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# CGTA09

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My talk at http://www.kma.zcu.cz/cgta2009/ in Pilsen, 2009

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### CGTA09

1. 1. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workComputing the distance of closest approach between ellipses and ellipsoids L. Gonzalez-Vega, G. R. Quintana Departamento de MATemáticas, EStadística y COmputación University of Cantabria, Spain Conference on Geometry: Theory and Applications Dedicated to the memory of Prof. Josef Hoschek Pilsen, Czech Republic, June 29 - July 2, 2009 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
2. 2. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workContents 1 Problem 2 Distance of closest approach of two ellipses 3 Distance of closest approach of two ellipsoids 4 Future work L. Gonzalez-Vega, G. R. Quintana CGTA 2009
3. 3. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workIntroduction 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. L. Gonzalez-Vega, G. R. Quintana CGTA 2009
4. 4. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workIntroduction 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. It 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. L. Gonzalez-Vega, G. R. Quintana CGTA 2009
5. 5. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workPrevious work A description of a method for solving the problem in the case of two arbitrary hard ellipses can be found in X. Z HENG , P. PALFFY-M UHORAY, Distance of closest approach of two arbitrary hard ellipses in two dimensions, Physical Review, E 75, 061709,2007. An analytic expression for that distance is given as a function of their orientation relative to the line joining their centers. L. Gonzalez-Vega, G. R. Quintana CGTA 2009
6. 6. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workPrevious work Steps of the previous approach: 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. L. Gonzalez-Vega, G. R. Quintana CGTA 2009
7. 7. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workPrevious work Steps of the previous approach: 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. L. Gonzalez-Vega, G. R. Quintana CGTA 2009
8. 8. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workPrevious 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 avoid that computation. L. Gonzalez-Vega, G. R. Quintana CGTA 2009
9. 9. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workOur approach We use the results shown in: F. E TAYO, L. G ONZÁLEZ -V EGA , N. DEL R ÍO, 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. L. Gonzalez-Vega, G. R. Quintana CGTA 2009
10. 10. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workOur approach Following their notation we deﬁne the characteristic polynomial of the pencil determined by two ellipses(resp. ellipsoids) 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 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
11. 11. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workOur approach 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. Results about the intersection of two ellipsoids: a complete characterization, in terms of the sign of the real roots of the characteristic polynomial, of the separation case. L. Gonzalez-Vega, G. R. Quintana CGTA 2009
12. 12. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workOur approach More precisely: Two ellipsoids are separated if and only if their characteristic polynomial has two distinct positive roots. The characteristic equation always has at least two negative roots. The ellipsoids touch each other externally if and only if the characteristic equation has a positive double root. L. Gonzalez-Vega, G. R. Quintana CGTA 2009
13. 13. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workOur approach F. E TAYO, L. G ONZÁLEZ -V EGA , N. DEL R ÍO, A new approach to characterizing the relative position of two ellipses depending on one parameter, Computed Aided Geometric Desing 23, 324-350, 2006. An equivalent characterization is given for the case of two coplanar ellipses. In fact the ten relative positions of two ellipses are characterized by using several tools coming from Real Algebraic Geometry, Computer Algebra and Projective Geometry (Sturm-Habicht sequences and the classiﬁcation of pencils of conics in P2 (R)). Each one is determined by a set of equalities and inequalities depending only on the matrices of the conics. L. Gonzalez-Vega, G. R. Quintana CGTA 2009
