A curve is given parametrically by r(t) = f(t) i + g(t) j. Show that, if s is the length of an arc measured from a fixed point P₀ on the curve so that s increases as t increases, then

Deduce that dr/ds is a unit tangent vector to the curve at r(t) and that dr/ds and d²r/ds² are perpendicular. Show that

where κ is the curvature of the curve at that point.

dt sp P