In the calculus of plane curves, one learns that the curvature k of the curve y = y(x) at the point (x, y) is given by

and that the curvature of a circle of radius r is k = 1/r.

Conversely, substitute ρ = y' to derive a general solution of the second-order differential equation

(with r constant) in the form

Thus a circle of radius r (or a part thereof) is the only plane curve with constant curvature 1/r.

Transcribed Image Text:

K= |y" (x)] [1 + y'(x)2]3/2¹