14. 14. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workOur 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 smallest real root provides the distance of closest approach. We will see that it extends in a natural way to the case of two ellipsoids. L. Gonzalez-Vega, G. R. Quintana CGTA 2009
15. 15. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workWe consider the two coplanar ellipses given by the equations: x2 y2 E1 = (x, y) ∈ R2 : + −1=0 a bE2 = (x, y) ∈ R2 : a11 x2 + a22 y 2 + 2a12 xy + 2a13 x + 2a23 y + a33 = 0 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
16. 16. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workConﬁguration of the ellipses L. Gonzalez-Vega, G. R. Quintana CGTA 2009
17. 17. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workEquation of a moving ellipse E1 (t) along the line deﬁned by thecenters: (x − pt)2 (y − qt)2 E1 (t) = (x, y) ∈ R2 : + −1=0 a bwhere a22 a13 − a12 a23 p= a2 − a11 a22 12 a11 a23 − a12 a13 q= a2 − a11 a22 12 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
18. 18. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workThe characteristic polynomial of the pencil λA2 + A1 (t):H(t; λ) = det(λA2 + A1 (t)) = h3 (t)λ3 + h2 (t)λ2 + h1 (t)λ + h0 (t)External tangent situation is produced when H(t; λ) has adouble positive root: the equation which gives us the searchedvalue of t, t0 , is S(t) = 0 whereS(t) = discλ H(t; λ) = s8 t8 +s7 t7 +s6 t6 +s5 t5 +s4 t4 +s3 t6 +s2 t4 +s1 t2 +s0 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
19. 19. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workDistance 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 p2 + q 2 where t0 is the smallest positive real root of S(t) = discλ H(t; λ), H(t; λ) is the characteristic polynomial of the pencil determined by them and (p, q) is the center of E2 . L. Gonzalez-Vega, G. R. Quintana CGTA 2009
20. 20. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workExample Let A and B be the ellipses: 1 A := (x, y) ∈ R2 : x2 + y 2 − 1 = 0 2 B := (x, y) ∈ R2 : 9x2 + 4y 2 − 54x − 32y + 109 = 0 1 A centered at the origin and semi-axes of length 1 and √ . 2 B centered at (3, 4) with semi-axes of length 2 and 3. L. Gonzalez-Vega, G. R. Quintana CGTA 2009
21. 21. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workPosition of the ellipses A (blue) and B (green) L. Gonzalez-Vega, G. R. Quintana CGTA 2009
22. 22. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workExample We make the center of the ﬁrst one to move along the line determined by the centers. (y − 4t)2 A(t) := (x, y) ∈ R2 : (x − 3t)2 + −1=0 2 Characteristic polynomial of the pencil λB + A(t): B 17 17 5 HA(t) (t; λ) = λ3 + − 36 t2 + 18 t − 24 λ2 + 23 145 2 145 1 − 648 − 2592 t + 1296 t λ + 2592 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
23. 23. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workExample Polynomial whose smallest real root gives the instant t = t0 when the ellipses are tangent: 251243 115599091 1478946641 SA(t) (t) = − 80621568 t + 8707129344 t2 + 34828517376 t4 − B 266704681 3 55471163 6 158971867 5 8707129344 t + 2902376448 t − 4353564672 t + 6076225 8 6076225 7 40111 8707129344 t − 1088391168 t + 136048896 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
24. 24. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workExample Polynomial whose smallest real root gives the instant t = t0 when the ellipses are tangent: 251243 115599091 1478946641 SA(t) (t) = − 80621568 t + 8707129344 t2 + 34828517376 t4 − B 266704681 3 55471163 6 158971867 5 8707129344 t + 2902376448 t − 4353564672 t + 6076225 8 − 6076225 t7 + 40111 8707129344 t 1088391168 136048896 B The four real roots of SA(t) (t) are: t0 = 0.2589113100, t1 = 0.7450597195, t2 = 1.254940281, t3 = 1.741088690 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
25. 25. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workPositions of A(t) (blue) and B (green) t = t0 t = t1 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
