Post on 04-Jul-2015
description
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Computing the distance of closest approachbetween ellipses and ellipsoids
L. Gonzalez-Vega, G. R. Quintana
Departamento de MATemáticas, EStadística y COmputaciónUniversity of Cantabria, Spain
2009 SIAM/ACM Joint Conference on Geometric andPhysical Modeling, October 5-8, 2009
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Contents
1 Problem
2 Distance of closest approach of two ellipses
3 Distance of closest approach of two ellipsoids
4 Conclusions
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Introduction
The distance of closest approach of two arbitrary separatedellipses (resp. ellipsoids) is the distance among their centerswhen they are externally tangent, after moving them throughthe line joining their centers.
That distance appears when we study the problem ofdetermining the distance of closest approach of hard particleswhich is a key topic in some physical questions like modelingand simulating systems of anisometric particles, such as liquidcrystals, or in the case of interference analysis of molecules.
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Previous work
A description of a method for solving the problem in the case oftwo arbitrary hard ellipses (resp. ellipsoids) can be found in
X. ZHENG, P. PALFFY-MUHORAY, Distance of closestapproach of two arbitrary hard ellipses in two dimensions,Phys. Rev., E 75, 061709, 2007.X. ZHENG, W. IGLESIAS, P. PALFFY-MUHORAY, Distance ofclosest approach of two arbitrary hard ellipsoids, Phys.Rev. E, 79, 057702, 2009.
An analytic expression for that distance is given as a function oftheir orientation relative to the line joining their centers.
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Previous work (Zheng,Palffy-Muhoray)
Ellipses case:1 Two ellipses initially distant are given.2 One ellipse is translated toward the other along the line
joining their centers until they are externally tangent.3 PROBLEM: to find the distance d between the centers at
that time.4 Transformation of the two tangent ellipses into a circle and
an ellipse.5 Determination of the distance d′ of closest approach of the
circle and the ellipse.6 Determination of the distance d of closest approach of the
initial ellipses by inverse transformation.
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Previous work (Zheng,Palffy-Muhoray)
Ellipses case:1 Two ellipses initially distant are given.2 One ellipse is translated toward the other along the line
joining their centers until they are externally tangent.3 PROBLEM: to find the distance d between the centers at
that time.4 Transformation of the two tangent ellipses into a circle and
an ellipse. ⇒ Anisotropic scaling5 Determination of the distance d′ of closest approach of the
circle and the ellipse.6 Determination of the distance d of closest approach of the
initial ellipses by inverse transformation.
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Previous work (Zheng,Iglesias,Palffy-Muhoray)
Ellipsoids case:1 Two ellipsoids initially distant are given.2 Plane containing the line joining the centers of the two
ellipsoids.3 Equations of the ellipses formed by the intersection of this
plane and the ellipsoids.4 Determining the distance of closest approach of the
ellipses5 Rotating the plane until the distance of closest approach of
the ellipses is a maximum6 The distance of closest approach of the ellipsoids is this
maximum distance
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Previous work
To deal with anisotropic scaling and the inverse transformationinvolves the calculus of the eigenvectors and eigenvalues of thematrix of the transformation.
Our goal is to find when that computation is not required and ifit is, to simplify it. The way in which we do that extends in anatural way the ellipsoids case.
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Our approach
We use the results shown in:
F. ETAYO, L. GONZÁLEZ-VEGA, N. DEL RÍO, A new approach tocharacterizing the relative position of two ellipses depending onone parameter, Computed Aided Geometric Desing 23,324-350, 2006.
W. WANG, R. KRASAUSKAS, Interference analysis of conics andquadrics, Contemporary Math. 334, 25-36,2003.
W. WANG, J. WANG, M. S. KIM, An algebraic condition for theseparation of two ellipsoids, Computer Aided Geometric Desing18, 531-539, 2001.
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Our approach
Following their notation we define
DefinitionLet A and B be two ellipses (resp. ellipsoids) given by the equationsXTAX = 0 and XTBX = 0 respectively, the degree three (resp.four) polynomial
f(λ) = det(λA+B)
is called the characteristic polynomial of the pencil λA+B
Two ellipses (or ellipsoids) are separated if and only if theircharacteristic polynomial has two distinct positive roots.
The ellipses (or ellipsoids) touch each other externally if andonly if the characteristic equation has a positive double root.
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Our approach
We use the previous characterization in order to obtain thesolution of the problem.
We give a closed formula for the polynomial S(t) (dependingpolynomially on the ellipse parameters) whose biggest real rootprovides the distance of closest approach:
Ellipses case: d = t0√x20 + y20
Ellipsoids case: d = t0√x20 + y20 + z20
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
We consider the two coplanar ellipses given by the equations:
E1 ={(x, y) ∈ R2 : a22x
2 + a33y2 + 2a23xy + 2a31x+ 2a32y + a11 = 0
}E2 =
{(x, y) ∈ R2 : b22x
2 + b33y2 + 2b23xy + 2b31x+ 2b32y + b11 = 0
}We change the reference frame in order to have E1 centered at theorigin and E2 centered at (x0, y0) with axis parallel to the coordinateones:
E1 =
{(x, y) ∈ R2 :
(x cos (α) + y sin (α))2
a+
(x sin (α)− y cos (α))2
b= 1
}
E2 =
{(x, y) ∈ R2 :
(x− x0)2
c+
(y − y0)2
d= 1
}
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Let A1 and A2 be the matrices associated to E1 and E2.Characteristic polynomial of the pencil λA2 +A1:
H(λ) = det(λA2 +A1) = h3λ3 + h2λ
2 + h1λ+ h0
Compute the discriminant of H(λ), and introduce the change ofvariable (x0, y0) = (x0t, y0t). The equation which gives us thesearched value of t, t0, is S(t) = 0 where:
S(t) = discλH(λ) |(x0,y0)=(x0t,y0t)= s4t8 + s3t
6 + s2t4 + s1t
2 + s0
Making T = t2:
S(T ) = s′4T4 + s′3T
3 + s′2T2 + s′1T + s′0
Searched value of t: square root of the biggest real root of S(T )
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Distance of closest approach of two separated ellipses
TheoremGiven two separated ellipses E1 and E2 the distance of theirclosest approach is given as
d = t0
√x20 + y20
where t0 is the square root of the biggest positive real root ofS(T ) = S(t) |T=t2= (discλH(λ) |(x0,y0)=(x0t,y0t)) |T=t2 , whereH(λ) is the characteristic polynomial of the pencil determinedby them and (x0, y0) is the center of E2.
