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Topic: ELLIPSE (Read 1146 times) |
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Pietro K.C.
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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.
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| « Last Edit: Dec 10th, 2002, 8:37am by Pietro K.C. » |
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"I always wondered about the meaning of life. So I looked it up in the dictionary under 'L' and there it was --- the meaning of life. It was not what I expected." (Dogbert)
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Eigenray
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Re: ELLIPSE
« Reply #1 on: Dec 10th, 2005, 4:14pm » |
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| hidden: | Parameterize the ellipse by
(x, y) = (b cos t, a sin t).
Its total circumference is
L = [int] (dx2 + dy2)1/2
= [int]02pi 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]02pi 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. |
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?
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Grimbal
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Re: ELLIPSE
« Reply #2 on: Dec 11th, 2005, 11:54am » |
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Pythagoras?
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