Determining the "length of an irregular arc segment" is also called rectification of a curve. Historically, many methods were used for specific curves. The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases. A curve in the plane can be approximated by connecting a finite number of points on the curve using line segments to create a polygonal path. Since it is straightforward to calculate the length of each linear segment , the total length of the approximation can be found by summing the lengths of each linear segment; that approximation is known as the " chordal distance".
If the curve is not already a polygonal path, better approximations to the curve can be obtained by following the shape of the curve increasingly more closely. The approach is to use an increasingly larger number of segments of smaller lengths. The lengths of the successive approximations do not decrease and will eventually keep increasing—possibly indefinitely, but for smooth curves this will tend to a limit as the lengths of the segments get arbitrarily small.
For some curves there is a smallest number "L" that is an upper bound on the length of any polygonal approximation. If such a number exists, then the curve is said to be "rectifiable" and the curve is defined to have "arc length" "L".
Let "C" be a curve in Euclidean space "X" = "R" "n" , so "C" is the image of a continuous function "f" : → "X" of the interval into "X".
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