- 1. U NIVERSIDAD DE C ANTABRIA FACULTAD DE C IENCIAS D PTO . DE
M ATEMTICAS , E STADSTICA Y C OMPUTACINClosed formulae for distance
functionsinvolving ellipses. T ESIS DEL M STER EN M ATEMTICAS Y C
OMPUTACIN REALIZADA POR G EMA R. Q UINTANA P ORTILLABAJO LA
DIRECCIN DE LOS PROFESORES D. F ERNANDO E TAYOG ORDEJUELA Y D. L
AUREANO G ONZLEZ -V EGA DURANTE ELCURSO 2008-2009
2. 2 3. ContentsContents3List of Figures51 Introduction 71.1
General considerations . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 71.2 Description of the contents . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .82 The distance between
two ellipses112.1 Introduction . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 112.2 The distance between
one point and an ellipse . . . . . . . . . . . . . . . . . . .112.3
The distance between two ellipses . . . . . . . . . . . . . . . . .
. . . . . . . .143 Closest approach of two ellipses or ellipsoids
173.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 173.2 Two ellipses case . . . . . . . . .
. . . . . .. . . . . . . . . . . . . . . . . . . 18 3.2.1 Example .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
213.3 Distance of closest approach of two ellipsoids . . . . . . .
. . . . . . . . . . . . 25 3.3.1 Example . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 274 Conclusions and
future work314.1 Conclusions . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . .. . . . . 314.2 Future work . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 314.2.1
Using ellipses to check safety regions . . . . . . . . . . . . . .
. . . . . 324.2.2 Haussdorf distance computations between ellipses
and ellipsoids. . . . . 33Bibliography 35 4. 4 CONTENTS 5. List of
Figures 2.1Analyzing graphically d(E0 , E1 ). . . . . . . . . . . .
. . . . . . . . . . . . . . . 15 2.2Analyzing the implicit curve
determined by d(E0 , E1 ). . . . . . . . . . . . . . . .16
3.1Distance of closest approach of two ellipses in two dimensions.
. . . . . . . . . .19 3.2Conguration of the two ellipses. . . . . .
. . . . . . . . . . . . . . . . . . . . .21 3.3Position of the
ellipses A (blue) and B (green). . . . . . . . . . . . . . . . . .
. 22 3.4Position of the ellipses A(t) (blue) and B (green) at the
instant t = t0 . . . . . . . 23 3.5Position of the ellipses A(t)
(blue) and B (green) at the instant t = t1 . . . . . . . 23
3.6Position of the ellipses A(t) (blue) and B (green) at the
instant t = t2 . . . . . . . 24 3.7Position of the ellipses A(t)
(blue) and B (green) at the instant t = t3 . . . . . . . 24
3.8Conguration of the two ellipsoids. . . . . . . . . . . . . . . .
. . . . . . . . . .26 3.9Conguration of the two ellipsoids E1
(blue)and E2 (green). . . . . . . . . . . .27 3.10 Conguration of
the two ellipsoids E1 (t) (blue)and E2 (green) at the instant t =
t0 . 28 3.11 Conguration of the two ellipsoids E1 (t) (blue)and E2
(green) at the instant t = t1 . 29 3.12 Conguration of the two
ellipsoids E1 (t) (blue)and E2 (green) at the instant t = t2 . 29
3.13 Conguration of the two ellipsoids E1 (t) (blue)and E2 (green)
at the instant t = t3 . 30 6. 6 LIST OF FIGURES 7. Chapter
1Introduction1.1 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 elds 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 afne 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 uids. The ideas of the second chapter of the
thesis have been exposed in a short-talk given by theauthor of the
thesis in the Seventh International Workshop on Automated Deduction
in Geometry2008 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 conferenceit
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 WenpingWang. Another
short-talk about the same topic was also given by the author of the
Master Thesisin the XI Encuentro de lgebra Computacional y
Aplicaciones which took place in Granada from10th September 12th to
in 2008. 8. 8 CHAPTER 1. INTRODUCTIONThe 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 Conferenceon Geometry: Theory and Applications
that has been held from June 29 to July 2, 2009 at Pilsen inCzech
Republic, dedicated to the memory of the Professor Josef Hoschek.
