The distance of closest approach of two objects is the distance between their centers when they are externally
tangent. The objects may be
geometric shapes or
physical particles with well-defined boundaries. The distance of closest approach is sometimes referred to as the contact distance.
For the simplest objects,
spheres, the distance of closest approach is simply the sum of their radii. For non-spherical objects, the distance of closest approach is a function of the orientation of the objects, and its calculation can be difficult. The maximum
packing density
A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. In simplest terms, this is the ratio of the volume of bodies in a space to the volume of the space itself. I ...
of hard particles, an important problem of ongoing interest, depends on their distance of closest approach.
The interactions of particles typically depend on their separation, and the distance of closest approach plays an important role in determining the behavior of
condensed matter
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the su ...
systems.
Excluded volume
The excluded volume of particles (the volume excluded to the centers of other particles due to the presence of one) is a key parameter in such descriptions,; the distance of closest approach is required to calculate the excluded volume. The excluded volume for identical spheres is just four times the volume of one
sphere. For other
anisotropic objects, the excluded volume depends on orientation, and its calculation can be surprising difficult. The simplest shapes after spheres are ellipses and ellipsoids; these have receive
considerable attention yet their excluded volume is not known. Vieillard Baron was able to provide an overlap criterion for two ellipses. His results were useful for computer simulations of hard particle systems and for
packing problem
Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few conta ...
s using
Monte Carlo simulations.
The one anisotropic shape whose excluded volume can be expressed analytically is the
spherocylinder; the solution of this problem is a classic work by Onsager. The problem was tackled by considering the distance between two line segments, which are the center lines of the capped cylinders. Results for other shapes are not readily available.
The orientation dependence of the distance of closest approach has surprising consequences. Systems of hard particles, whose interactions are only entropic, can become ordered. Hard spherocylinders form not only orientationally ordered nematic, but also positionally ordered smectic phases. Here, the system gives up some (orientational and even positional) disorder to gain disorder and
entropy elsewhere.
Case of two ellipses
Vieillard Baron first investigated this problem, and although he did not obtain a result for the distance of closest approaches , he derived the overlap criterion for two ellipses. His final results were useful for the study of the phase behavior of hard particles and for the
packing problem
Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers. The goal is to either pack a single container as densely as possible or pack all objects using as few conta ...
using
Monte Carlo simulations. Although overlap criteria have been developed, 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
Fortran 90 subroutine is provided in Ref.
The procedure consists of three steps:
#
Transformation of the two
tangent ellipses
and
, whose centers are joined by the
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
, into a
circle and an ellipse
, whose centers are joined by the vector
. The circle
and the ellipse
remain tangent after the transformation.
# Determination of the distance
of closest approach of
and
analytically. It requires the appropriate solution of a
quartic equation
In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is
:ax^4+bx^3+cx^2+dx+e=0 \,
where ''a'' ≠ 0.
The quartic is the highest order polynomi ...
. The normal
is calculated.
# Determination of the distance
of closest approach and the location of the point of contact of
and
by the inverse transformations of the vectors
and
.
Input:
*
lengths of the semiaxes
,
*
unit vectors
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vec ...
,
along major
axes
Axes, plural of '' axe'' and of '' axis'', may refer to
* ''Axes'' (album), a 2005 rock album by the British band Electrelane
* a possibly still empty plot (graphics)
See also
* Axess (disambiguation)
*Axxess (disambiguation) Axxess may refer to ...
of both ellipses, and
*
unit vector joining the centers of the two ellipses.
Output:
*distance
between the centers when the ellipses
and
are
externally tangent, and
*location of point of contact in terms of
,
.
Case 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
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
. 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. Subroutines are provided in two formats: Fortran90 and C.
C subroutine for distance of closest approach of ellipsoids
/ref>
The algorithm consists of three steps.
# 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.
# 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.
# 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.
See also
* Apsis
* Impact parameter
References
{{DEFAULTSORT:Distance Of Closest Approach Of Ellipses And Ellipsoids
Conic sections
Distance