CGTA09

Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work

Computing the distance of closest approach
between ellipses and ellipsoids

L. Gonzalez-Vega, G. R. Quintana

Departamento de MATemáticas, EStadística y COmputación
University of Cantabria, Spain

Conference on Geometry: Theory and Applications
Dedicated to the memory of Prof. Josef Hoschek
Pilsen, Czech Republic, June 29 - July 2, 2009

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

Problem
Future work

Contents

1 Problem

2 Distance of closest approach of two ellipses

3 Distance of closest approach of two ellipsoids

4 Future work


Problem
Future work

Introduction

The distance of closest approach of two arbitrary separated
ellipses (resp. ellipsoids) is the distance among their centers
when they are externally tangent, after moving them through
the line joining their centers.


Problem
Future work

Introduction

The distance of closest approach of two arbitrary separated
ellipses (resp. ellipsoids) is the distance among their centers
when they are externally tangent, after moving them through
the line joining their centers.

It appears when we study the problem of determining the
distance of closest approach of hard particles which is a key
topic in some physical questions like modeling and simulating
systems of anisometric particles such as liquid crystals or in the
case of interference analysis of molecules.


Problem
Future work

Previous work

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

X. Z HENG , P. PALFFY-M UHORAY, Distance of closest
approach 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 of
their orientation relative to the line joining their centers.


Problem
Future work

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.
initial ellipses by inverse transformation.


Problem
Future work

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 scaling
circle and the ellipse.
initial ellipses by inverse transformation.


Problem
Future work

Previous work

To deal with anisotropic scaling and the inverse transformation
involves the calculus of the eigenvectors and eigenvalues of the
matrix of the transformation.

Our goal is to avoid that computation.


Problem
Future work

Our approach

We use the results shown in:
F. E TAYO, L. G ONZÁLEZ -V EGA , N. DEL R ÍO, A new approach to
characterizing the relative position of two ellipses depending on
one parameter, Computed Aided Geometric Desing 23,
324-350, 2006.
W. WANG , R. K RASAUSKAS, Interference analysis of conics and
quadrics, Contemporary Math. 334, 25-36,2003.
W. WANG , J. WANG , M. S. K IM, An algebraic condition for the
separation of two ellipsoids, Computer Aided Geometric Desing
18, 531-539, 2001.


Problem
Future work

Our approach

Following their notation we deﬁne the characteristic polynomial
of the pencil determined by two ellipses(resp. ellipsoids)
Deﬁnition
Let A and B be two ellipses (resp. ellipsoids) given by the
equations X T AX = 0 and X T BX = 0 respectively, the degree
three (resp. four) polynomial

f (λ) = det(λA + B)

is called the characteristic polynomial of the pencil λA + B


Problem
Future work

Our approach

W. WANG , R. K RASAUSKAS, Interference analysis of conics and
quadrics, Contemporary Math. 334, 25-36,2003.
W. WANG , J. WANG , M. S. K IM, An algebraic condition for the
separation of two ellipsoids, Computer Aided Geometric Desing
18, 531-539, 2001.

Results about the intersection of two ellipsoids: a complete
characterization, in terms of the sign of the real roots of the
characteristic polynomial, of the separation case.


Problem
Future work

Our approach

More precisely:

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


Problem
Future work

Our approach

F. E TAYO, L. G ONZÁLEZ -V EGA , N. DEL R ÍO, A new approach to
characterizing the relative position of two ellipses depending on one
parameter, 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 using
several tools coming from Real Algebraic Geometry, Computer Algebra and
Projective Geometry (Sturm-Habicht sequences and the classiﬁcation of
pencils of conics in P2 (R)). Each one is determined by a set of equalities and
inequalities depending only on the matrices of the conics.


Problem
Future work

Our approach

We use the previous characterization in order to obtain the
solution of the problem.

We give a closed formula for the polynomial S(t) (depending
polynomially on the ellipse parameters) whose smallest real
root provides the distance of closest approach. We will see that
it extends in a natural way to the case of two ellipsoids.


Problem
Future work

We consider the two coplanar ellipses given by the equations:

x2 y2
E1 = (x, y) ∈ R2 : + −1=0
a b

E2 = (x, y) ∈ R2 : a11 x2 + a22 y 2 + 2a12 xy + 2a13 x + 2a23 y + a33 = 0


Problem
Future work

Conﬁguration of the ellipses


Problem
Future work

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

(x − pt)2 (y − qt)2
E1 (t) = (x, y) ∈ R2 : + −1=0
a b
where
a22 a13 − a12 a23
p=
a2 − a11 a22
12
a11 a23 − a12 a13
q=
a2 − a11 a22
12


Problem
Future work

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 a
double positive root: the equation which gives us the searched
value of t, t0 , is S(t) = 0 where

S(t) = discλ H(t; λ) = s8 t8 +s7 t7 +s6 t6 +s5 t5 +s4 t4 +s3 t6 +s2 t4 +s1 t2 +s0