It also has been recentlyaccepted for presentation at the 2009
SIAM/ACM Joint Conference on Geometric and PhysicalModeling
conference, which will be celebrated from 5th to 8th October, 2009,
at the Hilton SanFrancisco Financial District in San Francisco
(California). The talk will be given in the sessiondedicated to the
eld of Geometric Algorithms.1.2 Description of the contentsThe
thesis is divided in four chapters, beginning with an introduction
to the topics which aregoing to be developed in it.The second
chapter deals with the problem of computing the minimum distance
between twoseparated ellipses in the plane. Our goal is to obtain
that using a closed formula. Using eliminationtheory we nd a
polynomial, which depends only on the parameters that determine the
denitionof the ellipses, that provides us the square of the minimum
distance as its smallest positive realroot. 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 eld of problems: what we dorst is to introduce a formula for
the distance between a given point and one ellipse. All the
processis 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 takesit maximum. That is, to
nd (using geometric and optimization techniques) the closest
regionsbetween the conics: those regions where the points in which
the minimum distance is reachedare going to be contained. Then, the
problem is reduced to the calculus of distances from point topoint.
We avoid this calculus, obtaining the searched distance in a direct
way. In order to do that wehave 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 determinein a similar way (avoiding the calculus of
footpoints) the distance between two given ellipses, justmaking the
exterior point to belong to another ellipse. The main advantage our
method presents isthat 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 thelines of our future work. This
case takes special importance in robotics when you are interested
incomputing 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
separatedellipses in the plane. That is the distance among their
centers when they are externally tangent, 9. 1.2 Description of the
contents 9after moving them through the line joining the centers of
the ellipses. That distance in the case ofhard particles modeled as
ellipses is a key parameter of their interaction and plays an
important rolein the resulting phase behavior. In [15], the paper
that encouraged us to deal with the study of thistopic, the authors
obtain it in a complicate way. That way involves the calculus of
the eigenvaluesand eigenvectors of a matrix of a linear
transformation. What we do is to propose an alternativeway to
obtain that which do not require that calculus: the searched
distance is provided as thesmallest real root of a polynomial. That
is, we obtain, again, a closed formula. Our method is basedon the
results given in [7], [13] and [14] that characterize the positions
of two separated ellipsesand ellipsoids. The algorithm we have
developed for the case of two coplanar ellipses is adaptedeasily to
the case of two ellipsoids in R3 , obtaining another closed formula
for the distance we areinterested in. The main advantage this
method presents is that we avoid the calculus of eigenvectorsor
eigenvalues which is known to be a difcult task in numerical and
symbolic algorithms.The last chapter of this document presents an
resume of the conclusions obtained in it, and italso contains some
comments about the topics we hope to study and to obtain
interesting resultsabout which will conform our future work
guideline. 10. 10 CHAPTER 1. INTRODUCTION 11. Chapter 2The distance
between two ellipses2.1 Introduction Since we are interested in the
practical applications of the calculus of the distance between
twoconics 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 whosesmallest
nonnegative real root provides the square of the distance between a
given point and anellipse. The coefcients 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 thecomputation of footpoints
being this task a source of numerical problems since they do not
behavecontinuously like the distance does. This problem has been
analyzed by several authors like [3], [4], [9], [11], [12] but all
