Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 3 down vote favorite
1

Image

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

share|improve this question
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? – Cameron Buie Aug 1 '12 at 19:56

1 Answer

up vote 4 down vote accepted

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

share|improve this answer
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. – joriki Aug 1 '12 at 21:13
@joriki: Really interesting observation that I missed. – enzotib Aug 1 '12 at 21:32
2  
In short, the circle is a point-glissette of the ellipse. See also this animation from here. – J. M. Aug 2 '12 at 4:46
@J.M.... that web page... it's hypnotizing... *swoons* – Rahul Narain Aug 2 '12 at 8:30
@Rahul, shoulda put in a warning, I guess... – J. M. Aug 2 '12 at 11:53

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.