An application of the maximum principle to the geometry of plane curves

The maximum principle of control theory is used to find necessary and sufficient conditions for a plane curve, which has bounded piecewise continuous curvature and prescribed initial and terminal points and directions, to have minimal length. This result is used to prove that such a closed curve having length L and curvature k satisfying Iki IK can be contained in a circle of radius R, where R O, (xo, yo, uo) and (x1, Yl, u1) be given. Then among the curves (x(s),y(s)), Oline segments and arcs of circles of radius 1/K. Received by the editors March 22, 1973 and, in revised form, September 10, 1973. AMS (MOS) subject classifications (1970). Primary 53A05, 52A40, 49B10.