Problem
Future work

Distance of closest approach of two separated ellipses

Theorem
Given two separated ellipses E1 and E2 the distance of their
closest approach is given as

d = t0 p2 + q 2

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


Problem
Future work

Example

Let A and B be the ellipses:

1
A := (x, y) ∈ R2 : x2 + y 2 − 1 = 0
2

B := (x, y) ∈ R2 : 9x2 + 4y 2 − 54x − 32y + 109 = 0

1
A centered at the origin and semi-axes of length 1 and √ .
2

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


Problem
Future work

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


Problem
Future work

Example

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

(y − 4t)2
A(t) := (x, y) ∈ R2 : (x − 3t)2 + −1=0
2

Characteristic polynomial of the pencil λB + A(t):
B 17 17 5
HA(t) (t; λ) = λ3 + − 36 t2 + 18 t − 24 λ2 +
23 145 2 145 1
− 648 − 2592 t + 1296 t λ + 2592


Problem
Future work

Example

Polynomial whose smallest real root gives the instant t = t0
when the ellipses are tangent:

251243 115599091 1478946641
SA(t) (t) = − 80621568 t + 8707129344 t2 + 34828517376 t4 −
B
266704681 3 55471163 6 158971867 5
8707129344 t + 2902376448 t − 4353564672 t +
6076225 8 6076225 7 40111
8707129344 t − 1088391168 t + 136048896


Problem
Future work

Example

Polynomial whose smallest real root gives the instant t = t0
when the ellipses are tangent:

251243 115599091 1478946641
SA(t) (t) = − 80621568 t + 8707129344 t2 + 34828517376 t4 −
B
266704681 3 55471163 6 158971867 5
8707129344 t + 2902376448 t − 4353564672 t +
6076225 8 − 6076225 t7 + 40111
8707129344 t 1088391168 136048896

B
The four real roots of SA(t) (t) are:

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


Problem
Future work

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

t = t0 t = t1


Problem
Future work

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

t = t2 t = t3


Problem
Future work

Let A1 and A2 be the symmetric deﬁnite positive matrices deﬁning
the separated ellipsoids E1 and E2 as X T A1 X = 0 and X T A2 X = 0
where X T = (x, y, z, 1), and

1
   
a 0 0 0 a11 a12 a13 a14
1  a12
 0
b 0 0  a22 a23 a24 
A1 =  1
 A2 = 
 a13

 0 0 c 0  a23 a33 a34 
0 0 0 −1 a14 a24 a34 a44

i.e.,

x2 y2 z2
E1 = (x, y) ∈ R2 : + + −1=0
a b c

a11 x2 + a22 y 2 + a33 z 2 + 2a12 xy + 2a13 xz+
E2 = (x, y) ∈ R2 :
2a23 yz + 2a14 x + 2a24 y + 2a34 z + a44 = 0


Problem
Future work

Conﬁguration of the two ellipsoids


Problem
Future work

Characteristic polynomial

(x − txc )2 (y − tyc )2 (z − tzc )2
E1 (t) = (x, y) ∈ R2 : + + −1=0
a b c

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

S(t) = discλ (H(t, λ)) = s12 t12 + ... + s0


Problem
Future work


Theorem
Given two separated ellipsoids E1 and E2 the distance of their
closest approach is given as

d = t0 x 2 + yc + zc
c
2 2

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


Problem
Future work

Example

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

1 2 1 2
E1 := (x, y, z) ∈ R3 : x + y + z2 − 1 = 0
4 2

1 2 1 51 1
E2 := (x, y, z) ∈ R3 : x − 2 x + y2 − 3 y + + z2 − 5 z = 0
5 4 2 2

1 2 1 2 5 197 2
E1 (t) := (x, y, z) ∈ R3 : x + y + z 2 − tx − 6 ty − 10 tz − 1 + t =0
4 2 2 4


Problem
Future work

Conﬁguration of the two ellipsoids E1 (blue)and E2
(green)


Problem
Future work

Example

Characteristic polynomial of E2 and E1 (t):

E2
HE1 (t) (t; λ) = λ4 − 43 λ3 − 197 λ3 t2 − 301 λ2 − 659 λ2 t2 +
4 2 4
197
2 λ3 t−
237
2 λ − 265 λ t2 + 659 λ2 t + 5 + 265 λ t
2 2


Problem
Future work

Example

E2
Polynomial SE1 (t) (t) whose its smallest real root corresponds to the instant
t = t0 when the ellipsoids are tangent:

E2
SE1 (t) (t) = 16641
1024
− 1)4 (2725362025t8 − 21802896200t7 + 75970256860t6 −
(t
150580994360t5 + 185680506596t4 − 145836126384t3 +
71232102544t2 − 19777044480t + 2388833408)
E2
The four real roots of SE1 (t) (t) that determine the four tangency points are all
provided by the factor of degree 8:

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


Problem
Future work

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


Problem
Future work



Problem
Future work

Some geometric conﬁgurationsof the quadrics or conics we are
studying seem to be related with specially simple
decompositions of the polynomials involved in the calculus of
the minimum distance between them or of the closest approach
of them.

We are working in the algebraic-geometric interpretation of this
situation.


CGTA09

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