ADG08

ProblemOur approach

ExampleFuture work

Closed formulae for distance functionsinvolving ellipses.

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

1Departamento de Matemáticas, Estadística y ComputaciónUniversidad de Cantabria

2Department of Computer ScienceUniversity of Hong Kong

VII International Workshop on Automated Deduction inGeometry

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

ProblemOur approach

ExampleFuture work

Contents

1 Problem

2 Our approach

3 Example

4 Future work


ProblemOur approach

ExampleFuture work

Introduction

We want to compute the distance between two coplanarellipses.

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

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


ProblemOur approach

ExampleFuture work

Applications

The computation of the minimum distance between two ellipses(static or dynamic) is a fundamental task in variousapplications:

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 computationalphysics and chemistry,etc.


ProblemOur approach

ExampleFuture work

Previous works

I. Z. EMIRIS, E. TSIGARIDAS, G. M. TZOUMAS. Thepredicates for the Voronoi diagram of ellipses. Proc. ACMSymp. Comput. Geom., 2006.I. Z. EMIRIS, G. M. TZOUMAS. A Real-time and ExactImplementation of the predicates for the Voronoi Diagramfor parametric ellipses. Proc. ACM Symp. Solid PhysicalModelling, 2007.C. LENNERZ, E. SCHÖMER. Efficient DistanceComputation for Quadratic Curves and Surfaces.Geometric Modelling and Processing Proceedings, 2002.


ProblemOur approach

ExampleFuture work

Previous works

J.-K. SEONG, D. E. JOHNSON, E. COHEN. A HigherDimensional Formulation for Robust and InteractiveDistance Queries. Proc. ACM Solid and PhysicalModeling, 2006.K.A. SOHN, B. JÜTTLER, M.S. KIM, W. WANG.Computing the Distance Between Two Surfaces via LineGeometry. Proc. Tenth Pacific Conference on ComputerGraphics and Applications, 236-245, IEEE Press, 2002.

Common aspect: the problem is always solved by determining,first, the footpoints and then the searched distance.


ProblemOur approach

ExampleFuture work

Our approach

We do not make the minimum distance computation dependingon the determination of the footpoints. We study the ellipseseparation problem by analyzing the univariate polynomialproviding the distance.

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


ProblemOur approach

ExampleFuture work

Our approach


Parameters of our problem: center coordinates, axes length,inclination of the axes.Is there any advantage?


ProblemOur approach

ExampleFuture work

Our approach


Parameters of our problem: center coordinates, axes length,inclination of the axes.Is there any advantage?

Indeed: the distance behaves continuously but footpoints donot.


ProblemOur approach

ExampleFuture 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 positivevalue, 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


ProblemOur approach

ExampleFuture work


We want to solve a system of equations:fd(t) = 0

∂fd

∂t(t) = 0


ProblemOur approach

ExampleFuture work


We want to solve a system of equations:fd(t) = 0

∂fd

∂t(t) = 0

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


ProblemOur approach

ExampleFuture work


Rational change of variable:

cos t = 1−t21+t2

sin t = 2t1+t2

Disadvantage: more complicated.


ProblemOur approach

ExampleFuture work


Since z = cos t+ i sin t, z = 1z and we can use the complex

change of variable:

sin t = z− 1z

2i

cos t = z+ 1z

2


ProblemOur approach

ExampleFuture work


The new system:

(b− a)z4 + 2(x0

√a− iy0

√b)z3 − 2(x0

√a+ iy0

√b)z + a− b = 0

(b− a)z4 − 4(x0√a− iy0

√b)z3 − 2(2(x2

0 + y20 − d))z2+

+4(x0√a+ iy0

√b)z + b− a = 0

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


ProblemOur approach

ExampleFuture work


TheoremIf d0 is the distance of a point (x0, y0) to the ellipse ε0 withcenter (0, 0) and semiaxes of length

√a and

√b then d = d2

0 isthe smallest nonnegative real root of the polynomial F [x0,y0]

[a,b] (d).


