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        riddles >> putnam exam (pure math) >> ELLIPSE
        
(Message started by: Pietro K.C. on Oct 16^th, 2002, 9:34am)

Title: ELLIPSE
Post by Pietro K.C. on Oct 16^th, 2002, 9:34am

This one made me say, "ooh, pretty!" :)

An ellipse, whose semi-axes have lengths a and b, rolls without slipping on the curve y = c*sin(x/a). How are a,b,c related, given that the ellipse completes one revolution when it traverses one period of the curve?

Note: Taken from one of the actual Putnam Exams.

Title: Re: ELLIPSE
Post by Eigenray on Dec 10^th, 2005, 4:14pm

[hideb]Parameterize the ellipse by
(x, y) = (b cos t, a sin t).
Its total circumference is
L = [int] (dx² + dy²)^1/2
= [int]₀^2pi sqrt( b²sin² t + a²cos²t ) dt.
Similarly, if we parametrize the curve by
(x, y) = (at, c sin t),
we get the length along one period is
L = [int]₀^2pi sqrt(a² + c²sin² t) dt
= [int] sqrt(a²cos²t + (a²+c²)sin² t) dt.
Since this is monotonic in c, it follows we must have
b² = a² + c².[/hideb]

Well, that's necessary for the lengths to work out right. But if the ellipse is physically rolling around on top of the curve, it needs to actually fit. What are the conditions for this?

Title: Re: ELLIPSE
Post by Grimbal on Dec 11^th, 2005, 11:54am

[hide]Pythagoras?[/hide]