Master's Thesis: Closed formulae for distance functions involving ellipses.

U NIVERSIDAD DE C ANTABRIA
FACULTAD DE C IENCIAS
D PTO . DE M ATEMÁTICAS , E STADÍSTICA Y C OMPUTACIÓN

Closed formulae for distance functions
involving ellipses.

T ESIS DEL M ÁSTER EN M ATEMÁTICAS Y C OMPUTACIÓN
REALIZADA POR G EMA R. Q UINTANA P ORTILLA
BAJO LA DIRECCIÓN DE LOS PROFESORES D. F ERNANDO E TAYO
G ORDEJUELA Y D. L AUREANO G ONZÁLEZ -V EGA DURANTE EL
CURSO 2008-2009

Contents

Contents 3

List of Figures 5

1 Introduction 7
1.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Description of the contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 The distance between two ellipses 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 The distance between one point and an ellipse . . . . . . . . . . . . . . . . . . . 11
2.3 The distance between two ellipses . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Closest approach of two ellipses or ellipsoids 17
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Two ellipses case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Distance of closest approach of two ellipsoids . . . . . . . . . . . . . . . . . . . 25
3.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Conclusions and future work 31
4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2.1 Using ellipses to check safety regions . . . . . . . . . . . . . . . . . . . 32
4.2.2 Haussdorf distance computations between ellipses and ellipsoids . . . . . 33

Bibliography 35

List of Figures

2.1 Analyzing graphically d(E0 , E1 ). . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Analyzing the implicit curve determined by d(E0 , E1 ). . . . . . . . . . . . . . . . 16

3.1 Distance of closest approach of two ellipses in two dimensions. . . . . . . . . . . 19
3.2 Configuration of the two ellipses. . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Position of the ellipses A (blue) and B (green). . . . . . . . . . . . . . . . . . . 22
3.4 Position of the ellipses A(t) (blue) and B (green) at the instant t = t0 . . . . . . . 23
3.8 Configuration of the two ellipsoids. . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.9 Configuration of the two ellipsoids E1 (blue)and E2 (green). . . . . . . . . . . . 27
3.10 Configuration of the two ellipsoids E1 (t) (blue)and E2 (green) at the instant t = t0 . 28

Chapter 1

Introduction

1.1 General considerations
This thesis deals with two main problems:

• the computation of the minimum distance between two coplanar ellipses; and

• the calculus of the closest approach of two arbitrary separated ellipses.

Both of them are special important issues in some fields related to mathematics:

• The problem of detecting the collisions or overlap of two ellipses or ellipsoids is of inter-
est to robotics, CAD/CAM, computer animation, computer vision, etc., where ellipses or
ellipsoids are often used for modeling (or enclosing) the shape of the objects one wants to
analyze. What we are looking for is to obtain a closed formula which gives us the minimum
distance between two ellipses in the two dimensional real affine space.

• Computing the distance of the closest approach of two ellipses is one topic which has a
lot of importance in some areas of the Physics and Chemistry. It appears, for example, in
modeling 2D liquid crystals or in modeling the phase behavior of isotropic fluids.

The ideas of the second chapter of the thesis have been exposed in a short-talk given by the
author of the thesis in the Seventh International Workshop on Automated Deduction in Geometry
2008 that was hosted by the East China Normal University (ECNU) at its campus in Shanghai,
China, from September 22nd to September 24th in 2008. In the proceedings of this conference
it appears a short resume of the content of the talk: “Closed formulae for distance functions in-
volving ellipses” by Fernando Etayo, Laureano Gonzalez-Vega, Gema R. Quintana and Wenping
Wang. Another short-talk about the same topic was also given by the author of the Master Thesis
in the XI Encuentro de Álgebra Computacional y Aplicaciones which took place in Granada from
10th September 12th to in 2008.

8 CHAPTER 1. INTRODUCTION

The third chapter constitutes the main topic of a short-talk under the title “Computing the dis-
tance of closest approach between ellipses and ellipsoids” given by the author at the Conference
on Geometry: Theory and Applications that has been held from June 29 to July 2, 2009 at Pilsen in
Czech Republic, dedicated to the memory of the Professor Josef Hoschek. It also has been recently
accepted for presentation at the 2009 SIAM/ACM Joint Conference on Geometric and Physical
Modeling conference, which will be celebrated from 5th to 8th October, 2009, at the Hilton San
Francisco Financial District in San Francisco (California). The talk will be given in the session
dedicated to the field of Geometric Algorithms.

1.2 Description of the contents
The thesis is divided in four chapters, beginning with an introduction to the topics which are
going to be developed in it.

