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CIEM 07

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First talk at the seminar http://www.ciem.unican.es/encuentros/wcagmi/
when I was undergraduated.

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CIEM 07

  1. 1. Problem Our approach Example Future work Computing the distance between two ellipses in the same plane. Fernando Etayo, L. González-Vega, Gema R. Quintana Universidad de Cantabria Workshop on Computer Algebra in Geometric Modeling and Industry, CIEM 2007Fernando Etayo, L. González-Vega, Gema R. Quintana
  2. 2. Problem Our approach Example Future workContents 1 Problem 2 Our approach 3 Example 4 Future work Fernando Etayo, L. González-Vega, Gema R. Quintana
  3. 3. Problem Our approach Example Future workProblem Computing the minimum distance between two coplanar ellipses without computing the foot points. The distance between two separated ellipses is an algebraic number: our goal is to determine the polynomial with the minimum distance between the given two ellipses as a real root. Fernando Etayo, L. González-Vega, Gema R. Quintana
  4. 4. Problem Our approach Example Future workApplications Collision detection Orbit analysis (non-coplanar ellipses) Fernando Etayo, L. González-Vega, Gema R. Quintana
  5. 5. Problem Our approach Example Future workPrevious work Efficient Distance Computation for Quadratic Curves and Surfaces. C. L ENNERZ , E. S CHÖMER . Computing the Distance Between Two Surfaces via Line Geometry. K.A. S OHN , B. J ÜTTLER , M.S. K IM , W. WANG . Minimum Distance Between Two Sphere-swept Surfaces. K. L EE , J.K. S EONG , K.J. K IM , S.J. H ONG . The common aspect in all these works is that the problem is always solved using foot points. Fernando Etayo, L. González-Vega, Gema R. Quintana
  6. 6. Problem Our approach Example Future workOur approach We do not want to make the minimum distance computation depending on the foot points since our goal is to study the ellipse separation problem when they move by analyzing the univariate polynomial providing by the distance. We consider the following cases: static case: parallel axes non-parallel axes continuous motion case The ellipses are supposed to be given in a non concrete way: center coordinates, axes length, etc. are parameters to our problem. Fernando Etayo, L. González-Vega, Gema R. Quintana
  7. 7. Problem Our approach Example Future workWe consider the parametric equations of an ellipse √ √ xe = xc + a cos t, ye = yc + b sin tin order to construct a function fd which gives the distancebetween a point (x0 , y0 ) and the ellipse: √ √ fd := (x0 − a cos s)2 + (y0 − b sin s)2 − dTo simplify the expression of the function we use the followingsubstitution: 1 1 z−z z+z sin t = , cos t = 2i 2and then using resultants we eliminate the variable z.Fernando Etayo, L. González-Vega, Gema R. Quintana
  8. 8. Problem Our approach Example Future workFernando Etayo, L. González-Vega, Gema R. Quintana
  9. 9. Problem Our approach Example Future workTo finish, we make the point (x0 , y0 ) to belong to the otherellipse and continue like we did before. We obtain a polynomialonly in the variable d, Pd . The minimum distance is given by theminimum real root of Pd .Fernando Etayo, L. González-Vega, Gema R. Quintana
  10. 10. Problem Our approach Example Future workExample Let E1 and E2 be two ellipses in R2 . E1 with center (0, 0) and semi-axes of length 3 and 2. E2 centered in (7, 5) and with semi-axes 4 and 2. E1 is moving along the y-axis. When t = 5 the two ellipses intersect. Fernando Etayo, L. González-Vega, Gema R. Quintana
  11. 11. Problem Our approach Example Future workIn this case the minimum distance is given by computing thereal roots of a polynomial Pd of degree 60 which factorizes inpolynomials of a degree of at most 12: two double factors ofdegree 2, a triple factor of degree 12 and a simple factor ofdegree 12; all of them multiplied by d4 .Evaluating Pd in t = 5 we obtain that the distance is 0, as weexpected.Fernando Etayo, L. González-Vega, Gema R. Quintana
  12. 12. Problem Our approach Example Future workFuture work Continue studying the continuous motion case. Generalize to ellipsoids. Non-coplanar ellipses. Fernando Etayo, L. González-Vega, Gema R. Quintana
  13. 13. Problem Our approach Example Future work Thank you!Fernando Etayo, L. González-Vega, Gema R. Quintana

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