We define the curvature of a path by ||r(s)||, where r(s) is the arc-length parametrization of...

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We define the curvature of a path by ||r"(s)||, where r(s) is the arc-length parametrization of the path. Given a path r(t), we let r(s) be its arc-length parametrization so thats = S,lr'(t)||dt. (a) Show that r' (t) x r"(1) = (r(s) x r"(s). (b) Hence, or otherwise, show that the curvature can be expressed in terms of t. Give the explicit form of the curvature function. We define the curvature of a path by ||r"(s)||, where r(s) is the arc-length parametrization of the path. Given a path r(t), we let r(s) be its arc-length parametrization so thats = S,lr'(t)||dt. (a) Show that r' (t) x r"(1) = (r(s) x r"(s). (b) Hence, or otherwise, show that the curvature can be expressed in terms of t. Give the explicit form of the curvature function.