ADG 2008

Problem
Our approach
Example
Future work

Closed formulae for distance functions
involving ellipses.

F. Etayo1 , L. Gonzalez-Vega1 , G. R. Quintana1 , W. Wang2

1 Departamento de Matemáticas, Estadística y Computación
Universidad de Cantabria
2 Department of Computer Science
University of Hong Kong

VII International Workshop on Automated Deduction in
Geometry

F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008

Problem
Our approach
Example
Future work

Contents

1 Problem

2 Our approach

3 Example

4 Future work


Problem
Our approach
Example
Future work

Introduction

We want to compute the distance between two coplanar
ellipses.

The minimum distance between a given point and one ellipse is
a positive algebraic number: our goal is to determine a
polynomial with this number as a real root.

This way of presenting the distance is independent of the
corresponding footpoints and provides the distance directly. We
can use this formula for analyzing the Ellipses Moving Problem
(EMP).


Problem
Our approach
Example
Future work

Applications

The computation of the minimum distance between two ellipses
(static or dynamic) is a fundamental task in various
applications:
collision detection in robotics,
interference avoidance in CAD/CAM,
interactions in virtual reality,
computer games,
orbit analysis (non-coplanar ellipses),
interference analysis of molecules in computational
physics and chemistry,
etc.


Problem
Our approach
Example
Future work

Previous works

I. Z. E MIRIS , E. T SIGARIDAS , G. M. T ZOUMAS . The
predicates for the Voronoi diagram of ellipses. Proc. ACM
Symp. Comput. Geom., 2006.
I. Z. E MIRIS , G. M. T ZOUMAS . A Real-time and Exact
Implementation of the predicates for the Voronoi Diagram
for parametric ellipses. Proc. ACM Symp. Solid Physical
Modelling, 2007.
C. L ENNERZ , E. S CHÖMER . Efﬁcient Distance
Computation for Quadratic Curves and Surfaces.
Geometric Modelling and Processing Proceedings, 2002.


Problem
Our approach
Example
Future work

Previous works

J.-K. S EONG , D. E. J OHNSON , E. C OHEN . A Higher
Dimensional Formulation for Robust and Interactive
Distance Queries. Proc. ACM Solid and Physical
Modeling, 2006.
K.A. S OHN , B. J ÜTTLER , M.S. K IM , W. WANG .
Computing the Distance Between Two Surfaces via Line
Geometry. Proc. Tenth Paciﬁc Conference on Computer
Graphics and Applications, 236-245, IEEE Press, 2002.
Common aspect: the problem is always solved by determining,
ﬁrst, the footpoints and then the searched distance.


Problem
Our approach
Example
Future work

Our approach

We do not make the minimum distance computation depending
on the determination of the footpoints. We study the ellipse
separation problem by analyzing the univariate polynomial
providing the distance.

Parameters of our problem: center coordinates, axes length,
inclination of the axes.


Problem
Our approach
Example
Future work

Our approach


Is there any advantage?


Problem
Our approach
Example
Future work

Our approach


Is there any advantage?

Indeed: the distance behaves continuously but footpoints do
not.


Problem
Our approach
Example
Future work

The distance of a point to an ellipse

We consider the parametric equations of an ellipse, ε0 :
√ √
x = a cos t, y = b sin t, t ∈ [0, 2π)

in order to construct a function fd whose minimum positive
value, d, gives the square of the distance between a point
(x0 , y0 ) and the ellipse:
√ √
fd := (x0 − a cos t)2 + (y0 − b sin t)2 − d


Problem
Our approach
Example
Future work


We want to solve a system of equations:

 fd (t) = 0
∂fd
(t) =0

∂t


Problem
Our approach
Example
Future work


We want to solve a system of equations:

 fd (t) = 0
∂fd
(t) =0

∂t

There are two posibilities:
rational change of variable
complex change of variable


Problem
Our approach
Example
Future work


Rational change of variable:
1−t2
cos t = 1+t2

2t
sin t = 1+t2

Disadvantage: more complicated.


Problem
Our approach
Example
Future work


1
Since z = cos t + i sin t, z = z and we can use the complex
change of variable:
1
z− z
sin t = 2i

1
z+ z
cos t = 2


Problem
Our approach
Example
Future work


The new system:

√ √ √ √
 (b − a)z 4 + 2(x0 √a − iy0 √b)z 3 − 2(x0 a + iy0 b)z + a − b = 0


(b − a)z 4 − 4(x0 a − iy0 b)z 3 − 2(2(x2 + y0 − d))z 2 +
0
2
√
√
+4(x0 a + iy0 b)z + b − a = 0


Using resultants we eliminate the variable z
(and, as a by-product, i disappears).


Problem
Our approach
Example
Future work


Theorem
If d0 is the distance of a point (x0 , y0 ) to the ellipse ε0 with
√
√
center (0, 0) and semiaxes of length a and b then d = d2 is 0
[x0 ,y ]
the smallest nonnegative real root of the polynomial F[a,b] 0 (d).


