ABSTRACT
Since x(t + At) = x(t) + x.(t) A t H---- , the vector x spans the direction of the tangent of x at t. Moreover, the length of x( t )A t is an approximation of the length of the curve segment from t to t -I-At; |x(£)| dt represents the arc element of the curve and
is the arc length of the curve segment x[a, to]. The arc length is indepen dent of the parametrization and is itself the natural Euclidean parametriza tion of the curve. Namely on reparametrization by t = t(s) and with x = x(t(s)) as a function of s one has
where the prime denotes differentiation with respect to the arc length s.