of theirapproximations 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 aand b parallel to the
coordinate axis. Let (x0 , y0 ) be a point exterior to E0 . The
distance betweenthem, d, is given by:d = min (x x0 )2 + (y y0 )2 :
(x, y) E0In order to get a closed formula giving d in terms of the
parameters a, b, x0 and y0 , the ellipse 12. 12 CHAPTER 2. THE
DISTANCE BETWEEN TWO ELLIPSESE0 is characterized by the usual
parametrization: x = a cos t, y = b sin t, t [0, 2)In this way the
square of the minimum distance from (x0 , y0 ) to E0 , D = d2 , is
attained at avalue of the parameter t0 where the function 2 2f (t)
= x0 a cos t + y0 b sin ttakes 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 = 0and 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 ) = 0To perform this elimination, and in order to make
cos(to ) and sin(t0 ) disappear, we introducethe change of
variable2 :1 1 cos(t0 ) = 2 z + z11sin(t0 ) = 2i z z The principal
advantage of this change of variable is given by the fact that if z
= cos(t0 ) + 1i sin(t0 ) then z = z . In this way it is concluded
that the searched D veries that there exists z Csuch 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 o2What 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. Its known that in that
cal-culus strange factors may appear3 . In our case the factor
which appears and that we have removed 1This 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. Lets
remember that an offsetcurve is the set of all points that lie a
perpendicular distance d from a given curve in R2 .The scalar d is
called the offsetradius. If the parametric equation of the given
curve is P (t) = (x(t), y(t)) then the offset curve with offset
radius d isgiven by(y (t), x (t)) (d, P (t)) = P (t) + d x (t)2 + y
(t)2Note that in this denition, if d is positive, the offset is on
our right as we walk along the base curve in the direction
ofincreasing parameter value.2In [10] one can see how this
transformation is used to solve some kinematic equations3This is
due to the implementation of the scientic software we are using:
Maple 12 13. 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]0which 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 semiaxesa and b then D = d2 is the smallest nonnegative real
root of the polynomial[x ,y0 ]0F[a,b] (D) = (a b)2 D4 + 2(a b)(b2 +
2x2 b + y0 b 2ay0 a2 x2 a)D3 + 022 0+ (y0 b2 8y0 ba2 6b2 a2 + 6a3
y0 2x2 a3 + a4 + 6x2 y0 b2 2y0 b3 + 6y0 a2 + 4x2 a2 b+42 2 0 0224 0
+ 2b3 a + 6x2 y0 a2 + 2a3 b 6x4 ab + 4y0 b2 a + 6x4 b2 + x4 a2 +
6b3 x2 10x2 y0 ab + b4 02 02 0 0 00 2 8x2 ab2 6y0 ab)D2 04 2(ab4 +
y0 a4 a2 b3 + a4 b + 2y0 a2 + 2b2 x6 a3 b2 bx2 ay0 bx4 ay0 + 3x2
ay0 b2 +2 60 040202+ 3x2 a2 y0 b by0 a + b2 y0 x2 + 3x4 b3 + 3y0 a3
+ x2 b4 + x4 a2 y0 bx6 a 5x4 ab2 +02 6 4 0 040 020 0 + 3b2 y0 x4 +
3y0 a 2x2 a3 y0 + 3x4 a2 b + 3x2 b2 y0 2x2 ab3 2y0 a3 b 3y0 ab3 3x2
a3 b2 0 4 0 2 004022 0 2x2 b3 y0 5y0 a2 b + 4x2 a2 b2 + 4y0 a2 b2
)D+02 4 0 2+ (x4 + 2x2 b + b2 2x2 a 2ba + a2 + y0 + 2x2 y0 2y0 b +
2ay0 ) (bx2 + ay0 ba)2 =0 00 4 02 2202 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 0maximum
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 ab.In this case [
ab,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 abmakes the function f (t) to vanish. The other two
solutions 2 D = 2a b 2 a(a b) =a ab 2 D = 2a b + 2 a(a b) =a+
abproduce, respectively, the minimum and the maximum distance from
the foci to the ellipse E0 . 14. 14 CHAPTER 2. THE DISTANCE BETWEEN
TWO ELLIPSESRemark 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 + x220 2 0 d2 + 2Rd + R2 y0 x2 20 d2 2Rd + R2 y0 x22 0with
real rootsd1 = R +y0 + x22 0d2 = R y0 + x2 20d3 = R + y0 + x22 0d4
= R y0 + x22 0Thus:d = min {di : di 0} = R y0 + x2 . 20It is
important to quote here that the formula presented in Theorem 2.2.1
provides the minimumdistance without requiring the availability or
previous computation of the footpoints (i.e. the pointswhere 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 = (s), y = (s) and s [0, 2) Thend(E0 , E1 )
= min (x1 x0 )2 + (y1 y0 )2 : (x0 , y0 ) E0 , (x1 , y1 ) E1is the
square root of the smallest nonnegative real root of the family of
univariate polynomials [(s),(s)]F[a,b] (D). The foci question