Problem
Our approach
Example
Future work

[x ,y ]
F[a,b] 0 (d) =
0

= (a − b)2 d4 + 2(a − b)(b2 + 2x2 b + y0 b − 2ay0 − a2 − x2 a)d3
0
2 2
0
+(y0 b − 8y0 ba − 6b a + 6a y0 − 2x0 a + a + 6x2 y0 b2 − 2y0 b3
4 2 2 2 2 2 3 2 2 3 4
0
2 2
4 2 2 2 3 2 2 2 3 4 2 2
+6y0 a + 4x0 a b + 2b a + 6x0 y0 a + 2a b − 6x0 ab + 4y0 b a
+6x4 b2 + 4x4 a2 + 6b3 x2 − 10x2 y0 ab + b4 − 8x2 ab2 − 6y0 ab)d2
0 0 0 0
2
0
4
4 4 2 3 4 6 2 2 6 3 2 2 4
−2(ab + y0 − a b + a b + 2y0 a + 2b x0 − a b − bx0 ay0
4 2 2 2 2 2 2 2 6 2 4 2 4 3
−bx0 ay0 + 3x0 ay0 b + 3x0 a y0 b − by0 a + b y0 x0 + 3x0 b
+3y0 a3 + x2 b4 + x4 a2 y0 − bx6 a − 5x4 ab2 + 3b2 y0 x4 + 3y0 ab2
4
0 0
2
0 0
2
0
4
2 3 2 4 2 2 2 2 2 3 2 3 2 3
−2x0 a u0 + 3x0 a b + 3x0 b y0 − 2x0 ab − 2y0 a b − 3y0 ab
−3x2 a3 b − 2x2 b3 y0 − 5y0 a2 b + 4x2 a2 b2 + 4y0 a2 b2 )d
0 0
2 4
0
2

+(x0 + 2x0 b + b − 2x0 a − 2ba + a + y0 + 2x2 y0 − 2y0 b + 2ay0 )·
4 2 2 2 2 4
0
2 2 2
2 2 2
(bx0 + ay0 − ba) =
4 [a,b]
= k=0 hk (x0 , y0 )dk


Problem
Our approach
Example
Future work

Remarks to the theorem

[x ,y ]
The biggest real root of F[a,b] 0 (d) is the square of the
0

maximum distance between (x0 , y0 ) and the points in ε0 .
If x0 is a focus of ε0
√
[ a−b,0]
F[a,b] (d) = (a − b)2 d2 (d2 + 2(b − 2a)d + b2 )
√ √ √ √
⇒ d = ( a − a − b)2 , ( a + a − b)2

In the case of a circle a = b = R2 and if d = d2
0
√
[ a−b,0]
F[a,b] (d2 ) = R4 (y0 + x2 )2 ·
0
2
0
2 + 2Rd + R2 − y 2 − x2 )(d2 − 2Rd + R2 − y 2 − x2 )
· (d0 0 0 0 0 0 0 0
⇒ d0 = R − y0 + x2
2
0


Problem
Our approach
Example
Future work

The distance between two ellipses

Let ε1 be an ellipse disjoint with ε0 , presented by the
parametrization x = α(s), y = β(s), s ∈ [0, 2π). Then

d(ε0 , ε1 ) = min{ (x1 − x0 )2 + (y1 − y0 )2 : (xi , yi ) ∈ εi , i = 1, 2}

is the square root of the smallest nonnegative real root of
[α(s),β(s)]
the family of univariate polynomials F[a,b] (d).


Problem
Our approach
Example
Future work

The distance between two ellipses

In order to determine d(ε0 , ε1 ) we are analyzing two posibilities:
d is determined as the smallest positive real number s.t.
there exist s ∈ [0, 2π) solving
[α(s),β(s)] 4 [a,b]
F[a,b] = k=0 hk (α(s), β(s))dk = 0
¯ [α(s),β(s)] :=
F[a,b]
4 ∂ [a,b]
(α(s), β(s))dk =
k=0 ∂s hk 0

d is determined by analyzing the implicit curve
[α(s),β(s)]
F[a,b] = 0.


Problem
Our approach
Example
Future work

First case

Since α(s) and β(s) are linear forms on cos(s) and sin(s) this
question is converted into an algebraic problem in the same
way we have proceeded in the case point-ellipse, by performing
the change of variable

1 1 1 1
cos s = w+ , sin s = w−
2 w 2i w

and then using resultants to eliminate w.
We obtain a univariate polynomial of degree 60, Gε1 , whose
ε0
smallest positive real root is the square of d(ε0 , ε1 ).
Gε1 depends polynomially on the parameters of ε0 and ε1 .
ε0


Problem
Our approach
Example
Future work

Second case

[α(s),β(s)]
d is determined by analyzing the implicit curve F[a,b] = 0 in
the region d ≥ 0 and s ∈ [0, 2π). In order to aply the algorithm
by L. G ONZALEZ -V EGA , I. N ÉCULA , Efﬁcient topology
determination of implicitly deﬁned algebraic plane curves.
Computer Aided Geometric Design, 19: 719-743, 2002, we use
the change of coordinates:

1 − u2 2u
cos s = 2
sin s =
1+u 1 + u2
[α(s),β(s)]
and the real algebraic plane curve F[a,b] = 0 is analyzed in
d ≥ 0, u ∈ R.


Problem
Our approach
Example
Future work

Example

We consider ε0 and ε1 . ε0 with center (0, 0) and semi-axes of
length 3 and 2. ε1 centered in (2, −3) and with semi-axes,
parallel to the coordinate axes, of length 2 and 1.


Problem
Our approach
Example
Future work

Example


Problem
Our approach
Example
Future work

Example

In this case the minimum distance is given by computing the
real roots of the polynomial:
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.

The non multiple factor of degree 12 is the one providing
the smallest and the biggest nonnegative real roots of
Gε1 (d). It is not still clear if this pattern appears in a
ε0
general way.


Problem
Our approach
Example
Future work

Future work

Continue studying the continuous motion case.
Generalize to ellipsoids.
Non-coplanar ellipses.
Other conics.


Problem
Our approach
Example
Future work

Thank you!


ADG 2008

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ADG 2008