The second chapter deals with the problem of computing the minimum distance between two
separated ellipses in the plane. Our goal is to obtain that using a closed formula. Using elimination
theory we find a polynomial, which depends only on the parameters that determine the definition
of the ellipses, that provides us the square of the minimum distance as its smallest positive real
root. The same polynomial gives us the maximum distance as the square positive root of its max-
imum real root. Our approach provides a new point of view in this field of problems: what we do
first is to introduce a formula for the distance between a given point and one ellipse. All the process
is completed in a way totally independent of footpoints. Note that in all the previous works (see
[3],[4],[9],[12]) the computation of this distance requires the previous calculus of these points.
That points are the ones in which the probability of the minimum distance of being reached takes
it maximum. That is, to find (using geometric and optimization techniques) the closest regions
between the conics: those regions where the points in which the minimum distance is reached
are going to be contained. Then, the problem is reduced to the calculus of distances from point to
point. We avoid this calculus, obtaining the searched distance in a direct way. In order to do that we
have used several tools coming from Real Algebraic Geometry and Computer Algebra. The for-
mula we obtain for the calculus of the distance from a point to an ellipse is then used to determine
in a similar way (avoiding the calculus of footpoints) the distance between two given ellipses, just
making the exterior point to belong to another ellipse. The main advantage our method presents is
that we can generalize this method to other conics, like hyperbolas, and to quadrics, like ellipsoids,
in an easy way. It is also easy to apply it to the continuous motion case, that constitutes one of the
lines of our future work. This case takes special importance in robotics when you are interested in
computing the safety regions in which your automata can move avoiding crashes between them,
for example.

The third chapter contains the computation of the closest approach of two arbitrary separated
ellipses in the plane. That is the distance among their centers when they are externally tangent,

1.2 Description of the contents 9

after moving them through the line joining the centers of the ellipses. That distance in the case of
hard particles modeled as ellipses is a key parameter of their interaction and plays an important role
in the resulting phase behavior. In [15], the paper that encouraged us to deal with the study of this
topic, the authors obtain it in a complicate way. That way involves the calculus of the eigenvalues
and eigenvectors of a matrix of a linear transformation. What we do is to propose an alternative
way to obtain that which do not require that calculus: the searched distance is provided as the
smallest real root of a polynomial. That is, we obtain, again, a closed formula. Our method is based
on the results given in [7], [13] and [14] that characterize the positions of two separated ellipses
and ellipsoids. The algorithm we have developed for the case of two coplanar ellipses is adapted
easily to the case of two ellipsoids in R3 , obtaining another closed formula for the distance we are
interested in. The main advantage this method presents is that we avoid the calculus of eigenvectors
or eigenvalues which is known to be a difﬁcult task in numerical and symbolic algorithms.
The last chapter of this document presents an resume of the conclusions obtained in it, and it
also contains some comments about the topics we hope to study and to obtain interesting results
about which will conform our future work guideline.

Chapter 2

The distance between two ellipses

2.1 Introduction
Since we are interested in the practical applications of the calculus of the distance between two
conics we are going to assume that the distance between a given point and one ellipse is a posi-
tive real algebraic number. We show how to determine and study the univariate polynomial whose
smallest nonnegative real root provides the square of the distance between a given point and an
ellipse. The coefﬁcients of this polynomial are polynomials in the different parameters character-
izing the given ellipse (center coordinates, axes length and orientation) and the given point.

The minimum distance presented in this way does not depend on the footpoints giving the dis-
tance directly and thus we can use this formula for analyzing the Ellipses Moving Problem (EMP).
This is a critical problem in Computer Graphics and previous solutions to this problem require the
computation of footpoints being this task a source of numerical problems since they do not behave
continuously like the distance does.

This problem has been analyzed by several authors like [3], [4], [9], [11], [12] but all of their
approximations are based on the footpoints determination with the drawbacks mentioned before.

2.2 The distance between one point and an ellipse
√
Let E0 be an ellipse with center at the origin of coordinates (0,0) and semiaxes of lenghts a
√
and b parallel to the coordinate axis. Let (x0 , y0 ) be a point exterior to E0 . The distance between
them, d, is given by:

d = min (x − x0 )2 + (y − y0 )2 : (x, y) ∈ E0

In order to get a closed formula giving d in terms of the parameters a, b, x0 and y0 , the ellipse

12 CHAPTER 2. THE DISTANCE BETWEEN TWO ELLIPSES

E0 is characterized by the usual parametrization:
√ √
x = a cos t, y = b sin t, t ∈ [0, 2π)
In this way the square of the minimum distance from (x0 , y0 ) to E0 , D = d2 , is attained at a
value of the parameter t0 where the function
√ 2 √ 2
f (t) = x0 − a cos t + y0 − b sin t

takes its minimum value1 .
Thus t0 is a solution of the equation:
√ √
g(t) = f (t) = 2(b − a) cos t sin t + 2x0 a sin t − 2y0 b cos t = 0
and d2 = f (t0 ).