pointed out in Remark 2.2.2 needs to be taken into account hereif
one of the foci of E0 belongs to E1 . That is not a problem because
that question is very easy tocheck and to deal with when computing
d(E0 , E1 ). [(s),(s)] In order to determine the smallest positive
real root of F[a,b] (D) we are analyzing twopossibilities. In the
rst one D is determined as the smallest positive real number such
that thereexists s [0, 2] such that4 [(s),(s)][a,b]F[a,b] = hk
((s), (s))Dk = 0 , k=0 15. 2.3 The distance between two ellipses 15
4[(s),(s)] def [a,b]F [a,b] = h ((s), (s))Dk = 0s kk=0Since (s) and
(s) are linear forms on cos(s) and sin(s) then this question is
converted into analgebraic problem in the same way we have
proceeded in Section 2.2 by performing the change ofvariables in
both equations: 1 1 1 1cos(s) = w+ , sin(s) =w . 2 w 2iw 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 theE0 [x ,y ]square of d(E0
, E1 ). This polynomial needs to be computed once (like F[a,b] 0
(D) as shown in 0Theorem 2.2.1) and depends polynomially on the
parameters dening E0 and E1 . [(s),(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, 2). In order to apply the algorithm in [8] the
change ofcoordinates1 u2 2u cos(s) =, sin(s) =1 + u2 1 +
u2[(u),(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) parallelto 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 andthe cylinder over E1 are the
distances to E0 of the points in E1 being d(E0 , E1 ) the smallest
height(in green). 16. 16CHAPTER 2. THE DISTANCE BETWEEN TWO
ELLIPSESFigure 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, 2). The point in green representsd(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-E0nomial of
multiplicity one, one degree 12 polynomial with multiplicity three
and other multiplefactors of lower degree being the non multiple
factor of degree 12 the one providing the smallestand the biggest
real roots of GE1 (D): E0G1 (d) = k1 d4 (d12 216d11 + ...)(d2 54d +
1053)2 (d2 52d + 1700)2 (k2 d12 + k3 d11 + ...)3 0where ki are real
numbers.It is not still clear if this factorization pattern appears
in a general way, in the case of ellipseswith 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 nding the geometric interpretation of thisfactorization pattern
hoping that it would help us to prove the conjecture. 17. Chapter
3Closest approach of two ellipses orellipsoids3.1 IntroductionThe
distance of closest approach of two arbitrary separated ellipses
(resp. ellipsoids) is thedistance among their centers when they are
externally tangent, after moving them through the linejoining their
centers.That distance modelizes the problem of nding the distance
of closest approach of hard parti-cles which is a key topic in some
physical questions: short-range repulsive forces between atomsand
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
usuallydominates the potential energy, the distance of closest
approach is a key parameter in statisticaldescriptions of condensed
phases.For non-spherical molecules, such as the constituents of
liquid crystals1 , the distance dependson orientation and its
calculation is surprisingly difcult: at rst glance, this problem
seems simpleenough 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
CrystalGordon Conference in 1983 (attended by W. M. Gelbart and R.
B. Meyer); this, however, did notgenerate a solution. J.
Vieillard-Baron, an early worker on this problem, was reportedly
greatlydisturbed by the difculties he encountered.The simplest
smooth non-spherical shapes are the ellipse and the ellipsoid. In
[15], the authorsdescribe a method for solving the problem of
determining the distance of closest approach of thecenters of two
arbitrary hard ellipses, nding 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
workperformed so far, and focuses on the still unanswered questions
which will determine the future challenges in the eld. 18. 18
CHAPTER 3. CLOSEST APPROACH OF TWO ELLIPSES OR ELLIPSOIDSof 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 theyare in point contact externally 3. PROBLEM:
to nd the distance d between the centers when the ellipses are so
tangent, thatis, to nd the distance of closest approach. 4.
