User:Xyzheng/distance of closest approach of two hard ellipses

Two externally tangent ellipses

The distance of closest approach of hard particles is a key parameter of their interaction and plays an important role in the resulting phase behavior. For non-spherical particles, the distance of closest approach depends on orientation, and its calculation can be surprisingly difficult. Although overlap criteria have been developed for use in computer simulations , analytic solutions for the distance of closest approach and the location of the point of contact have only recently become available . The details of the calculations are provided in Ref. . The Fortran90 subroutine are provided in Ref.

The Method

The procedure constists of three steps:

1. Transformation of the two tangent ellipses $E_{1}$ and $E_{2}$, whose centers are joined by the vector $d$, into a circle $C_{1}'$ and an ellipse $E_{2}'$, whose centers are joined by the vector $d'$. The circle $C_{1}'$ and the ellipse $E_{2}'$ remain tangent after the transformation.
2. Determination of the distance $d'$ of closest approach of $C_{1}'$ and $E_{2}'$ analytically. It requires the appropriate solution of a quartic equation. The normal $n'$ is calculated.
3. Determination of the distance $d$ of closest approach and the location of the point of contact of $E_{1}$ and $E_{2}$ by the inverse transformations of the vectors $d'$ and $n'$.

Input:

• lengths of the semiaxes $a_{1},b_{1},a_{2},b_{2}$,
• unit vectors $k_{1}$,$k_{2}$ along major axes of both ellipses, and
• unit vector $d$ joining the centers of the two ellipses.

Output:

• distance $d$ between the centers when the ellipses $E_{1}$ and $E_{2}$ are externally tangent, and
• location of point of contact in terms of $k_{1}$,$k_{2}$.