To get the searched formula for d we have to eliminate t0 from the system of equations:
f (t0 ) − D = 0
g(t0 ) = 0
To perform this elimination, and in order to make cos(to ) and sin(t0 ) disappear, we introduce
the change of variable2 :
1 1
cos(t0 ) = 2 z + z
1 1
sin(t0 ) = 2i z − z
The principal advantage of this change of variable is given by the fact that if z = cos(t0 ) +
¯ 1
i sin(t0 ) then z = z . In this way it is concluded that the searched D verifies that there exists z ∈ C
such that |z| = 1 and
√ √ √ √
(b − a)z 4 + 2(x0 a − iy0 b)z 3 − 2(x0 a − iy0 b)z + a − b = 0
√ √ √ √
(b − a)z 4 + 2(x0 a − iy0 b)z 3 − 2(2(x2 + y0 − D) + a + b)z 2 + 4(x0 a + iy0 b)z + b − a = 0
o
2

What we have to do now is to solve the previous system. The way we do that consists in com-
puting the resultant of the two equations with respect to the variable z. It’s known that in that cal-
culus strange factors may appear3 . In our case the factor which appears and that we have removed
1
This derivation has the following geometric interpretation: it is essentially computing the envelope of circles cen-
tered on the ellipses. The equation obtained is the offset curve of distance d to the ellipse. Let’s remember that an offset
curve is the set of all points that lie a perpendicular distance d from a given curve in R2 .The scalar d is called the offset
radius. If the parametric equation of the given curve is P (t) = (x(t), y(t)) then the offset curve with offset radius d is
given by
(y (t), −x (t))
Ω(d, P (t)) = P (t) + d
x (t)2 + y (t)2
Note that in this definition, if d is positive, the offset is on our right as we walk along the base curve in the direction of
increasing parameter value.
2
In [10] one can see how this transformation is used to solve some kinematic equations
3
This is due to the implementation of the scientific software we are using: Maple 12

2.2 The distance between one point and an ellipse 13

[x ,y ]
is 256(a − b)2 . Once we have done that we obtain a polynomial F[a,b] 0 (D) in Z[a, b, x0 , y0 ][D]
0

which provides the desired formulae for d as shown by the next theorem.
Theorem 2.2.1. If d is the distance of a point (x0 , y0 ) to the ellipse with center (0, 0) and semiaxes
a and b then D = d2 is the smallest nonnegative real root of the polynomial
[x ,y0 ]
0
F[a,b] (D) = (a − b)2 D4
+ 2(a − b)(b2 + 2x2 b + y0 b − 2ay0 − a2 − x2 a)D3 +
0
2 2
0
+ (y0 b2 − 8y0 ba2 − 6b2 a2 + 6a3 y0 − 2x2 a3 + a4 + 6x2 y0 b2 − 2y0 b3 + 6y0 a2 + 4x2 a2 b+
4 2 2
0 0
2 2 4
0
+ 2b3 a + 6x2 y0 a2 + 2a3 b − 6x4 ab + 4y0 b2 a + 6x4 b2 + x4 a2 + 6b3 x2 − 10x2 y0 ab + b4 −
0
2
0
2
0 0 0 0
2

− 8x2 ab2 − 6y0 ab)D2 −
0
4

− 2(ab4 + y0 a4 − a2 b3 + a4 b + 2y0 a2 + 2b2 x6 − a3 b2 − bx2 ay0 − bx4 ay0 + 3x2 ay0 b2 +
2 6
0 0
4
0
2
0
2

+ 3x2 a2 y0 b − by0 a + b2 y0 x2 + 3x4 b3 + 3y0 a3 + x2 b4 + x4 a2 y0 − bx6 a − 5x4 ab2 +
0
2 6 4
0 0
4
0 0
2
0 0
+ 3b2 y0 x4 + 3y0 a − 2x2 a3 y0 + 3x4 a2 b + 3x2 b2 y0 − 2x2 ab3 − 2y0 a3 b − 3y0 ab3 − 3x2 a3 b−
2
0
4
0
2
0 0
4
0
2 2
0
− 2x2 b3 y0 − 5y0 a2 b + 4x2 a2 b2 + 4y0 a2 b2 )D+
0
2 4
0
2

+ (x4 + 2x2 b + b2 − 2x2 a − 2ba + a2 + y0 + 2x2 y0 − 2y0 b + 2ay0 ) · (bx2 + ay0 − ba)2 =
0 0 0
4
0
2 2 2
0
2

4
[a,b]
= hk (x0 , y0 )Dk (2.1)
k=0
[x ,y ]
when (x0 , y0 ) is not a foci of E0 . The biggest real root of F[a,b] 0 (D) = 0 is the square of the
0

maximum distance between (x0 , y0 ) and the points in the ellipse E0 .
Remark 2.2.2. If √
(x0 = ± a − b , y0 = 0)
is a foci of E0 (and for simplicity assuming a > b) then
√ √
d= a− a−b.
In this case √
[ a−b,0]
F[a,b] (D) = (a − b)2 D2 (D2 + 2(b − 2a)D + b2 ) .
The solution D = 0 comes from the fact that the complex (and non real) value
√
a
t = arc cos √
a−b
makes the function f (t) to vanish. The other two solutions
√ √ 2
D = 2a − b − 2 a(a − b) = a− a−b
√ √ 2
D = 2a − b + 2 a(a − b) = a+ a−b
produce, respectively, the minimum and the maximum distance from the foci to the ellipse E0 .