Transformation of the two tangent ellipses into a circle and an
ellipse. The circle and theellipse 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 thetransformation. It is known the difculties 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
positionsof two ellipses by using several tools coming from Real
Algebraic Geometry, computer Alge-bra and Projective Geometry
(Sturm-Habicht sequences and the classication of pencils of
conicsin P2 (R)). Each relative position is exclusively
characterized by a set of equalities and inequal-ities depending
only on the matrices dening the two considered ellipses and does
not requirein advance the computation or knowledge of the
intersection points between them. We use thecharacterization of
externally tangent ellipses and ellipsoids provided in [13] and
[14].3.2 Two ellipses caseDenition 3.2.1. Given two arbitrary
separated ellipses E1 and E2 we dene the distance oftheir closest
approach as the distance among their centers when they are
externally tangent, aftermoving them through the line joining their
centers. 2The anisotropy is the property of being directionally
dependent(i.e. opposed to isotropy, which means homogeneityin all
directions). It can be dened as a difference in a physical property
(absorbance, refractive index, density, etc.) forsome material when
measured along different axes. An example is the light coming
through a polarizing lens.3By anisotropic scaling an ellipse can be
transformed into a unit circle, that is to make the isotropy
disappear, asyou can see in [15]. 19. 3.2 Two ellipses case19
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 = 0be the equation of an ellipse. As usual it can be
rewritten asX T AX = 0where X T = (x, y, 1) and A = (aij ) is the
symmetric positive denite matrix of the coefcients.Following the
notation in [13] and [14] we dene the characteristic polynomial of
the pencil de-termined by two ellipses as follows.Denition 3.2.2.
Let A and B be two ellipses given by the equations X T AX = 0 and X
T BX = 0respectively, the degree three polynomial f () = det(A +
B)is called the characteristic polynomial of the pencil A + BIn
[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
characteristicpolynomial, of the separation case: i.e when the two
ellipsoids can be separated by a plane. Moreprecisely 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 isthe characterization we are going to use in order to obtain
the solution of the problem without 20. 20 CHAPTER 3. CLOSEST
APPROACH OF TWO ELLIPSES OR ELLIPSOIDSusing geometric
transformations which involve the calculus of eigenvalues or
eigenvectors. Thepresented approach provides a closed formula for
the polynomial S(t) (depending polynomiallyon the ellipse
parameters) whose smallest real root provides the distance of
closest approach. Wewill see that the presented approach extend in
a natural way to the distance of closest approach fortwo
ellipsoids.Remark 3.2.3. In order to simplify the computation, we
consider the two coplanar ellipses givenby 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 thecoordinate axes,
x and y, respectively; and the other one located in a general
position, as shown ingure 3.2. Let A2 be the matrix associated to
E2 : a11 a12 a13 A2 = a12 a22 a23 a13 a23 a33The center of E2 is
