1. The document discusses computing the distance of closest approach between two ellipses or two ellipsoids.
2. Previous work transformed the ellipses into a circle and ellipse to determine the distance, but this involved complex eigenvector/eigenvalue calculations.
3. The authors propose using the characteristic polynomial of the pencil determined by the ellipses/ellipsoids to characterize their separation vs tangency.
4. They provide a closed formula for the polynomial whose smallest positive root gives the distance of closest approach between two separated ellipses or ellipsoids.
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CGTA09
1. 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 MATemĂĄticas, EStadĂstica y COmputaciĂłn
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
2. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Contents
1 Problem
2 Distance of closest approach of two ellipses
3 Distance of closest approach of two ellipsoids
4 Future work
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
3. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
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.
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
4. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
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.
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
5. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
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.
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
6. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
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 ïŹnd 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 CGTA 2009
7. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
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 ïŹnd the distance d between the centers at
that time.
4 Transformation of the two tangent ellipses into a circle and
an ellipse. â Anisotropic scaling
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 CGTA 2009
8. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
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.
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
9. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Our approach
We use the results shown in:
F. E TAYO, L. G ONZĂLEZ -V EGA , N. DEL R ĂO, 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.
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
10. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Our approach
Following their notation we deïŹne the characteristic polynomial
of the pencil determined by two ellipses(resp. ellipsoids)
DeïŹnition
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
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
11. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
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.
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
12. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
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.
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
13. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Our approach
F. E TAYO, L. G ONZĂLEZ -V EGA , N. DEL R ĂO, 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 classiïŹcation 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.
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
14. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
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.
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
15. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
We consider the two coplanar ellipses given by the equations:
x2 y2
E1 = (x, y) â R2 : + â1=0
a b
E2 = (x, y) â R2 : a11 x2 + a22 y 2 + 2a12 xy + 2a13 x + 2a23 y + a33 = 0
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
16. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
ConïŹguration of the ellipses
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
17. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Equation of a moving ellipse E1 (t) along the line deïŹned by the
centers:
(x â pt)2 (y â qt)2
E1 (t) = (x, y) â R2 : + â1=0
a b
where
a22 a13 â a12 a23
p=
a2 â a11 a22
12
a11 a23 â a12 a13
q=
a2 â a11 a22
12
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
18. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
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
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
19. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
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 .
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
20. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Example
Let A and B be the ellipses:
1
A := (x, y) â R2 : x2 + y 2 â 1 = 0
2
B := (x, y) â R2 : 9x2 + 4y 2 â 54x â 32y + 109 = 0
1
A centered at the origin and semi-axes of length 1 and â .
2
B centered at (3, 4) with semi-axes of length 2 and 3.
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
21. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Position of the ellipses A (blue) and B (green)
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
22. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Example
We make the center of the ïŹrst one to move along the line
determined by the centers.
(y â 4t)2
A(t) := (x, y) â R2 : (x â 3t)2 + â1=0
2
Characteristic polynomial of the pencil λB + A(t):
B 17 17 5
HA(t) (t; λ) = λ3 + â 36 t2 + 18 t â 24 λ2 +
23 145 2 145 1
â 648 â 2592 t + 1296 t λ + 2592
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
23. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Example
Polynomial whose smallest real root gives the instant t = t0
when the ellipses are tangent:
251243 115599091 1478946641
SA(t) (t) = â 80621568 t + 8707129344 t2 + 34828517376 t4 â
B
266704681 3 55471163 6 158971867 5
8707129344 t + 2902376448 t â 4353564672 t +
6076225 8 6076225 7 40111
8707129344 t â 1088391168 t + 136048896
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
24. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Example
Polynomial whose smallest real root gives the instant t = t0
when the ellipses are tangent:
251243 115599091 1478946641
SA(t) (t) = â 80621568 t + 8707129344 t2 + 34828517376 t4 â
B
266704681 3 55471163 6 158971867 5
8707129344 t + 2902376448 t â 4353564672 t +
6076225 8 â 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
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
25. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Positions of A(t) (blue) and B (green)
t = t0 t = t1
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
26. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Positions of A(t) (blue) and B (green)
t = t2 t = t3
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
27. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Let A1 and A2 be the symmetric deïŹnite positive matrices deïŹning
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
ïŁ« ïŁ¶ ïŁ« ïŁ¶
a 0 0 0 a11 a12 a13 a14
1 ïŁŹ a12
ïŁŹ 0
b 0 0 ïŁ· a22 a23 a24 ïŁ·
A1 = ïŁŹ 1
ïŁ· A2 = ïŁŹ
ïŁ a13
ïŁ·
ïŁ 0 0 c 0 ïŁž a23 a33 a34 ïŁž
0 0 0 â1 a14 a24 a34 a44
i.e.,
x2 y2 z2
E1 = (x, y) â R2 : + + â1=0
a b c
a11 x2 + a22 y 2 + a33 z 2 + 2a12 xy + 2a13 xz+
E2 = (x, y) â R2 :
2a23 yz + 2a14 x + 2a24 y + 2a34 z + a44 = 0
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
28. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
ConïŹguration of the two ellipsoids
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
29. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Characteristic polynomial
(x â txc )2 (y â tyc )2 (z â tzc )2
E1 (t) = (x, y) â R2 : + + â1=0
a b c
In order to ïŹnd 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, ïŹnd the roots of the polynomial of degree 12:
S(t) = discλ (H(t, λ)) = s12 t12 + ... + s0
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
30. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Distance of closest approach of two ellipsoids
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 .
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
31. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Example
Let E1 (t) and E2 be the two ellipsoids given as follows:
1 2 1 2
E1 := (x, y, z) â R3 : x + y + z2 â 1 = 0
4 2
1 2 1 51 1
E2 := (x, y, z) â R3 : x â 2 x + y2 â 3 y + + z2 â 5 z = 0
5 4 2 2
1 2 1 2 5 197 2
E1 (t) := (x, y, z) â R3 : x + y + z 2 â tx â 6 ty â 10 tz â 1 + t =0
4 2 2 4
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
32. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
ConïŹguration of the two ellipsoids E1 (blue)and E2
(green)
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
33. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Example
Characteristic polynomial of E2 and E1 (t):
E2
HE1 (t) (t; λ) = λ4 â 43 λ3 â 197 λ3 t2 â 301 λ2 â 659 λ2 t2 +
4 2 4
197
2 λ3 tâ
237
2 λ â 265 λ t2 + 659 λ2 t + 5 + 265 λ t
2 2
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
34. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
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
â 1)4 (2725362025t8 â 21802896200t7 + 75970256860t6 â
(t
150580994360t5 + 185680506596t4 â 145836126384t3 +
71232102544t2 â 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
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
35. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Positions of E1 (blue) and E2 (green) t = t0
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
36. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Positions of E1 (blue) and E2 (green) t = t1
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
37. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Positions of E1 (blue) and E2 (green) t = t2
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
38. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Positions of E1 (blue) and E2 (green) t = t3
L. Gonzalez-Vega, G. R. Quintana CGTA 2009
39. Problem
Distance of closest approach of two ellipses
Distance of closest approach of two ellipsoids
Future work
Some geometric conïŹgurationsof 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.
L. Gonzalez-Vega, G. R. Quintana CGTA 2009