Remark 2.2.3. If a = R2 , b = R2 (the ellipse E0 becomes a circumference) and D = d2 then:
[x ,y ] 2
F[R0 ,R2 ] (d2 ) = R4 y0 + x2
2
0 2
0 · d2 + 2Rd + R2 − y0 − x2 ·
2
0

· d2 − 2Rd + R2 − y0 − x2
2
0

with real roots

d1 = −R + y0 + x2
2
0 d2 = −R − y0 + x2
2
0

d3 = R + y0 + x2
2
0 d4 = R − y0 + x2
2
0

Thus:
d = min {di : di ≥ 0} = R − y0 + x2 .
2
0

It is important to quote here that the formula presented in Theorem 2.2.1 provides the minimum
distance without requiring the availability or previous computation of the footpoints (i.e. the points
where the searched distance is attained).

2.3 The distance between two ellipses
Let E0 be the ellipse given by the equation
√ √
x= a cos(t), y = b sin(t)

and E1 any other ellipse, disjoint with E0 , presented by the parameterization

x = α(s), y = β(s) and s ∈ [0, 2π)

Then

d(E0 , E1 ) = min (x1 − x0 )2 + (y1 − y0 )2 : (x0 , y0 ) ∈ E0 , (x1 , y1 ) ∈ E1

is the square root of the smallest nonnegative real root of the family of univariate polynomials
[α(s),β(s)]
F[a,b] (D). The foci question pointed out in Remark 2.2.2 needs to be taken into account here
if one of the foci of E0 belongs to E1 . That is not a problem because that question is very easy to
check and to deal with when computing d(E0 , E1 ).
[α(s),β(s)]
In order to determine the smallest positive real root of F[a,b] (D) we are analyzing two
possibilities. In the ﬁrst one D is determined as the smallest positive real number such that there
exists s ∈ [0, 2π] such that
4
[α(s),β(s)] [a,b]
F[a,b] = hk (α(s), β(s))Dk = 0 ,
k=0

2.3 The distance between two ellipses 15

4
[α(s),β(s)] def ∂ [a,b]
F [a,b] = h (α(s), β(s))Dk = 0
∂s k
k=0

Since α(s) and β(s) are linear forms on cos(s) and sin(s) then this question is converted into an
algebraic problem in the same way we have proceeded in Section 2.2 by performing the change of
variables in both equations:
1 1 1 1
cos(s) = w+ , sin(s) = w− .
2 w 2i w
Computing the resultant of these two equations with respect to w (both have degree 16 in w)
produces an univariate polynomial GE1 (D) of degree 60 whose smallest positive real root is the
E0
[x ,y ]
square of d(E0 , E1 ). This polynomial needs to be computed once (like F[a,b] 0 (D) as shown in
0

Theorem 2.2.1) and depends polynomially on the parameters deﬁning E0 and E1 .
[α(s),β(s)]
In the second possibility D is determined by analyzing the implicit curve F[a,b] (D) =
0 in the region D ≥ 0 and s ∈ [0, 2π). In order to apply the algorithm in [8] the change of
coordinates
1 − u2 2u
cos(s) = , sin(s) =
1 + u2 1 + u2
¯
[α(u),β(u)]
¯
is used and the real algebraic plane curve F[a,b] (D) = 0 analyzed in D ≥ 0, u ∈ R.

Example 2.3.1. We consider a = 3, b = 2 (for E0 ) and E1 the ellipse with center (2, −3) parallel
to the coordinate axis and with a = 2 and b = 1.

Figure 2.1: Analyzing graphically d(E0 , E1 ).

The picture at Figure 2.1 shows the surface where the height is the smallest nonnegative real
[x0 ,y ]
root of F[3,2] 0 (D) for any (x0 , y0 ): the heights of the intersection points between this surface and
the cylinder over E1 are the distances to E0 of the points in E1 being d(E0 , E1 ) the smallest height
(in green).


Figure 2.2: Analyzing the implicit curve determined by d(E0 , E1 ).

The picture √ Figure 2.2 shows the square root of the smallest and the biggest positive
at
[ 2 cos(s)+2,sin(s)−3]
real roots of F[3,2] (D) for any given s ∈ [0, 2π). The point in green represents
d(E0 , E1 ) and the point in blue the maximum distance between the ellipses E0 and E1 .

The degree 60 polynomial GE1 (D) factors, in this case, as the product of one degree 12 poly-
E0
nomial of multiplicity one, one degree 12 polynomial with multiplicity three and other multiple
factors of lower degree being the non multiple factor of degree 12 the one providing the smallest
and the biggest real roots of GE1 (D):
E0

Gε1 (d) = k1 d4 (d12 − 216d11 + ...)(d2 − 54d + 1053)2 (d2 − 52d + 1700)2 (k2 d12 + k3 d11 + ...)3
ε0

where ki are real numbers.