the point (p, q) whose coordinates are given in terms of the
elements of A2as follows:a22 a13 a12 a23p= a2 a11 a22 12 a11 a23
a12 a13q=a2 a11 a2212 The equation of the moving ellipse E1 (t)
obtained making the rst one move along the linewhich joins the
centers of the two ellipses yields:(x pt)2 (y qt)2E1 (t) = (x, y)
R2 : +1=0a b Now we consider the characteristic polynomial of the
pencil A2 + A1 (t):H(t; ) = det(A2 + A1 (t)) = h3 (t)3 + h2 (t)2 +
h1 (t) + 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; ) has a double positive root. So the equation which
gives us thesearched value of t, t0 is S(t) = 0 where S(t) = disc
H(t; ) = s8 t8 + s7 t7 + s6 t6 + s5 t5 + s4 t4 + s3 t6 + s2 t4 + s1
t2 + s0 21. 3.2 Two ellipses case 21If t0 is the smallest positive
real root of S(t) then the searched distance of the closest
approachof our ellipses is equal to t0 p2 + q 2 Figure 3.2:
Conguration of the two ellipses.Theorem 3.2.4. Given two separated
ellipses E1 and E2 dened as in the remark 3.2.3 the dis-tance of
their closest approach is given asd = t0p2 + q 2where t0 is the
smallest positive real root of S(t) = disc H(t; ), H(t; ) is the
characteristicpolynomial of the pencil determined by them and (p,
q) is the center of E2 .3.2.1 ExampleIn order to show the aspect
the polynomials involved in the previous calculus we are goingto
consider the following example: let A and B be the ellipses given
by the following equations,respectively:1A := (x, y) R2 : x2 + 2 y
2 1 = 0B := (x, y) R2 : 9x2 + 4y 2 54x 32y + 109 = 0This initial
conguration is shown in the gure 3.3.That is, both of them have
axes parallel tothe coordinate axes (in fact, A has its axes
contained in them). The center of A is the origin of 1coordinates
and the lengths of its semi-axes are 1 and 2 . B is centered in the
point (3, 4) withsemi-axes of lengths 2 and 3, resp. 22. 22CHAPTER
3. CLOSEST APPROACH OF TWO ELLIPSES OR ELLIPSOIDSFigure 3.3:
Position of the ellipses A (blue) and B (green). We make the center
of the rst one to move along the line determined by the centers of
theellipses. That gives us the equation of a moving ellipse,
depending on the parameter t:(y 4t)2A(t) := (x, y) R2 : (x 3t)2 +
1=02The characteristic polynomial of the pencil B + A(t), once
turned monic, results:17 2 17523 145 2 145 1 HA(t) (t; ) = 3 + B t
+t 2 + t +t +36 1824 648 2592 12962592And computing the resultant
of this polynomial with respect to we can determine the poly-
Bnomial SA(t) (t) whose its smallest real root represents the
instant t = t0 when the ellipses aretangent: 251243 115599091
1478946641 266704681 55471163 BSA(t) (t) = 80621568 t + 8707129344
t2 + 34828517376 t4 8707129344 t3 + 2902376448 t6158971867 56076225
6076225 40111 4353564672 t + 8707129344 t8 1088391168 t7 +
136048896BThe four real roots of SA(t) (t) are:t0 = 0.2589113100,
t1 = 0.7450597195, t2 = 1.254940281, t3 = 1.741088690The situations
associated to each value of the parameter t = ti , i = 0, 1, 2, 3
are shown in thefollowing gures: 23. 3.2 Two ellipses case 23
Figure 3.4: Position of the ellipses A(t) (blue) and B (green) at
the instant t = t0 . Figure 3.5: Position of the ellipses A(t)
(blue) and B (green) at the instant t = t1 . 24. 24CHAPTER 3.
CLOSEST APPROACH OF TWO ELLIPSES OR ELLIPSOIDS Figure 3.6: Position
of the ellipses A(t) (blue) and B (green) at the instant t = t2 .
