining the contact distance between two SpheroEllipses:




Here is some Fortran code (version 12.11)
[untar -> read README -> play!]
 
F. Vesely Nov-2012