26. 26. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workPositions of A(t) (blue) and B (green) t = t2 t = t3 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
27. 27. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workLet A1 and A2 be the symmetric deﬁnite positive matrices deﬁningthe separated ellipsoids E1 and E2 as X T A1 X = 0 and X T A2 X = 0where X T = (x, y, z, 1), and 1     a 0 0 0 a11 a12 a13 a14 1  a12  0 b 0 0  a22 a23 a24  A1 =  1  A2 =   a13   0 0 c 0  a23 a33 a34  0 0 0 −1 a14 a24 a34 a44i.e., x2 y2 z2 E1 = (x, y) ∈ R2 : + + −1=0 a b c a11 x2 + a22 y 2 + a33 z 2 + 2a12 xy + 2a13 xz+E2 = (x, y) ∈ R2 : 2a23 yz + 2a14 x + 2a24 y + 2a34 z + a44 = 0 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
28. 28. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workConﬁguration of the two ellipsoids L. Gonzalez-Vega, G. R. Quintana CGTA 2009
29. 29. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workCharacteristic polynomial (x − txc )2 (y − tyc )2 (z − tzc )2 E1 (t) = (x, y) ∈ R2 : + + −1=0 a b c In order to ﬁnd the value of t, t0 , for which the ellipsoids are externally tangent we have to to check if the polynomial H(t; λ) = det(E1 (t) + λE2 ), which has degree four, has a double real root. That is, ﬁnd the roots of the polynomial of degree 12: S(t) = discλ (H(t, λ)) = s12 t12 + ... + s0 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
30. 30. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workDistance 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 + yc + zc c 2 2 where t0 is the smallest positive real root of S(t) = discλ H(t; λ), H(t; λ) is the characteristic polynomial of the pencil determined by them, and (xc , yc , zc ) is the center of E2 . L. Gonzalez-Vega, G. R. Quintana CGTA 2009
31. 31. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workExample Let E1 (t) and E2 be the two ellipsoids given as follows: 1 2 1 2 E1 := (x, y, z) ∈ R3 : x + y + z2 − 1 = 0 4 2 1 2 1 51 1 E2 := (x, y, z) ∈ R3 : x − 2 x + y2 − 3 y + + z2 − 5 z = 0 5 4 2 2 1 2 1 2 5 197 2 E1 (t) := (x, y, z) ∈ R3 : x + y + z 2 − tx − 6 ty − 10 tz − 1 + t =0 4 2 2 4 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
32. 32. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workConﬁguration of the two ellipsoids E1 (blue)and E2(green) L. Gonzalez-Vega, G. R. Quintana CGTA 2009
33. 33. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workExample Characteristic polynomial of E2 and E1 (t): E2 HE1 (t) (t; λ) = λ4 − 43 λ3 − 197 λ3 t2 − 301 λ2 − 659 λ2 t2 + 4 2 4 197 2 λ3 t− 237 2 λ − 265 λ t2 + 659 λ2 t + 5 + 265 λ t 2 2 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
34. 34. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workExample E2 Polynomial SE1 (t) (t) whose its smallest real root corresponds to the instant t = t0 when the ellipsoids are tangent: E2 SE1 (t) (t) = 16641 1024 − 1)4 (2725362025t8 − 21802896200t7 + 75970256860t6 − (t 150580994360t5 + 185680506596t4 − 145836126384t3 + 71232102544t2 − 19777044480t + 2388833408) E2 The four real roots of SE1 (t) (t) that determine the four tangency points are all provided by the factor of degree 8: t0 = 0.6620321914, t1 = 0.6620321914 t2 = 1.033966297, t3 = 1.337967809 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
35. 35. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workPositions of E1 (blue) and E2 (green) t = t0 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
36. 36. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workPositions of E1 (blue) and E2 (green) t = t1 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
37. 37. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workPositions of E1 (blue) and E2 (green) t = t2 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
38. 38. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workPositions of E1 (blue) and E2 (green) t = t3 L. Gonzalez-Vega, G. R. Quintana CGTA 2009
39. 39. Problem Distance of closest approach of two ellipses Distance of closest approach of two ellipsoids Future workSome geometric conﬁgurationsof the quadrics or conics we arestudying seem to be related with specially simpledecompositions of the polynomials involved in the calculus ofthe minimum distance between them or of the closest approachof them.We are working in the algebraic-geometric interpretation of thissituation. L. Gonzalez-Vega, G. R. Quintana CGTA 2009