ProblemOur approach

ExampleFuture work

F[x0,y0][a,b] (d) =

= (a− b)2d4 + 2(a− b)(b2 + 2x20b+ y2

0b− 2ay20 − a2 − x2

0a)d3

+(y40b

2 − 8y20ba

2 − 6b2a2 + 6a3y20 − 2x2

0a3 + a4 + 6x2

0y20b

2 − 2y20b

3

+6y40a

2 + 4x20a

2b+ 2b3a+ 6x20y

20a

2 + 2a3b− 6x40ab+ 4y2

0b2a

+6x40b

2 + 4x40a

2 + 6b3x20 − 10x2

0y20ab+ b4 − 8x2

0ab2 − 6y4

0ab)d2

−2(ab4 + y40 − a2b3 + a4b+ 2y6

0a2 + 2b2x6

0 − a3b2 − bx20ay

40

−bx40ay

20 + 3x2

0ay20b

2 + 3x20a

2y20b− by6

0a+ b2y40x

20 + 3x4

0b3

+3y40a

3 + x20b

4 + x40a

2y20 − bx6

0a− 5x40ab

2 + 3b2y20x

40 + 3y4

0ab2

−2x20a

3u20 + 3x4

0a2b+ 3x2

0b2y2

0 − 2x20ab

3 − 2y20a

3b− 3y20ab

3

−3x20a

3b− 2x20b

3y20 − 5y4

0a2b+ 4x2

0a2b2 + 4y2

0a2b2)d

+(x40 + 2x2

0b+ b2 − 2x20a− 2ba+ a2 + y4

0 + 2x20y

20 − 2y2

0b+ 2ay20)·

(bx20 + ay2

0 − ba)2 ==∑4k=0 h

[a,b]k (x0, y0)dk


ProblemOur approach

ExampleFuture work

Remarks to the theorem

The biggest real root of F [x0,y0][a,b] (d) is the square of the

maximum distance between (x0, y0) and the points in ε0.If x0 is a focus of ε0

F[√a−b,0]

[a,b] (d) = (a− b)2d2(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 = d20

F[√

a−b,0][a,b]

(d20) = R4(y20 + x20)2·

· (d20 + 2Rd0 +R2 − y20 − x20)(d20 − 2Rd0 +R2 − y20 − x2

0)

⇒ d0 =

∣∣∣∣R−√y20 + x20

∣∣∣∣


ProblemOur approach

ExampleFuture work

The distance between two ellipses

Let ε1 be an ellipse disjoint with ε0, presented by theparametrization 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 ofthe family of univariate polynomials F

[α(s),β(s)][a,b] (d).


ProblemOur approach

ExampleFuture 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{

F[α(s),β(s)][a,b] =

∑4k=0 h

[a,b]k (α(s), β(s))dk = 0

F̄[α(s),β(s)][a,b] :=

∑4k=0

∂∂sh

[a,b]k (α(s), β(s))dk = 0

d is determined by analyzing the implicit curveF

[α(s),β(s)][a,b] = 0.


ProblemOur approach

ExampleFuture work

First case

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

cos s =12

(w +

1w

), sin s =

12i

(w − 1

w

)and then using resultants to eliminate w.We obtain a univariate polynomial of degree 60, Gε1ε0 , whosesmallest positive real root is the square of d(ε0, ε1).Gε1ε0 depends polynomially on the parameters of ε0 and ε1.


ProblemOur approach

ExampleFuture work

Second case

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

the region d ≥ 0 and s ∈ [0, 2π). In order to aply the algorithmby L. GONZALEZ-VEGA, I. NÉCULA, Efficient topologydetermination of implicitly defined algebraic plane curves.Computer Aided Geometric Design, 19: 719-743, 2002, we usethe change of coordinates:

cos s =1− u2

1 + u2sin s =

2u1 + u2

and the real algebraic plane curve F [α(s),β(s)][a,b] = 0 is analyzed in

d ≥ 0, u ∈ R.


ProblemOur approach

ExampleFuture work

Example

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


ProblemOur approach

ExampleFuture work

Example


ProblemOur approach

ExampleFuture work

Example

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

Gε1ε0

(d) = k1d4(d12−216d11+...)(d2−54d+1053)2(d2−52d+1700)2(k2d

12+k3d11+...)3

where ki are real numbers.

The non multiple factor of degree 12 is the one providingthe smallest and the biggest nonnegative real roots ofGε1ε0

(d). It is not still clear if this pattern appears in ageneral way.


ProblemOur approach

ExampleFuture work

Future work

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


ProblemOur approach

ExampleFuture work

Thank you!


ADG08

Education

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