Distance of closest approach of two 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 [1][2], analytic solutions for the distance of closest approach and the location of the point of contact have only recently become available [3][4]. The details of the calculations are provided in Ref. [5]. The Fortran 90 subroutine is provided in Ref.[6].

The method

The procedure consists of three steps:
1. Transformation of the two tangent ellipses E1 and E2, whose centers are joined by the vector d, into a circle C1' and an ellipse E2', whose centers are joined by the vector d'. The circle C1' and the ellipse E2' remain tangent after the transformation.
2. Determination of the distance d' of closest approach of C1' and E2' 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 E1 and E2 by the inverse transformations of the vectors d' and n'.

Input:

* lengths of the semiaxes a1,b1,a2,b2,
* unit vectors k1,k2 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 E1 and E2 are externally tangent, and
* location of point of contact in terms of k1,k2.

Distance of closest approach of two ellipsoids

Consider two ellipsoids, each with a given shape and orientation, whose centers are on a line with given direction. We wish to determine the distance between centers when the ellipsoids are in point contact externally. This distance of closest approach is a function of the shapes of the ellipsoids and their orientation. There is no analytic solution for this problem, since solving for the distance requires the solution of a sixth order polynomial equation. Here an algorithm is developed to determine this distance, based on the analytic results for the distance of closest approach of ellipses in 2D, which can be implemented numerically. Details are given in publications [8][9]. Subroutines are provided in two formats: Fortran90 [10] and C [11].

Method

The algorithm consists of three steps.
1. Constructing a plane containing the line joining the centers of the two ellipsoids, and finding the equations of the ellipses formed by the intersection of this plane and the ellipsoids.
2. Determining the distance of closest approach of the ellipses; that is the distance between the centers of the ellipses when they are in point contact externally.
3. Rotating the plane until the distance of closest approach of the ellipses is a maximum. The distance of closest approach of the ellipsoids is this maximum distance.

References

  1. J. Vieillard-Baron, "Phase transition of the classical hard ellipse system", J. Chem. Phys., 56(10), 4729 (1972).
  2. J. W. Perram and M. S. Wertheim, "Statistical mechanics of hard ellipsoids. I. overlap algorithm and the contact function", J. Comput. Phys., 58, 409 (1985).
  3. X. Zheng and P. Palffy-Muhoray. "Distance of closest approach of two arbitrary hard ellipses in two dimensions", http://www.e-lc.org/docs/2007 01 17 00 46 52, (2007).
  4. X. Zheng and P. Palffy-Muhoray. "Distance of closest approach of two arbitrary hard ellipses in two dimensions", Phys. Rev. E, 75,061709 (2007).
  5. Complete version containing contact point algorithm and corrected typos , May 4, 2009.
  6. Fortran90 subroutine including the contact coordinates
  7. Fortran90 subroutine in quad precision without the contact coordinates
  8. Animation for ellipses
  9. Animation for ellipsoids
  10. X. Zheng, W. Iglesias, P. Palffy-Muhoray, "Distance of closest approach of two arbitrary hard ellipsoids", Phys. Rev. E, 79, 057702 (2009).
  11. X. Zheng, W. Iglesias, P. Palffy-Muhoray, "Distance of closest approach of two arbitrary hard ellipsoids", electronic Liquid Crystal Communications, 2008.
  12. Fortran90 subroutine for distance of closest approach of ellipsoids
  13. C subroutine for distance of closest approach of ellipsoids