This document discusses computing the minimum distance between two coplanar ellipses without computing foot points. It presents an approach using resultants to eliminate variables and obtain a polynomial whose minimum real root gives the distance. An example computes the distance between two moving ellipses. Future work includes further studying continuous motion, generalizing to ellipsoids, and handling non-coplanar ellipses.

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