CGTA09

ProblemDistance of closest approach of two ellipses

Distance of closest approach of two ellipsoidsFuture work

Computing the distance of closest approachbetween ellipses and ellipsoids

L. Gonzalez-Vega, G. R. Quintana

Departamento de MATemáticas, EStadística y COmputaciónUniversity of Cantabria, Spain

Conference on Geometry: Theory and ApplicationsDedicated to the memory of Prof. Josef HoschekPilsen, Czech Republic, June 29 - July 2, 2009

L. Gonzalez-Vega, G. R. Quintana CGTA 2009



Contents

1 Problem

2 Distance of closest approach of two ellipses

3 Distance of closest approach of two ellipsoids

4 Future work




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.




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.

It appears when we study the problem of determining thedistance of closest approach of hard particles which is a keytopic in some physical questions like modeling and simulatingsystems of anisometric particles such as liquid crystals or in thecase of interference analysis of molecules.




Previous work

A description of a method for solving the problem in the case oftwo arbitrary hard ellipses can be found in

X. ZHENG, P. PALFFY-MUHORAY, Distance of closestapproach of two arbitrary hard ellipses in two dimensions,Physical Review, E 75, 061709,2007.

An analytic expression for that distance is given as a function oftheir orientation relative to the line joining their centers.




Previous work

Steps of the previous approach: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.




Previous work

Steps of the previous approach: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.




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 avoid that computation.




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.




Our approach

Following their notation we define the characteristic polynomialof the pencil determined by two ellipses(resp. ellipsoids)

DefinitionLet A and B be two ellipses (resp. ellipsoids) given by theequations XTAX = 0 and XTBX = 0 respectively, the degreethree (resp. four) polynomial

f(λ) = det(λA+B)

is called the characteristic polynomial of the pencil λA+B




Our approach

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.

Results about the intersection of two ellipsoids: a completecharacterization, in terms of the sign of the real roots of thecharacteristic polynomial, of the separation case.




Our approach

More precisely:

Two ellipsoids are separated if and only if theircharacteristic polynomial has two distinct positive roots.The characteristic equation always has at least twonegative roots.The ellipsoids touch each other externally if and only if thecharacteristic equation has a positive double root.




Our approach

F. ETAYO, L. GONZÁLEZ-VEGA, N. DEL RÍO, A new approach tocharacterizing the relative position of two ellipses depending on oneparameter, Computed Aided Geometric Desing 23, 324-350, 2006.

An equivalent characterization is given for the case of two coplanar ellipses.

In fact the ten relative positions of two ellipses are characterized by usingseveral tools coming from Real Algebraic Geometry, Computer Algebra andProjective Geometry (Sturm-Habicht sequences and the classification ofpencils of conics in P2(R)). Each one is determined by a set of equalities andinequalities depending only on the matrices of the conics.




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 smallest realroot provides the distance of closest approach. We will see thatit extends in a natural way to the case of two ellipsoids.




We consider the two coplanar ellipses given by the equations:

E1 ={

(x, y) ∈ R2 :x2

a+y2

b− 1 = 0

}

E2 ={(x, y) ∈ R2 : a11x

2 + a22y2 + 2a12xy + 2a13x+ 2a23y + a33 = 0

}




Configuration of the ellipses




Equation of a moving ellipse E1(t) along the line defined by thecenters:

E1(t) ={

(x, y) ∈ R2 :(x− pt)2

a+

(y − qt)2

b− 1 = 0

}where

p =a22a13 − a12a23

a212 − a11a22

q =a11a23 − a12a13

a212 − a11a22




The characteristic polynomial of the pencil λA2 +A1(t):

H(t;λ) = det(λA2 +A1(t)) = h3(t)λ3 + h2(t)λ2 + h1(t)λ+ h0(t)

External tangent situation is produced when H(t;λ) has adouble positive root: the equation which gives us the searchedvalue of t, t0, is S(t) = 0 where

S(t) = discλH(t;λ) = s8t8+s7t7+s6t6+s5t5+s4t4+s3t6+s2t4+s1t2+s0




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√p2 + q2

where t0 is the smallest positive real root of S(t) = discλH(t;λ),H(t;λ) is the characteristic polynomial of the pencil determinedby them and (p, q) is the center of E2.




Example

Let A and B be the ellipses:

A :={

(x, y) ∈ R2 : x2 +12y2 − 1 = 0

}B :=

{(x, y) ∈ R2 : 9x2 + 4y2 − 54x− 32y + 109 = 0

}A centered at the origin and semi-axes of length 1 and 1√

2.

B centered at (3, 4) with semi-axes of length 2 and 3.




Position of the ellipses A (blue) and B (green)




Example

We make the center of the first one to move along the linedetermined by the centers.