It is not still clear if this factorization pattern appears in a general way, in the case of ellipses
with parallel axis, and can be used in practice. But our conjecture is that it would be possibly true.
We will see that one part of our future work consists in ﬁnding the geometric interpretation of this
factorization pattern hoping that it would help us to prove the conjecture.

Chapter 3

Closest approach of two ellipses or
ellipsoids

3.1 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.

That distance modelizes the problem of finding the distance of closest approach of hard parti-
cles which is a key topic in some physical questions: short-range repulsive forces between atoms
and molecules in soft condensed matter are often modeled by an effective hard core, which gov-
erns the proximity of neighbors. Since the attractive interaction wit a few nearest neighbors usually
dominates the potential energy, the distance of closest approach is a key parameter in statistical
descriptions of condensed phases.

For non-spherical molecules, such as the constituents of liquid crystals1 , the distance depends
on orientation and its calculation is surprisingly difficult: at first glance, this problem seems simple
enough for high-school geometry homework assignment. Further consideration shows, however,
that it s not simple at all. A prize for its solution was informally announced at the Liquid Crystal
Gordon Conference in 1983 (attended by W. M. Gelbart and R. B. Meyer); this, however, did not
generate a solution. J. Vieillard-Baron, an early worker on this problem, was reportedly greatly
disturbed by the difficulties he encountered.

The simplest smooth non-spherical shapes are the ellipse and the ellipsoid. In [15], the authors
describe a method for solving the problem of determining the distance of closest approach of the
centers of two arbitrary hard ellipses, finding an analytic expression for that distance as a function
1
An introduction to this topic can be found in [2]: a review article which gives an overview of the simulation work
performed so far, and focuses on the still unanswered questions which will determine the future challenges in the field.

18 CHAPTER 3. CLOSEST APPROACH OF TWO ELLIPSES OR ELLIPSOIDS

of their orientation relative to the line joining their centers. Their approach proceeds via the steps:

1. They consider two ellipses initially distant so that they have no point in common.

2. One ellipse is then translated toward the other along the line joining their centers until they
are in point contact externally

3. PROBLEM: to find the distance d between the centers when the ellipses are so tangent, that
is, to find the distance of closest approach.

4. Transformation of the two tangent ellipses into a circle and an ellipse. The circle and the
ellipse remain tangent after the transformation.

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 trans-
formation.

The problem here is that we have to deal with anisotropic2 scaling3 and the inverse trans-
formation, and this implies the calculus of the eigenvectors and eigenvalues of the matrix of the
transformation. It is known the difficulties this could involve numerically.

Because of that, we introduce a new approach to the problem using the results shown in [7].
The authors of that paper introduce a new approach for characterizing the ten relative positions
of two ellipses by using several tools coming from Real Algebraic Geometry, computer Alge-
bra and Projective Geometry (Sturm-Habicht sequences and the classification of pencils of conics
in P2 (R)). Each relative position is exclusively characterized by a set of equalities and inequal-
ities depending only on the matrices defining the two considered ellipses and does not require
in advance the computation or knowledge of the intersection points between them. We use the
characterization of externally tangent ellipses and ellipsoids provided in [13] and [14].

3.2 Two ellipses case
Definition 3.2.1. Given two arbitrary separated ellipses E1 and E2 we define the distance of
their closest approach as the distance among their centers when they are externally tangent, after
moving them through the line joining their centers.
2
The anisotropy is the property of being directionally dependent(i.e. opposed to isotropy, which means homogeneity
in all directions). It can be defined as a difference in a physical property (absorbance, refractive index, density, etc.) for
some material when measured along different axes. An example is the light coming through a polarizing lens.
3
By anisotropic scaling an ellipse can be transformed into a unit circle, that is to make the isotropy disappear, as
you can see in [15].

3.2 Two ellipses case 19

Figure 3.1: Distance of closest approach of two ellipses in two dimensions.

Let

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

be the equation of an ellipse. As usual it can be rewritten as

X T AX = 0

where X T = (x, y, 1) and A = (aij ) is the symmetric positive definite matrix of the coefficients.
Following the notation in [13] and [14] we define the characteristic polynomial of the pencil de-
termined by two ellipses as follows.

Definition 3.2.2. Let A and B be two ellipses given by the equations X T AX = 0 and X T BX = 0
respectively, the degree three polynomial

f (λ) = det(λA + B)

is called the characteristic polynomial of the pencil λA + B

In [13] and [14] the authors give some partial results about the intersection of two ellipsoids,
obtaining a complete characterization, in terms of the sign of the real roots of the characteristic
polynomial, of the separation case: i.e when the two ellipsoids can be separated by a plane. More
precisely they prove that:

• the two considered ellipsoids are separated if and only if their characteristic polynomial
(which has degree four in the case of ellipsoids) has two distinct positive roots;

• the characteristic equation always has at least two negative roots;

• and the ellipsoids touch each other externally if and only if the characteristic equation has a
positive double root.

