EACA2010
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1. The document discusses computing the distance of closest approach between ellipses and ellipsoids. 2. It summarizes previous work that transformed two tangent ellipses into a circle and ellipse to determine the distance. 3. The authors propose a new approach using the characteristic polynomial of the pencil to characterize when ellipses/ellipsoids are separated or touching.
EACA2010
- 1. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Computing the distance of closest approach between ellipses and ellipsoids F. Etayo, L. González-Vega, G. R. Quintana Departamento de MATemáticas, EStadística y COmputación University of Cantabria, Spain XII Encuentro de Álgebra Computacional y Aplicaciones, Santiago de Compostela, 19-21 de julio de 2010 F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 2. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Contents 1 Problem 2 Distance of closest approach of two ellipses 3 Distance of closest approach of two ellipsoids 4 Conclusions F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 3. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Introduction The distance of closest approach of two arbitrary separated ellipses (resp. ellipsoids) is the distance among their centers when they are externally tangent, after moving them through the line joining their centers. That distance appears when we study the problem of determining the distance of closest approach of hard particles which is a key topic in some physical questions like modeling and simulating systems of anisometric particles, such as liquid crystals, or in the case of interference analysis of molecules. F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 4. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Introduction Distance of closest approach of two ellipses. F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 5. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Previous work A description of a method for solving the problem in the case of two arbitrary hard ellipses (resp. ellipsoids) can be found in X. Z HENG , P. PALFFY-M UHORAY, Distance of closest approach of two arbitrary hard ellipses in two dimensions, Phys. Rev., E 75, 061709, 2007. X. Z HENG , W. I GLESIAS , P. PALFFY-M UHORAY, Distance of closest approach of two arbitrary hard ellipsoids, Phys. Rev. E, 79, 057702, 2009. An analytic expression for that distance is given as a function of their orientation relative to the line joining their centers. F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 6. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Previous work (Zheng,Palffy-Muhoray) Ellipses case: 1 Two ellipses initially distant are given. 2 One ellipse is translated toward the other along the line joining their centers until they are externally tangent. 3 PROBLEM: to find the distance d between the centers at that time. 4 Transformation of the two tangent ellipses into a circle and an ellipse. 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 transformation. F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 7. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Previous work (Zheng,Palffy-Muhoray) Ellipses case: 1 Two ellipses initially distant are given. 2 One ellipse is translated toward the other along the line joining their centers until they are externally tangent. 3 PROBLEM: to find the distance d between the centers at that time. 4 Transformation of the two tangent ellipses into a circle and an ellipse. ⇒ Anisotropic scaling 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 transformation. F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 8. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Previous work (Zheng,Iglesias,Palffy-Muhoray) Ellipsoids case: 1 Two ellipsoids initially distant are given. 2 Plane containing the line joining the centers of the two ellipsoids. 3 Equations of the ellipses formed by the intersection of this plane and the ellipsoids. 4 Determining the distance of closest approach of the ellipses 5 Rotating the plane until the distance of closest approach of the ellipses is a maximum 6 The distance of closest approach of the ellipsoids is this maximum distance F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 9. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Previous work To deal with anisotropic scaling and the inverse transformation involves the calculus of the eigenvectors and eigenvalues of the matrix of the transformation. Our goal is to find when that computation is not required and if it is, to simplify it. The way in which we do that extends in a natural way the ellipsoids case. F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 10. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Our approach We use the results shown in: F. E TAYO, L. G ONZ·LEZ -V EGA , N. DEL RÌ O, A new approach to characterizing the relative position of two ellipses depending on one parameter, Computed Aided Geometric Desing 23, 324-350, 2006. W. WANG , R. K RASAUSKAS, Interference analysis of conics and quadrics, Contemporary Math. 334, 25-36,2003. W. WANG , J. WANG , M. S. K IM, An algebraic condition for the separation of two ellipsoids, Computer Aided Geometric Desing 18, 531-539, 2001. F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 11. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Our approach Following their notation we define Definition Let A and B be two ellipses (resp. ellipsoids) given by the equations X T AX = 0 and X T BX = 0 respectively, the degree three (resp. four) polynomial f (λ) = det(λA + B) is called the characteristic polynomial of the pencil λA + B Two ellipses (or ellipsoids) are separated if and only if their characteristic polynomial has two distinct positive roots. The ellipses (or ellipsoids) touch each other externally if and only if the characteristic equation has a positive double root. F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 12. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Our approach We use the previous characterization in order to obtain the solution of the problem. We give a closed formula for the polynomial S(t) (depending polynomially on the ellipse parameters) whose biggest real root provides the distance of closest approach: Ellipses case: d = t0 x 2 + y0 0 2 Ellipsoids case: d = t0 x 2 + y0 + z0 0 2 2 F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 13. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions We consider the two coplanar ellipses given by the equations: E1 = (x, y) ∈ R2 : a22 x2 + a33 y 2 + 2a23 xy + 2a31 x + 2a32 y + a11 = 0 E2 = (x, y) ∈ R2 : b22 x2 + b33 y 2 + 2b23 xy + 2b31 x + 2b32 y + b11 = 0 We change the reference frame in order to have E1 centered at the origin and E2 centered at (x0 , y0 ) with axis parallel to the coordinate ones: (x cos (α) + y sin (α))2 (x sin (α) − y cos (α))2 E1 = (x, y) ∈ R2 : + =1 a b (x − x0 )2 (y − y0 )2 E2 = (x, y) ∈ R2 : + =1 c d F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 14. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Let A1 and A2 be the matrices associated to E1 and E2 . Characteristic polynomial of the pencil λA2 + A1 : H(λ) = det(λA2 + A1 ) = h3 λ3 + h2 λ2 + h1 λ + h0 Compute the discriminant of H(λ), and introduce the change of variable (x0 , y0 ) = (x0 t, y0 t). The equation which gives us the searched value of t, t0 , is S(t) = 0 where: S(t) = discλ H(λ) |(x0 ,y0 )=(x0 t,y0 t) = s4 t8 + s3 t6 + s2 t4 + s1 t2 + s0 Making T = t2 : S(T ) = s4 T 4 + s3 T 3 + s2 T 2 + s1 T + s0 Searched value of t: square root of the biggest real root of S(T ) F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 15. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Distance of closest approach of two separated ellipses Theorem Given two separated ellipses E1 and E2 the distance of their closest approach is given as d = t0 x 2 + y0 0 2 where t0 is the square root of the biggest positive real root of S(T ) = S(t) |T =t2 = (discλ H(λ) |(x0 ,y0 )=(x0 t,y0 t) ) |T =t2 , where H(λ) is the characteristic polynomial of the pencil determined by them and (x0 , y0 ) is the center of E2 . F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 16. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Example Let A and B be the ellipses: √ 2 7 2 3 5 A := (x, y) ∈ R : x + xy + y 2 = 10 8 4 8 1 2 3 1 8 109 B := (x, y) ∈ R2 : x − x + y2 − y = − 4 2 9 9 36 F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 17. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Example Polynomial whose biggest real root gives the square of the instant T = T0 when the ellipses are tangent: 466271425 √ √ B SA(T ) (T ) = + 9019725 3 T 4 + − 627564237 − 16904535 3 T 3 16 √ 32 √ 2 + 39363189 3 + 690647377 T 2 + − 1186083 16 256 16 3 − 58434963 T 128 + 4499761 256 B The two real roots of SA(T ) (T ) are: T0 = 0.5058481537; T1 = 0.07113873679 F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 18. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Positions of A and B(t) t0 = T0 t1 = T1 F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 19. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Let A1 and A2 be the matrices defining the separated ellipsoids E1 and E2 as X T A1 X = 0 and X T A2 X = 0 where X T = (x, y, z, 1), and A = (aij ), B = (bij ), i, j = 1..4 Change the reference frame to have E1 centered at the origin and E2 , at (x0 , y0 , z0 ) with axis parallel to the coordinate ones: P2 Q2 R2 E1 = (x, y, z) ∈ R3 : + 2 + 2 =1 a2 b c (x − x0 )2 (y − y0 )2 (z − z0 )2 E2 = (x, y, z) ∈ R3 : 2 + 2 + =1 d f g2 where P = x ux2 + 1 − ux 2 cos (α) + (ux uy (1 − cos (α)) − uz sin (α)) y + . . . Q = (ux uy (1 − cos (α)) + uz sin (α)) x + y uy 2 + 1 − uy 2 cos (α) + . . . R = (ux uz (1 − cos (α)) − uy sin (α)) x + (uyuz (1 − cos (α)) + ux sin (α)) y + . . . F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 20. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Characteristic polynomial of the pencil λA2 + A1 : H(λ) = det(λA2 + A1 ) = h4 λ4 + h3 λ3 + h2 λ2 + h1 λ + h0 Compute the discriminant of H(λ), and introduce the change of variable (x0 , y0 , z0 ) = (x0 t, y0 t, z0 t). The equation which gives us the searched value of t, t0 , is S(t) = 0 where: S(t) = discλ H(λ) |(x0 t,y0 t,z0 t) = s6 t12 +s5 t10 +s4 t8 +s3 t6 +s2 t4 +s1 t2 +s0 Making T = t2 : S(T ) = s6 t6 + s5 t5 + s4 T 4 + s3 T 3 + s2 T 2 + s1 T + s0 Searched value of t: square root of the biggest real root of S(T ) F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 21. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Distance of closest approach of two ellipsoids Theorem Given two separated ellipsoids E1 and E2 the distance of their closest approach is given as d = t0 x 2 + y0 + z0 0 2 2 where t0 is the square root of the biggest positive real root of S(T ) = S(t) |T =t2 = (discλ H(λ) |(x0 t,y0 t,z0 t) ) |T =t2 , where H(λ) is the characteristic polynomial of the pencil determined by them and (x0 , y0 , z0 ) is the center of E2 . F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 22. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Example Let A (blue) and B (green) be the ellipsoids: 1 2 1 2 A := (x, y, z) ∈ R3 : x + y + z2 = 1 4 2 1 2 1 1 51 B := (x, y, z) ∈ R3 : x − 2 x + y2 − 3 y + z2 − 5 z = − 5 4 2 2 F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 23. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Example Polynomial whose biggest real root gives the square of the instant T = T0 when the ellipsoids are tangent: SA(T ) (T ) = 16641 T 2 2725362025 T 4 − 339879840 T 3 + 3362446 T 2 − 11232 T + 9 B B The two real roots of SA(T ) (T ) are: T0 = 0.1142222397; T1 = 0.001153709353 F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 24. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Positions of A (blue) and B(t) (green) t0 = T0 t1 = T1 F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 25. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Ellipses case: Basic configuration: Compute the eigenvectors of a 2x2 matrix Compute the real roots of a 4-degree polynomial Other configurations: roots of a 8-degree polynomial Ellipsoids case: Basic configuration: Compute the eigenvectors of a 3x3 matrix Compute the real roots roots of a 6-degree polynomial Other configurations: roots of a 12-degree polynomial F. Etayo, L. González-Vega, G. R. Quintana EACA2010
- 26. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Conclusions Thank you! F. Etayo, L. González-Vega, G. R. Quintana EACA2010