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Example
Let A and B be the ellipses:
A :=
{(x, y) ∈ R2 :
7
8x2 +
√3
4xy +
5
8y2 = 10
}B :=
{(x, y) ∈ R2 :
1
4x2 − 3
2x+
1
9y2 − 8
9y = −109
36
}
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Example
Polynomial whose biggest real root gives the square of theinstant T = T0 when the ellipses are tangent:
SBA(T )(T ) =(466271425
16 + 9019725√3)T 4 +
(− 627564237
32 − 169045352
√3)T 3
+(39363189
16
√3 + 690647377
256
)T 2 +
(− 1186083
16
√3− 58434963
128
)T
+ 4499761256
The two real roots of SBA(T )(T ) are:
T0 = 0.5058481537; T1 = 0.07113873679
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Positions of A (blue) and B(t) (green)
t0 =√T0 t1 =
√T1
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Let A1 and A2 be the matrices defining the separated ellipsoids E1
and E2 as XTA1X = 0 and XTA2X = 0 where XT = (x, y, z, 1), andA = (aij), B = (bij), i, j = 1..4Change the reference frame to have E1 centered at the origin and E2,at (x0, y0, z0) with axis parallel to the coordinate ones:
E1 =
{(x, y, z) ∈ R3 :
P 2
a2+Q2
b2+R2
c2= 1
}
E2 =
{(x, y, z) ∈ R3 :
(x− x0)2
d2+
(y − y0)2
f2+
(z − z0)2
g2= 1
}where
P = x(ux2 +
(1− ux2
)cos (α)
)+ (uxuy (1− cos (α))− uz sin (α)) y + . . .
Q = (uxuy (1− cos (α)) + uz sin (α))x+ y(uy2 +
(1− uy2
)cos (α)
)+ . . .
R = (uxuz (1− cos (α))− uy sin (α))x+ (uyuz (1− cos (α)) + ux sin (α)) y + . . .
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Characteristic polynomial of the pencil λA2 +A1:
H(λ) = det(λA2 +A1) = h4λ4 + h3λ
3 + h2λ2 + h1λ+ h0
Compute the discriminant of H(λ), and introduce the change ofvariable (x0, y0, z0) = (x0t, y0t, z0t). The equation which gives us thesearched value of t, t0, is S(t) = 0 where:
S(t) = discλH(λ) |(x0t,y0t,z0t)= s6t12+s5t
10+s4t8+s3t
6+s2t4+s1t
2+s0
Making T = t2:
S(T ) = s′6t6 + s′5t
5 + s′4T4 + s′3T
3 + s′2T2 + s′1T + s′0
Searched value of t: square root of the biggest real root of S(T )
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Distance of closest approach of two ellipsoids
TheoremGiven two separated ellipsoids E1 and E2 the distance of theirclosest approach is given as
d = t0
√x20 + y20 + z20
where t0 is the square root of the biggest positive real root ofS(T ) = S(t) |T=t2= (discλH(λ) |(x0t,y0t,z0t)) |T=t2 , where H(λ)is the characteristic polynomial of the pencil determined bythem and (x0, y0, z0) is the center of E2.
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Example
Let A (blue) and B (green) be the ellipsoids:
A :=
{(x, y, z) ∈ R3
:1
4x2+
1
2y2+ z
2= 1
}
B :=
{(x, y, z) ∈ R3
:1
5x2 − 2 x +
1
4y2 − 3 y +
1
2z2 − 5 z = −
51
2
}
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Example
Polynomial whose biggest real root gives the square of theinstant T = T0 when the ellipsoids are tangent:
SBA(T )(T ) = 16641T 2 (2725362025T 4 − 339879840T 3 + 3362446T 2 − 11232T + 9
)The two real roots of SB
A(T )(T ) are:
T0 = 0.1142222397; T1 = 0.001153709353
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Positions of A (blue) and B(t) (green)
t0 =√T0 t1 =
√T1
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Ellipses case:Basic configuration:
Compute the eigenvectors of a 2x2 matrixCompute the real roots of a 4-degree polynomial
Other configurations: roots of a 8-degree polynomialEllipsoids case:
Basic configuration:Compute the eigenvectors of a 3x3 matrixCompute the real roots roots of a 6-degree polynomial
Other configurations: roots of a 12-degree polynomial
L. Gonzalez-Vega, G. R. Quintana GDSPM09
ProblemDistance of closest approach of two ellipses
Distance of closest approach of two ellipsoidsConclusions
Thank you!
L. Gonzalez-Vega, G. R. Quintana GDSPM09