In [7] an equivalent characterization is given for the case of two coplanar ellipses. That is
the characterization we are going to use in order to obtain the solution of the problem without


using geometric transformations which involve the calculus of eigenvalues or eigenvectors. The
presented approach provides 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 the presented approach extend in a natural way to the distance of closest approach for
two ellipsoids.

Remark 3.2.3. In order to simplify the computation, we consider the two coplanar ellipses given
by the equations:

x2 y2
E1 = (x, y) ∈ R2 : a + b −1=0
E2 = (x, y) ∈ R2 : a11 x2 + a22 y2 + 2a12 xy + 2a13 x + 2a23 y + a33 = 0
√ √
That is one of them centered at the origin with semi-axes of length a and b, along the
coordinate axes, x and y, respectively; and the other one located in a general position, as shown in
ﬁgure 3.2. Let A2 be the matrix associated to E2 :
 
a11 a12 a13
A2 =  a12 a22 a23 
a13 a23 a33

The center of E2 is the point (p, q) whose coordinates are given in terms of the elements of A2
as follows:
a22 a13 − a12 a23
p=
a2 − a11 a22
12

a11 a23 − a12 a13
q=
a2 − a11 a22
12

The equation of the moving ellipse E1 (t) obtained making the ﬁrst one move along the line
which joins the centers of the two ellipses yields:

(x − pt)2 (y − qt)2
E1 (t) = (x, y) ∈ R2 : + −1=0
a b

Now we consider 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)

Note that the case we are interested in is the externally tangent one. This situation is pro-
duced when the polynomial H(t; λ) has a double positive root. So 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


If t0 is the smallest positive real root of S(t) then the searched distance of the closest approach
of our ellipses is equal to
t0 p2 + q 2

Figure 3.2: Configuration of the two ellipses.

Theorem 3.2.4. Given two separated ellipses E1 and E2 defined as in the remark 3.2.3 the dis-
tance 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 .

3.2.1 Example
In order to show the aspect the polynomials involved in the previous calculus we are going
to consider the following example: let A and B be the ellipses given by the following equations,
respectively:
1
A := (x, y) ∈ R2 : x2 + 2 y 2 − 1 = 0
B := (x, y) ∈ R2 : 9x2 + 4y 2 − 54x − 32y + 109 = 0
This initial configuration is shown in the figure 3.3.That is, both of them have axes parallel to
the coordinate axes (in fact, A has its axes contained in them). The center of A is the origin of
1
coordinates and the lengths of its semi-axes are 1 and √2 . B is centered in the point (3, 4) with
semi-axes of lengths 2 and 3, resp.


Figure 3.3: Position of the ellipses A (blue) and B (green).

We make the center of the ﬁrst one to move along the line determined by the centers of the
ellipses. That gives us the equation of a moving ellipse, depending on the parameter t:

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

The characteristic polynomial of the pencil λB + A(t), once turned monic, results:

17 2 17 5 23 145 2 145 1
HA(t) (t; λ) = λ3 + −
B
t + t− λ2 + − − t + t λ+
36 18 24 648 2592 1296 2592

And computing the resultant of this polynomial with respect to λ we can determine the poly-
B
nomial SA(t) (t) whose its smallest real root represents the instant t = t0 when the ellipses are
tangent:

251243 115599091 1478946641 266704681 55471163
B
SA(t) (t) = − 80621568 t + 8707129344 t2 + 34828517376 t4 − 8707129344 t3 + 2902376448 t6
158971867 5 6076225 6076225 40111
− 4353564672 t + 8707129344 t8 − 1088391168 t7 + 136048896

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

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

The situations associated to each value of the parameter t = ti , i = 0, 1, 2, 3 are shown in the
following ﬁgures:


Figure 3.4: Position of the ellipses A(t) (blue) and B (green) at the instant t = t0 .





B
As one can see in the previous ﬁgures, the four real roots of SA(t) (t) give us the four instants
in which the two ellipses are tangent. Being t0 the root which gives us the closest approach of
them which is d = 5t0 = 1.294556550 in this case.

3.3 Distance of closest approach of two ellipsoids 25

3.3 Distance of closest approach of two ellipsoids
The presented approach in the previous section extends in a natural way to the distance of
closest approach for two ellipsoids4 .

Definition 3.3.1. A real ellipsoid is the quadric surface defined X T AX = 0 where X T =
(x, y, z, 1),
 
a11 a12 a13 a14
 a12 a22 a23 a24 
A=  a13 a23

a33 a34 
a14 a24 a34 a44

is non-singular, det(A) > 0 and the cofactor of the term a44 does not vanish.

Definition 3.3.2. The center (xc , yc , zc ) of a central quadric surface5 X T AX = 0 is given by the
equations:
A14 A24 A34
xc = ; yc = ; zc =
A44 A44 A44
where each Aij represents the cofactor of the element aij of the matrix A.

