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Talk I gave at EACA08 in Granada 2008 http://www.ugr.es/~eaca2008/es.php
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ProblemOur approach
ExampleFuture work
Closed formulae for distance functionsinvolving ellipses.
F. Etayo1, L. González-Vega1, G. R. Quintana1, W. Wang2
1Departamento de Matemáticas, Estadística y ComputaciónUniversidad de Cantabria
2Department of Computer ScienceUniversity of Hong Kong
XI Encuentro de Álgebra Computacional y Aplicaciones,Universidad de Granada 2008
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
ProblemOur approach
ExampleFuture work
Contents
1 Problem
2 Our approach
3 Example
4 Future work
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
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.
That distance does not depend on the footpoint. It gives thedistance directly. We can use this formula for analyzing theEllipses Moving Problem (EMP).
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
ProblemOur approach
ExampleFuture work
Applications
The EMP is a critical problem in Computer Graphics, withapplications like:
Collision detectionOrbit analysis (non-coplanar ellipses)
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
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.
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
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 using footpoints.
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
ProblemOur approach
ExampleFuture work
Our approach
We do not make the minimum distance computation dependingon the foot points. We study the ellipse separation problem byanalyzing the univariate polynomial provided by the distance.
Parameters of our problem: center coordinates, axes length...
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
ProblemOur approach
ExampleFuture work
Our approach
We do not make the minimum distance computation dependingon the foot points. We study the ellipse separation problem byanalyzing the univariate polynomial provided by the distance.
Parameters of our problem: center coordinates, axes length...Is there any advantage?
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
ProblemOur approach
ExampleFuture work
Our approach
We do not make the minimum distance computation dependingon the foot points. We study the ellipse separation problem byanalyzing the univariate polynomial provided by the distance.
Parameters of our problem: center coordinates, axes length...Is there any advantage?
Yes: the distance behaves continuously but footpoints don’t.
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
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 which gives the distancebetween a point (x0, y0) and the ellipse:
fd := (x0 −√a cos t)2 + (y0 −
√b sin t)2 − d
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
ProblemOur approach
ExampleFuture work
The distance of a point to an ellipse
We want to solve a system of equations:{fd(t) = 0∂fd∂t (t) = 0
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
ProblemOur approach
ExampleFuture work
The distance of a point to an ellipse
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
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
ProblemOur approach
ExampleFuture work
The distance of a point to an ellipse
Rational change of variable:
cos t = 1−t21+t2
sin t = 2t1+t2
Disadvantage: more complicated.
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
ProblemOur approach
ExampleFuture work
The distance of a point to an ellipse
Rational change of variable:
cos t = 1−t21+t2
sin t = 2t1+t2
Disadvantage: more complicated.
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
ProblemOur approach
ExampleFuture work
The distance of a point to an ellipse
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
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
ProblemOur approach
ExampleFuture work
The distance of a point to an ellipse
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(we also eliminate i).
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
ProblemOur approach
ExampleFuture work
The distance of a point to an ellipse
TheoremIf d0 is the distance of a point (x0, y0) to the ellipse ε0 withcenter (0, 0) and semiaxes a and b then d = d2
0 is the smallestnonnegative real root of the polynomial
F[x0,y0][a,b] (d) = (a− b)2d4 + 2(a− b)(b2 + 2x2
0b + y20b− 2ay20 − a2 − x2
0a)d3
+(y40b2 − 8y20ba
2 − 6b2a2 + 6a3y20 − 2x20a
3 + a4 + 6x20y
20b
2 − 2y20b3
+6y40a2 + 4x2
0a2b + 2b3a + 6x2
0y20a
2 + 2a3b− 6x40ab + 4y20b
2a
+6x40b
2 + 4x40a
2 + 6b3x20 − 10x2
0y20ab + b4 − 8x2
0ab2 − 6y40ab)d
2
−2(ab4 + y40 − a2b3 + a4b + 2y60a
2 + 2b2x60 − a
3b2 − bx20ay
40
−bx40ay
20 + 3x2
0ay20b
2 + 3x20a
2y20b− by60a + b2y40x
20 + 3x4
0b3
+3y40a3 + x2
0b4 + x4
0a2y20 − bx
60a− 5x4
0ab2 + 3b2y20x
40 + 3y40ab
2
−2x20a
3u20 + 3x4
0a2b + 3x2
0b2y20 − 2x2
0ab3 − 2y20a
3b− 3y20ab3
−3x20a
3b− 2x20b
3y20 − 5y40a2b + 4x2
0a2b2 + 4y20a
2b2)d
+(x40 + 2x2
0b + b2 − 2x20a− 2ba + a2 + y40 + 2x2
0y20 − 2y20b + 2ay20)·
(bx20 + ay20 − ba)
2 =∑4
k=0 h[a,b]k
(x0, y0)dk
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
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 circumference a = b = R2 and ifd = d2
0
F[√
a−b,0][a,b] (d20) = R4(y20 + x2
0)2(d20 + 2Rd0 + R2 − y20 − x20)(d20 − 2Rd0 + R2 − y20 − x
20)
⇒ d0 = |R−√y20 + x2
0|
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
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 : (x0, y0) ∈ ε0, (x1, y1) ∈ ε1}
is the square root of the smallest nonnegative real root ofthe family of univariate polynomials F
α(s),β(s)a,b (d).
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
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.
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
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.
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
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. GONZÁLEZ-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.
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
ProblemOur approach
ExampleFuture work
Example
We consider ε0 and ε1. E1 with center (0, 0) and semi-axes oflength 3 and 2. E2 centered in (2,−3) and with semi-axes,parallel to the coordinate axes, of length 2 and 1.
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
ProblemOur approach
ExampleFuture work
Example
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
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 real roots of Gε1
ε0(d). It is not
still clear if this pattern appears in a general way.
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
ProblemOur approach
ExampleFuture work
Future work
Continue studying the continuous motion case.Generalize to ellipsoids.Non-coplanar ellipses.Other conics.
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang
ProblemOur approach
ExampleFuture work
Thank you!
F. Etayo, L. González-Vega, G. R. Quintana, W. Wang