A(t) :={

(x, y) ∈ R2 : (x− 3t)2 +(y − 4t)2

2− 1 = 0

}

Characteristic polynomial of the pencil λB +A(t):

HBA(t)(t;λ) = λ3 +

(−17

36 t2 + 17

18 t−524

)λ2+(

− 23648 −

1452592 t

2 + 1451296 t

)λ+ 1

2592




Example

Polynomial whose smallest real root gives the instant t = t0when the ellipses are tangent:

SBA(t)(t) = − 25124380621568 t+ 115599091

8707129344 t2 + 1478946641

34828517376 t4−

2667046818707129344 t

3 + 554711632902376448 t

6 − 1589718674353564672 t

5+6076225

8707129344 t8 − 6076225

1088391168 t7 + 40111

136048896




Example

Polynomial whose smallest real root gives the instant t = t0when the ellipses are tangent:

SBA(t)(t) = − 25124380621568 t+ 115599091

8707129344 t2 + 1478946641

34828517376 t4−

2667046818707129344 t

3 + 554711632902376448 t

6 − 1589718674353564672 t

5+6076225

8707129344 t8 − 6076225

1088391168 t7 + 40111

136048896

The four real roots of SBA(t)(t) are:

t0 = 0.2589113100, t1 = 0.7450597195,t2 = 1.254940281, t3 = 1.741088690




Positions of A(t) (blue) and B (green)

t = t0 t = t1




Positions of A(t) (blue) and B (green)

t = t2 t = t3




Let A1 and A2 be the symmetric definite positive matrices definingthe separated ellipsoids E1 and E2 as XTA1X = 0 and XTA2X = 0where XT = (x, y, z, 1), and

A1 =

1a 0 0 00 1

b 0 00 0 1

c 00 0 0 −1

A2 =

a11 a12 a13 a14

a12 a22 a23 a24

a13 a23 a33 a34

a14 a24 a34 a44

i.e.,

E1 ={

(x, y) ∈ R2 :x2

a+y2

b+z2

c− 1 = 0

}

E2 ={

(x, y) ∈ R2 :a11x

2 + a22y2 + a33z

2 + 2a12xy + 2a13xz+2a23yz + 2a14x+ 2a24y + 2a34z + a44 = 0

}




Configuration of the two ellipsoids




Characteristic polynomial

E1(t) ={

(x, y) ∈ R2 :(x− txc)2

a+

(y − tyc)2

b+

(z − tzc)2

c− 1 = 0

}In order to find the value of t, t0, for which the ellipsoids are externallytangent we have to to check if the polynomialH(t;λ) = det(E1(t) + λE2), which has degree four, has a double realroot. That is, find the roots of the polynomial of degree 12:

S(t) = discλ(H(t, λ)) = s12t12 + ...+ s0




Distance of closest approach of two ellipsoids

TheoremGiven two separated ellipsoids E1 and E2 the distance of theirclosest approach is given as

d = t0√x2c + y2

c + z2c

where t0 is the smallest positive real root of S(t) = discλH(t;λ),H(t;λ) is the characteristic polynomial of the pencil determinedby them, and (xc, yc, zc) is the center of E2.




Example

Let E1(t) and E2 be the two ellipsoids given as follows:

E1 :=

{(x, y, z) ∈ R3 :

1

4x2 +

1

2y2 + z2 − 1 = 0

}

E2 :=

{(x, y, z) ∈ R3 :

1

5x2 − 2 x +

1

4y2 − 3 y +

51

2+

1

2z2 − 5 z = 0

}

E1(t) :=

{(x, y, z) ∈ R3 :

1

4x2 +

1

2y2 + z2 − 5

2tx− 6 ty − 10 tz − 1 +

197

4t2 = 0

}




Configuration of the two ellipsoids E1 (blue)and E2

(green)




Example

Characteristic polynomial of E2 and E1(t):

HE2E1(t)

(t;λ) = λ4 − 43λ3 − 1974 λ3t2 − 301

2 λ2 − 6594 λ2t2 + 197

2 λ3t−2372 λ− 265

2 λ t2 + 6592 λ2t+ 5 + 265λ t




Example

Polynomial SE2E1(t)(t) whose its smallest real root corresponds to the instant

t = t0 when the ellipsoids are tangent:

SE2E1(t)(t) = 16641

1024(t− 1)4(2725362025t8 − 21802896200t7 + 75970256860t6−

150580994360t5 + 185680506596t4 − 145836126384t3+71232102544t2 − 19777044480t + 2388833408)

The four real roots of SE2E1(t)(t) that determine the four tangency points are all

provided by the factor of degree 8:

t0 = 0.6620321914, t1 = 0.6620321914t2 = 1.033966297, t3 = 1.337967809




Positions of E1 (blue) and E2 (green) t = t0




Some geometric configurationsof the quadrics or conics we arestudying seem to be related with specially simpledecompositions of the polynomials involved in the calculus ofthe minimum distance between them or of the closest approachof them.

We are working in the algebraic-geometric interpretation of thissituation.


CGTA09

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