Remark 3.3.3. In order to make the computation more simple we consider the following configu-
ration of the quadrics we are studying:
Let A1 and A2 be the symmetric definite positive matrices defining the separated ellipsoidsE1 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
 0 1
b 0 0 
A1 = 
 0 1

0 c 0 
0 0 0 −1
 
a11 a12 a13 a14
 a12 a22 a23 a24 
A2 = 
 a13

a23 a33 a34 
a14 a24 a34 a44
i.e.,

x2 y2 z2
E1 = (x, y) ∈ R2 : a + b + c −1=0
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
4
In [5] it is shown the relation between this distance and molecular simulations.
5
Like the ellipsoid.


We act like we did in the case of the ellipses, assuming that one of the ellipsoids is centered
√ √ √
at the origin of coordinates with semi-axis of length a, b, c, along the directions described
by the coordinate axes, x, y and z, respectively; and the second one given in a general form, with
center at (xc , yc , zc ), determined by the entries of the matrix A2 in the way it is shown by the
definition 3.3.2. That configuration is shown in figure 3.8

Figure 3.8: Configuration of the two ellipsoids.

In the same way we proceeded in the case of the ellipses we make the center of E1 (and so all
the points in E1 do) move in the line which joins the centers of the two surfaces, obtaining

(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
study the roots of the characteristic polynomial associated to them, that is to check if the polyno-
mial H(t; λ) = det(E1 (t) + λE2 ), which has degree four, has a double real root, like the authors
do in [14]. This is done by computing the roots of the polynomial of degree 12:

S(t) = discλ (H(t, λ)) = s12 t12 + ... + s0
If t0 is the smallest positive real root of S(t) then the searched distance of closest approach is
equal to
t0 x2 + yc + zc
c
2 2

Theorem 3.3.4. Given two separated ellipsoids E1 and E2 defined as in 3.3.3 the distance of their
closest approach is given as
d = t0 x2 + 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 .


3.3.1 Example

In order to illustrate the previous theorem we are going to consider a practical example. Let
E1 and E2 be the two ellipsoids given as follows:

E1 := (x, y, z) ∈ R3 : 1 x2 + 1 y 2 + z 2 − 1 = 0
4 2
E2 := (x, y, z) ∈ R3 : 1 x2 − 2 x + 1 y 2 − 3 y + 51 + 1 z 2 − 5 z = 0
5 4 2 2

√
That is, E1 is centered at the origin of coordinates with semi-axis of lengths 2, 2 and 1 along
the x-axis, y-axis and z-axis, resp. And the point (5, 6, √ is the center of E2 whose axis are parallel
5) √
to the coordinate ones and have semi-lengths equal to 5, 2 and 2 with respect to the coordinate
frame. This situation can be observed in ﬁgure3.9.

Figure 3.9: Conﬁguration of the two ellipsoids E1 (blue)and E2 (green).

We make the center of the ellipsoid E1 to move along the line determined by the centers of
the two quadrics we are studying. That is the way in which the equation of a moving ellipsoid,
depending on the parameter t, is obtained:


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
The characteristic polynomial of E2 and E1 (t), once turned monic, results:
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
And computing the resultant of this polynomial with respect to the variable λ we can determine
E2 (t)
the polynomial SE1 (t) whose its smallest real root represents the instant t = t0 when the ellipsoids
are tangent:
E (t) 16641
2
SE1 (t) = − 1)4 (2725362025t8 − 21802896200t7 + 75970256860t6 − 150580994360t5 +
1024 (t
185680506596t4 − 145836126384t3 + 71232102544t2 − 19777044480t + 2388833408)
2 E (t)
The four real roots of SE1 (t) that determine the four tangency points are all provided by the
factor of degree 8 which appears in its decomposition. And they are the following:
t0 = 0.6620321914, t1 = 0.6620321914, t2 = 1.033966297, t3 = 1.337967809
The relative positions of the two ellipsoids associated to each value of the parameter t = ti ,
i = 0, 1, 2, 3 are shown in the following ﬁgures:

Figure 3.10: Conﬁguration of the two ellipsoids E1 (t) (blue)and E2 (green) at the instant t = t0 .



As it happened in the case of the ellipses the previous ﬁgures show that when t takes the four
E2 (t)
values determined by that four real roots of SE1 (t) the four instants in which the two ellipsoids
are tangent are produced. Being t0 the root which gives us the closest approach of the two quadric
√
surfaces, which is d = 11t0 = 2.195712378 for the example we are working with.

Chapter 4

Conclusions and future work

4.1 Conclusions
The main conclusions of the work we have developed are the following:

• A closed form solution for several problems involving ellipses or ellipsoids when dealing
with interference or distance computations has been presented.

• Closed form solutions try to concentrate at the very end the application of numerical tech-
niques.

