Homework Sourseshow that the curvature of a plane curve is k dphids where p
show that the curvature of a plane curve is k= |d(phi)/ds|, where (phi) is the angle between T and i; that is, (phi) is the angle of inclination of the tangent line. Please explain.
Solution
change of parameter =(s) then, by the chain rule, dsdf=dfddsd=(s)dfd dsdg=(s)dgd transformation for the determinant: dfds d2fds2 dsdg ds2d2g =(s)3 dfd d2fd2 dgd d2gd2 dsdf2+dsdg2=(s)2dfd2+dgd2 Therefore, the following quantity is invariant under both rotation and reparameterization : dfd d2fd2 dgd d2gd2 dfd2+dgd232
