CIEM 07

ProblemOur approach

ExampleFuture work

Computing the distance between two ellipsesin the same plane.

Fernando Etayo, L. González-Vega, Gema R. Quintana

Universidad de Cantabria

Workshop on Computer Algebra in Geometric Modeling andIndustry, CIEM 2007


ProblemOur approach

ExampleFuture work

Contents

1 Problem

2 Our approach

3 Example

4 Future work


ProblemOur approach

ExampleFuture work

Problem

Computing the minimum distance between two coplanarellipses without computing the foot points.

The distance between two separated ellipses is an algebraicnumber: our goal is to determine the polynomial with theminimum distance between the given two ellipses as a real root.


ProblemOur approach

ExampleFuture work

Applications

Collision detectionOrbit analysis (non-coplanar ellipses)


ProblemOur approach

ExampleFuture work

Previous work

Efficient Distance Computation for Quadratic Curves andSurfaces.C. LENNERZ, E. SCHÖMER.Computing the Distance Between Two Surfaces via LineGeometry.K.A. SOHN, B. JÜTTLER, M.S. KIM, W. WANG.Minimum Distance Between Two Sphere-swept Surfaces.K. LEE, J.K. SEONG, K.J. KIM, S.J. HONG.

The common aspect in all these works is that the problem isalways solved using foot points.


ProblemOur approach

ExampleFuture work

Our approach

We do not want to make the minimum distance computationdepending on the foot points since our goal is to study theellipse separation problem when they move by analyzing theunivariate polynomial providing by the distance.We consider the following cases:

static case:parallel axesnon-parallel axes

continuous motion caseThe ellipses are supposed to be given in a non concrete way:center coordinates, axes length, etc. are parameters to ourproblem.


ProblemOur approach

ExampleFuture 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 distancebetween 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 followingsubstitution:

sin t =z − 1

z

2i, cos t =

z + 1z

2and then using resultants we eliminate the variable z.


ProblemOur approach

ExampleFuture work


ProblemOur approach

ExampleFuture work

To 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.


ProblemOur approach

ExampleFuture work

Example

Let E1 and E2 be two ellipses in R2. E1 with center (0, 0) andsemi-axes of length 3 and 2. E2 centered in (7, 5) and withsemi-axes 4 and 2. E1 is moving along the y-axis. When t = 5the two ellipses intersect.


ProblemOur approach

ExampleFuture work

In 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.


ProblemOur approach

ExampleFuture work

Future work

Continue studying the continuous motion case.Generalize to ellipsoids.Non-coplanar ellipses.


ProblemOur approach

ExampleFuture work

Thank you!


CIEM 07

Education

Transcript of CIEM 07