• Closed form solutions are a critical step towards efficiency when dealing with moving ob-
jects (in order to define safety regions or to check that a collision is still far away). Do not
forget that we are looking for solutions easy to be found and checked, that is, for real-time
applications: software or hardware systems that are subject to real-time constraints (i.e.,
operational deadlines from event to system response).

• We have completely avoided the calculus of the footpoints. Avoiding footpoints computation
is very important in this context: the key is that distance is a continuous function of the
geometric data while footpoints are not.

4.2 Future work
In relation to the study of the minimum distance between two separated ellipses, the main
topic in which we are interested consist of obtaining the decomposition of the polynomial GE1 (D)
E0
of degree 60, in the general case. We just have it in particular cases. Once we have done that we
would be able to prove our conjecture and it also will help us with the study of the existing rela-
tionship between the geometric configurations of the two ellipses and the terms that appear in the
factorization of GE1 (D).
E0

32 CHAPTER 4. CONCLUSIONS AND FUTURE WORK

Particular geometric configurations1 of 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 currently working in
the algebraic-geometric interpretation of this situation in order to extract global conclusions.

We would also like to continue the study of the continuous motion case, in order to apply our
techniques to robotics and similar fields in which the objects of study are not static, they represent
bodies which are describing trajectories depending on the time.

In a more or less easy way the approach we have introduced for the computation of the min-
imum distance between two separated ellipses in the plane seems to be generalizable to the three
dimensional case: the study of the minimum distance between two separated ellipsoids in the
space. It will also be interesting to consider the case of non-coplanar ellipses, that is, when the two
separated ellipses are not contained in the same plane, and of course the case of other coplanar
and non-coplanar conics, like hyperbolas, for instance.

4.2.1 Using ellipses to check safety regions
Suppose that you have a collection of moving automata and you are interested in checking
the regions in which your robots do not collide each others. In other words, you are interested in
finding the set of the safety regions. This problem can be modelized using ellipses: we can assume
that each robot is contained into an ellipse and then ask questions like:

• When the distance between both ellipses is bigger than d?

• Is there any closed formulae which gives us that?

Note that collision detection is an important problem in animation, CAD and robotics. Col-
lision detection is usually used to improve reality in virtual environment by avoiding penetrating
between objects. Besides, it is used as path planning in robotics to calculate in advance paths of
moving robots to avoid collision during motions. There are many applications like robot or vehicle
path planning where the robots and vehicles are represented as 2D figures moving in 2D plane.
The answer to them is given by considering the following discriminant:

[α(s),β(s)]
H(D) = discs F[a,b] (D)

and asking when all the positive rald roots of H(D) are bigger than d2 . The way in which closed
formulae can be determined is using subresultants and similar techniques.
1
This has to be understood in the sense of extreme regularity (like the bi-quadratic polynomials case, for instance)
of the decomposition of the polynomials involved in the calculus of the minimum distance or closest approach.

4.2 Future work 33

4.2.2 Haussdorf distance computations between ellipses and ellipsoids
Similar approaches to deal with Haussdorf distance computations between ellipses and ellip-
soids are being analyzed: The field of the analysis and comparison of geometric shapes acquires
special importance in various application areas within Computer Science. For example pattern
recognition or computer vision. It does also in other disciplines concerned with the shape of ob-
jects such as cartography, molecular biology, medicine...

The general situation is that we are given two objects A, B modeled as subsets of the two or
three dimensional space and we are interested in knowing how much they resemble each other.
For this purpose we need a similarity measure defined on pairs of shapes indicating the degree of
resemblance of these shapes. As one can see in [6] and [1] a frequently used similarity measure
is the Hausdorff distance, which is defined for two arbitrary non-empty compact sets A and B.
It assigns to each point of one set the distance to its closest point in the other set and takes the
maximum over all these values.

Definition 4.2.1. The one-sided Hausdorff distance from A to B is

∂H (a, b) = {maxa∈A minb∈B } d(a, b)

where d(a, b) denotes a distance measure between points a and b.

The case which is directly related with the topics in which we have been working is the one
when A and B are planar shapes, i.e., A, B ⊂ R2 and d is the Euclidean distance. All the pre-
vious works in this area are again based on the calculus of the footpoints.The authors reduce the
problem of determining the Hausdorff distance from a curve a; a(t) = (xa (t), ya (t)) to a curve b;
b(s) = (xb (s), yb (s)) to determining the distance of constantly many candidate or critical points
on a to the curve b and then taking the maximum over these distances. To characterize all candi-
date points they make some theoretical considerations involving, for example, the computation of
the medial axis of the curve.

Our goal here is to obtain similar characterizations like the proposed in [6] or [1] but (of
course) avoiding the calculus of the footpoints, doing it in a direct way, using similar ideas that the
ones which have appear in the previous chapters of this document.

34 CHAPTER 4. CONCLUSIONS AND FUTURE WORK

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Master's Thesis: Closed formulae for distance functions involving ellipses.

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