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CIEM 07

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

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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 2007



Fernando Etayo, L. González-Vega, Gema R. Quintana
Problem
                                       Our approach
                                            Example
                                        Future work


Contents


  1    Problem

  2    Our approach

  3    Example

  4    Future work




  Fernando Etayo, L. González-Vega, Gema R. Quintana
Problem
                                      Our approach
                                           Example
                                       Future work


Problem

  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
Problem
                                       Our approach
                                            Example
                                        Future work


Applications




         Collision detection
         Orbit analysis (non-coplanar ellipses)




  Fernando Etayo, L. González-Vega, Gema R. Quintana
Problem
                                       Our approach
                                            Example
                                        Future work


Previous 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
Problem
                                      Our approach
                                           Example
                                       Future work


Our 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

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CIEM 07

  • 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 2007 Fernando Etayo, L. González-Vega, Gema R. Quintana
  • 2. Problem Our approach Example Future work Contents 1 Problem 2 Our approach 3 Example 4 Future work Fernando Etayo, L. González-Vega, Gema R. Quintana
  • 3. Problem Our approach Example Future work Problem 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. Problem Our approach Example Future work Applications Collision detection Orbit analysis (non-coplanar ellipses) Fernando Etayo, L. González-Vega, Gema R. Quintana
  • 5. Problem Our approach Example Future work Previous 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. Problem Our approach Example Future work Our 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. Problem Our approach Example Future work We consider the parametric equations of an ellipse √ √ xe = xc + a cos t, ye = yc + b sin t in order to construct a function fd which gives the distance between a point (x0 , y0 ) and the ellipse: √ √ fd := (x0 − a cos s)2 + (y0 − b sin s)2 − d To simplify the expression of the function we use the following substitution: 1 1 z−z z+z sin t = , cos t = 2i 2 and then using resultants we eliminate the variable z. Fernando Etayo, L. González-Vega, Gema R. Quintana
  • 8. Problem Our approach Example Future work Fernando Etayo, L. González-Vega, Gema R. Quintana
  • 9. Problem Our approach Example Future work To finish, we make the point (x0 , y0 ) to belong to the other ellipse and continue like we did before. We obtain a polynomial only in the variable d, Pd . The minimum distance is given by the minimum real root of Pd . Fernando Etayo, L. González-Vega, Gema R. Quintana
  • 10. Problem Our approach Example Future work Example 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. Problem Our approach Example Future work In this case the minimum distance is given by computing the real roots of a polynomial Pd of degree 60 which factorizes in polynomials of a degree of at most 12: two double factors of degree 2, a triple factor of degree 12 and a simple factor of degree 12; all of them multiplied by d4 . Evaluating Pd in t = 5 we obtain that the distance is 0, as we expected. Fernando Etayo, L. González-Vega, Gema R. Quintana
  • 12. Problem Our approach Example Future work Future work Continue studying the continuous motion case. Generalize to ellipsoids. Non-coplanar ellipses. Fernando Etayo, L. González-Vega, Gema R. Quintana
  • 13. Problem Our approach Example Future work Thank you! Fernando Etayo, L. González-Vega, Gema R. Quintana