3
$\begingroup$

Image

An ellipse slides between two perpendicular lines. To which family does the locus of the centre of the ellipse belong to?

  • 0
    By this, do you mean that the ellipse is rotated and translated continuously in the plane, so that at all times each of the two perpendicular lines is tangent to the ellipse, until the ellipse traverses through all possible such positions?2012-08-01

1 Answers 1

4

Given the parametric equation of a rotated ellipse $$ x(t)=x_0+a\cos\theta\cos{t}-b\sin\theta\sin{t}\\ y(t)=y_0+b\cos\theta\sin{t}+a\sin\theta\cos{t} $$ the conditions $\dot{x}(t)=x(t)=0$ for the contact point to the vertical line give $$ x_0=\sqrt{a^2\cos^2\theta+b^2\sin^2\theta} $$ and from $\dot{y}(t)=y(t)=0$ $$ y_0=\sqrt{a^2\sin^2\theta+b^2\cos^2\theta} $$ Here is an animated graphics.

enter image description here

  • 3
    +1, very nice! I didn't think the solution would be that simple. It's worth mentioning, though, that these points lie on the circle $x_0^2+y_0^2=a^2+b^2$ centred at the origin.2012-08-01
  • 1
    @joriki: Really interesting observation that I missed.2012-08-01
  • 2
    In short, the circle is a *[point-glissette](http://mathworld.wolfram.com/Glissette.html)* of the ellipse. See also [this animation](http://www.mathcurve.com/courbes2d/glissette/glissetteellipse.gif) from [here](http://www.mathcurve.com/courbes2d/glissette/glissette.shtml).2012-08-02
  • 0
    @J.M.... that web page... it's hypnotizing... *\*swoons\**2012-08-02
  • 0
    @Rahul, shoulda put in a warning, I guess...2012-08-02
  • 0
    i think that the solution is wrong because the locus stated above is not of the path traced by center of ellipse whereas it is the locus of the director circle (a circle formed by the points where two perpendicular tangent lines to the curve intersect) therefore it is not correct according to me2016-01-31