adg08
DESCRIPTION
Presentation at the conference ADG in Shanhai, China, in 2008TRANSCRIPT
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
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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).
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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.
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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.
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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.
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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.Is there any advantage?
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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.Is there any advantage?
Indeed: the distance behaves continuously but footpoints donot.
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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(and, as a by-product, i disappears).
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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 of length
√a and
√b then d = d2
0 isthe smallest nonnegative real root of the polynomial F [x0,y0]
[a,b] (d).
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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
∣∣∣∣
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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).
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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.
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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.
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
ProblemOur approach
ExampleFuture work
Example
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
ProblemOur approach
ExampleFuture work
Example
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
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.
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
ProblemOur approach
ExampleFuture work
Future work
Continue studying the continuous motion case.Generalize to ellipsoids.Non-coplanar ellipses.Other conics.
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008
ProblemOur approach
ExampleFuture work
Thank you!
F. Etayo, L. Gonzalez-Vega, G. R. Quintana, W. Wang ADG 2008