Distance of closest approach of ellipses and ellipsoids

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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.

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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 of hard particles, an important problem of ongoing interest, [1] 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 systems.

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,; [2] [3] 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. [4] The simplest shapes after spheres are ellipses and ellipsoids; these have received considerable attention, [5] 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 problems using Monte Carlo simulations.

The concept of excluded volume was introduced by Werner Kuhn in 1934 and applied to polymer molecules shortly thereafter by Paul Flory.

Sphere round geometrical and circular object in three-dimensional space; special case of spheroid

A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball.

Monte Carlo Quarter and ward of Monaco

Monte Carlo officially refers to an administrative area of the Principality of Monaco, specifically the ward of Monte Carlo/Spélugues, where the Monte Carlo Casino is located. Informally the name also refers to a larger district, the Monte Carlo Quarter, which besides Monte Carlo/Spélugues also includes the wards of La Rousse/Saint Roman, Larvotto/Bas Moulins, and Saint Michel. The permanent population of the ward of Monte Carlo is about 3,500, while that of the quarter is about 15,000. Monaco has four traditional quarters. From west to east they are: Fontvieille, Monaco-Ville, La Condamine, and Monte Carlo.

Two externally tangent ellipses Ellipses.png
Two externally tangent ellipses

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. [6] 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. [7] Here, the system gives up some (orientational and even positional) disorder to gain disorder and entropy elsewhere.

Entropy physical property of the state of a system, measure of disorder

In statistical mechanics, entropy is an extensive property of a thermodynamic system. It is closely related to the number Ω of microscopic configurations that are consistent with the macroscopic quantities that characterize the system. Under the assumption that each microstate is equally probable, the entropy is the natural logarithm of the number of microstates, multiplied by the Boltzmann constant kB. Formally,

Distance of closest approach of two ellipses

Vieillard Baron first investigated this problem, and although he did not obtain a result for the distance of closest approach, he derived the overlap criterion for two ellipses. His results were useful for the study of the phase behavior of hard particles and for the packing problem using Monte Carlo simulations. Although overlap criteria have been developed, [8] [9] analytic solutions for the distance of closest approach and the location of the point of contact have only recently become available. [10] [11] The details of the calculations are provided in Ref. [12] The Fortran 90 subroutine is provided in Ref. [13]

Method

The procedure consists of three steps:

  1. Transformation of the two tangent ellipses and , whose centers are joined by the vector , into a circle and an ellipse , whose centers are joined by the vector . The circle and the ellipse remain tangent after the transformation.
  2. Determination of the distance of closest approach of and analytically. It requires the appropriate solution of a quartic equation. The normal is calculated.
  3. 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:

Cartesian coordinate system coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

Unit vector vector of length one

In mathematics, a unit vector in a normed vector space is a vector of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat": . The term direction vector is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted as d. Two 2D direction vectors, d1 and d2 are illustrated. 2D spatial directions represented this way are numerically equivalent to points on the unit circle.

Output:

Tangent straight line touching a point in a curve

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f (x) at a point x = c on the curve if the line passes through the point (c, f ) on the curve and has slope f'(c) where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.

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. [14] [15] Subroutines are provided in two formats: Fortran90 [16] and C. [17]

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.

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Curvature physical quantity

In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object such as a surface deviates from being a flat plane, or a curve from being straight as in the case of a line, but this is defined in different ways depending on the context. There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space – in a way that relates to the radius of curvature of circles that touch the object – and intrinsic curvature, which is defined in terms of the lengths of curves within a Riemannian manifold.

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References

  1. S. Torquato and Y. Jiao, Nature 460, 876-879, 2009
  2. T.L. Hill, An Introduction to Statistical Thermodynamics (Addison Wesley, London, 1960)
  3. T.A. Witten, and P.A. Pincus, Structured Fluids (Oxford University Press, Oxford, 2004)
  4. Forces, Growth and Form in Soft Condensed Matter: At the Interface between Physics and Biology, ed. A.T. Skjeltrop and A.V. Belushkin, (NATO Science Series II: Mathematics, Physics and Chemistry, 2009),
  5. A. Donev, F.H. Stillinger, P.M. Chaikin and S. Torquato, Phys. Rev. Lett. 92, 255506 (2004)
  6. L. Onsager, Ann NY Acad Sci, 51, 627 (1949)
  7. D. Frenkel, J. Phys. Chem. 91, 4912-4916 (1987)
  8. J. Vieillard-Baron, "Phase transition of the classical hard ellipse system" J. Chem. Phys., 56(10), 4729 (1972).
  9. 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).
  10. X. Zheng and P. Palffy-Muhoray, "Distance of closest approach of two arbitrary hard ellipses in two dimensions", electronic Liquid Crystal Communications, 2007
  11. X. Zheng and P. Palffy-Muhoray, "Distance of closest approach of two arbitrary hard ellipses in two dimensions", Phys. Rev. E, 75, 061709 (2007).
  12. X. Zheng and P. Palffy-Muhoray, Complete version containing contact point algorithm, May 4, 2009.
  13. Fortran90 subroutine for contact distance and contact point for 2D ellipses by X. Zheng and P. Palffy-Muhoray, May 2009.
  14. X. Zheng, W. Iglesias, P. Palffy-Muhoray, "Distance of closest approach of two arbitrary hard ellipsoids", Phys. Rev. E, 79, 057702 (2009)
  15. X. Zheng, W. Iglesias, P. Palffy-Muhoray, "Distance of closest approach of two arbitrary hard ellipsoids", electronic Liquid Crystal Communications, 2008
  16. Fortran90 subroutine for distance of closest approach of ellipsoids
  17. C subroutine for distance of closest approach of ellipsoids