In the calculus of plane curves, one learns that the curvature k of the curve y = y(x) at...

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 [Edwards and Penney.Calculus: Early Transcendentals, 7th edition (Upper Saddle River, NJ: Prentice Hall, 2008)] Conversely, substitute ρ = y′ to derive a general solution of the second-order differential equation

ry″ = [1 + (y′)2]3/2

(with r constant) in the form

(x − a)2 + (y − b)2 = r2.

Thus a circle of radius r (or a part thereof) is theonly plan curve with constant curvature 1/r.