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Arc length is the distance between two points along a section of a curve.

Determining the length of an irregular arc segment is also called rectification of a curve. The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases.

General approach

Approximation by multiple linear segments

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 (using the Pythagorean theorem in Euclidean space, for example), the total length of the approximation can be found by summing the lengths of each linear segment; that approximation is known as the (cumulative) chordal distance.[1]

If the curve is not already a polygonal path, using a progressively larger number of segments of smaller lengths will result in better approximations. The lengths of the successive approximations will not decrease and may keep increasing indefinitely, but for smooth curves they will tend to a finite limit as the lengths of the segments get arbitrarily small.

For some curves there is a smallest number that is an upper bound on the length of any polygonal approximation. These curves are called rectifiable and the number

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 (using the Pythagorean theorem in Euclidean space, for example), the total length of the approximation can be found by summing the lengths of each linear segment; that approximation is known as the (cumulative) chordal distance.[1]

If the curve is not already a polygonal path, using a progressively larger number of segments of smaller lengths will result in better approximations. The lengths of the successive approximations will not decrease and may keep increasing indefinitely, but for smooth curves they will tend to a finite limit as the lengths of the segments get arbitrarily small.

For some curves there is a smallest number that is an upper bound on the length of any polygonal approximation. These curves are called rectifiable and the number is defined as the arc length.

Definition for a smooth curve

Let be a continuously differentiable function. The length of the curve defined by can be defined as the arbitrarily small.

For some curves there is a smallest number that is an upper bound on the length of any polygonal approximation. These curves are called rectifiable and the number is defined as the arc length.

Let be a continuously differentiable function. The length of the curve defined by can be defined as the limit of the sum of line segment lengths for a regular partition of as the number of segments approaches infinity. This means