Figure 3.7: Position of the ellipses A(t) (blue) and B (green) at
the instant t = t3 . BAs one can see in the previous gures, the
four real roots of SA(t) (t) give us the four instantsin which the
two ellipses are tangent. Being t0 the root which gives us the
closest approach ofthem which is d = 5t0 = 1.294556550 in this
case. 25. 3.3 Distance of closest approach of two ellipsoids253.3
Distance of closest approach of two ellipsoidsThe presented
approach in the previous section extends in a natural way to the
distance ofclosest approach for two ellipsoids4 .Denition 3.3.1. A
real ellipsoid is the quadricsurface dened X T AX = 0 where X T
=(x, y, z, 1),a11 a12 a13a14 a12 a22 a23a24 A= a13 a23 a33a34 a14
a24 a34a44is non-singular, det(A) > 0 and the cofactor of the
term a44 does not vanish.Denition 3.3.2. The center (xc , yc , zc )
of a central quadric surface5 X T AX = 0 is given by
theequations:A14 A24A34xc = ; yc = ; zc =A44 A44A44where 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 congu-ration of the quadrics we are studying:Let A1 and
A2 be the symmetric denite positive matrices dening the separated
ellipsoidsE1 andE2 as X T A1 X = 0 and X T A2 X = 0 where X T = (x,
y, z, 1), and 1a00 0 0 1 b00 A1 = 01 0c 0 0 00 1 a11 a12a13a14 a12
a22a23a24 A2 = a13a23a33a34 a14 a24a34a44i.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 = 04In
[5] it is shown the relation between this distance and molecular
simulations.5Like the ellipsoid. 26. 26CHAPTER 3. CLOSEST APPROACH
OF TWO ELLIPSES OR ELLIPSOIDS 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 describedby the coordinate axes, x, y and z,
respectively; and the second one given in a general form,
withcenter at (xc , yc , zc ), determined by the entries of the
matrix A2 in the way it is shown by thedenition 3.3.2. That
conguration is shown in gure 3.8 Figure 3.8: Conguration of the two
ellipsoids.In the same way we proceeded in the case of the ellipses
we make the center of E1 (and so allthe 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 cIn order
to nd the value of t, t0 , for which the ellipsoids are externally
tangent we have tostudy the roots of the characteristic polynomial
associated to them, that is to check if the polyno-mial H(t; ) =
det(E1 (t) + E2 ), which has degree four, has a double real root,
like the authorsdo in [14]. This is done by computing the roots of
the polynomial of degree 12:S(t) = disc (H(t, )) = s12 t12 + ... +
s0 If t0 is the smallest positive real root of S(t) then the
searched distance of closest approach isequal tot0 x2 + yc + zcc2
2Theorem 3.3.4. Given two separated ellipsoids E1 and E2 dened as
in 3.3.3 the distance of theirclosest approach is given asd = t0 x2
+ yc + zcc 2 2where t0 is the smallest positive real root of S(t) =
disc H(t; ), H(t; ) is the characteristicpolynomial of the pencil
determined by them, and (xc , yc , zc ) is the center of E2 . 27.
3.3 Distance of closest approach of two ellipsoids273.3.1 Example
In order to illustrate the previous theorem we are going to
consider a practical example. LetE1 and E2 be the two ellipsoids
given as follows:E1 := (x, y, z) R3 : 1 x2 + 1 y 2 + z 2 1 = 04 2E2
:= (x, y, z) R3 : 1 x2 2 x + 1 y 2 3 y + 51 + 1 z 2 5 z = 0 54 22
That is, E1 is centered at the origin of coordinates with semi-axis
of lengths 2, 2 and 1 alongthe x-axis, y-axis and z-axis, resp. And
the point (5, 6, is the center of E2 whose axis are parallel5) to
the coordinate ones and have semi-lengths equal to 5, 2 and 2 with
respect to the coordinateframe. This situation can be observed in
gure3.9. Figure 3.9: Conguration 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 ofthe two quadrics we are studying.
