On Monday, March 4, 2013 8:46:22 AM UTC+5:30, Brad Cooper wrote: > The "well behaved, smooth" function f(x) has endpoints f(0) = f(h) = 0. The > curve of the function has length s1. > An arc of a circle passing through (0, 0) and (0, h) has fixed curvature k > and its arc length is also s1. > It is required to show that a point must exist on f(x) where curvature is > also k.
Imagine an inextensible thread between two fixed pegs distance h apart. Plot arc length versus curvature for a circle and for any other arbitrarily displaced loose thread curve between the two pegged points (0,0) and (h,0) where s1 > h.
(If tensioned, it becomes like the thread of total length 2a between two foci 2c apart for drawing the ellipse by that property).
By mean value theorem there must be at least one intersection between the curve and its mean value.
The number of intersections can be arbitrarily large. HTH.