Paths, Arc Length

Arc Length

Let p(t) be a continuous function mapping a closed interval into n space. Let x be a Riemann net on the interval. Define a new type of Riemann sum for this net. Map the points of the net onto the path and measure the distances between successive points. The tiny line segments hug the curve, and the sum of their distances approximates the length of the path. If l is the least upper bound of the sum of the lengths of the segments, over all possible nets, then l is the path length.

Paths that have an upper bound, and hence a length, are called rectifyable. Not all paths are rectifyable. Consider the graph of y = x×cos(1/πx) as x runs from 0 to 1. Let y(0) = 0, so the path is continuous. Start with a net that has only the point 1. Then add 1/2, 1/3, 1/4, and so on, creating a series of nets. Each new point introduces another segment. The segment joining the nth point to its predecessor is at least 2/n in length. We are adding the reciprocols of the integers forever. This is the harmonic series, and it rises to infinity. Thus our continuous path has infinite length. Obviously this is pathological; most paths are well behaved.

If a path on [a,b] has length r and a path on [b,c] has length s, the combined path on [a,c] has length r+s. Given any partition on [a,c], add the point b, which can only increase the approximation by the triangular inequality. This is the sum of approximations over [a,b] and [b,c], so r+s remains an upper bound. At the same time, we can always find two subpartitions within ε/2 of r and s respectively, so r+s is the arc length.

If [a,c] is rectifyable, we can select any cutpoint b. Approximations on [a,b] and [b,c] are bounded, so both are rectifyable, and the sum of their lengths must be the total length.