That is the way in which the equation of a moving
ellipsoid,depending on the parameter t, is obtained: 28. 28CHAPTER
3. CLOSEST APPROACH OF TWO ELLIPSES OR ELLIPSOIDS 1 2 1 25197 2E1
(t) := (x, y, z) R3 : x + y + z 2 tx 6 ty 10 tz 1 +t =0 42 2 4The
characteristic polynomial of E2 and E1 (t), once turned monic,
results: E2HE1 (t) (t; ) = 4 43 3 197 3 t2 301 2 659 2 t2 +42 4
19723 t 2372 265 t2 + 659 2 t + 5 + 265 t22And computing the
resultant of this polynomial with respect to the variable we can
determineE2 (t)the polynomial SE1 (t) whose its smallest real root
represents the instant t = t0 when the ellipsoidsare tangent: E (t)
16641 2 SE1 (t) = 1)4 (2725362025t8 21802896200t7 + 75970256860t6
150580994360t5 +1024 (t 185680506596t4 145836126384t3 +
71232102544t2 19777044480t + 2388833408) 2E (t)The four real roots
of SE1 (t) that determine the four tangency points are all provided
by thefactor of degree 8 which appears in its decomposition. And
they are the following: t0 = 0.6620321914, t1 = 0.6620321914, t2 =
1.033966297, t3 = 1.337967809The 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 gures:Figure 3.10: Conguration
of the two ellipsoids E1 (t) (blue)and E2 (green) at the instant t
= t0 . 29. 3.3 Distance of closest approach of two
ellipsoids29Figure 3.11: Conguration of the two ellipsoids E1 (t)
(blue)and E2 (green) at the instant t = t1 .Figure 3.12:
Conguration of the two ellipsoids E1 (t) (blue)and E2 (green) at
the instant t = t2 . 30. 30CHAPTER 3. CLOSEST APPROACH OF TWO
ELLIPSES OR ELLIPSOIDSFigure 3.13: Conguration of the two
ellipsoids E1 (t) (blue)and E2 (green) at the instant t = t3 .As it
happened in the case of the ellipses the previous gures 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 ellipsoidsare
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. 31. Chapter
4Conclusions and future work4.1 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 efciency when dealing with moving ob-
jects (in order to dene 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 workIn
relation to the study of the minimum distance between two separated
ellipses, the maintopic in which we are interested consist of
obtaining the decomposition of the polynomial GE1 (D)E0of degree
60, in the general case. We just have it in particular cases. Once
we have done that wewould be able to prove our conjecture and it
also will help us with the study of the existing rela-tionship
between the geometric congurations of the two ellipses and the
terms that appear in thefactorization of GE1 (D). E0 32. 32CHAPTER
4. CONCLUSIONS AND FUTURE WORKParticular geometric congurations1 of
the quadrics or conics we are studying seem to berelated with
specially simple decompositions of the polynomials involved in the
calculus of theminimum distance between them or of the closest
approach of them. We are currently working inthe
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 ourtechniques to
robotics and similar elds in which the objects of study are not
static, they representbodies 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 threedimensional case: the study of the
minimum distance between two separated ellipsoids in thespace. It
will also be interesting to consider the case of non-coplanar
ellipses, that is, when the twoseparated ellipses are not contained
in the same plane, and of course the case of other coplanarand
non-coplanar conics, like hyperbolas, for instance.4.2.1 Using
ellipses to check safety regionsSuppose that you have a collection
of moving automata and you are interested in checkingthe regions in
which your robots do not collide each others. In other words, you
are interested innding the set of the safety regions. This problem
can be modelized using ellipses: we can assumethat 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 penetratingbetween objects. Besides, it is used as path
planning in robotics to calculate in advance paths ofmoving robots
to avoid collision during motions. There are many applications like
robot or vehiclepath planning where the robots and vehicles are
represented as 2D gures moving in 2D plane. The answer to them is
given by considering the following discriminant: [(s),(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 closedformulae can be
determined is using subresultants and similar techniques.1This 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. 33. 4.2 Future work 334.2.2 Haussdorf distance
computations between ellipses and ellipsoidsSimilar approaches to
deal with Haussdorf distance computations between ellipses and
ellip-soids are being analyzed: The eld of the analysis and
comparison of geometric shapes acquiresspecial importance in
various application areas within Computer Science. For example
patternrecognition 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
orthree dimensional space and we are interested in knowing how much
they resemble each other.For this purpose we need a similarity
measure dened on pairs of shapes indicating the degree
ofresemblance of these shapes. As one can see in [6] and [1] a
frequently used similarity measureis the Hausdorff distance, which
is dened 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 themaximum over all these
values.Denition 4.2.1. The one-sided Hausdorff distance from A to B
isH (a, b) = {maxaA minbB } 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 onewhen 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 theproblem 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 pointson 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 ofthe medial
axis of the curve. Our goal here is to obtain similar
characterizations like the proposed in [6] or [1] but (ofcourse)
avoiding the calculus of the footpoints, doing it in a direct way,
using similar ideas that theones which have appear in the previous
chapters of this document. 34. 34 CHAPTER 4. CONCLUSIONS AND FUTURE
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