Arc length

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

Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is also called curve rectification. A rectifiable curve has a finite number of segments in its rectification (so the curve has a finite length).

If a curve can be parameterized as an injective and continuously differentiable function (i.e., the derivative is a continuous function) {\displaystyle f\colon [a,b]\to \mathbb {R} ^{n}}{\displaystyle f\colon [a,b]\to \mathbb {R} ^{n}}, then the curve is rectifiable (i.e., it has a finite length).

The advent of infinitesimal calculus led to a general formula that provides closed-form solutions in some cases.

1 General approach

A curve in the plane can be approximated by connecting a finite number of points on the curve using (straight) 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 summation of 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, then using a progressively larger number of line segments of smaller lengths will result in better curve length approximations. Such a curve length determination by approximating the curve as connected (straight) line segments is called rectification of a curve. 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 {\displaystyle L}L that is an upper bound on the length of all polygonal approximations (rectification). These curves are called rectifiable and the arc length is defined as the number {\displaystyle L}L.

A signed arc length can be defined to convey a sense of orientation or “direction” with respect to a reference point taken as origin in the curve (see also: curve orientation and signed distance).[2]

Approximation to a curve by multiple linear segments, called rectification of a curve.

2 Formula for a smooth curve

See also: Curve § Length of a curve
Let {\displaystyle f\colon [a,b]\to \mathbb {R} ^{n}}{\displaystyle f\colon [a,b]\to \mathbb {R} ^{n}} be an injective and continuously differentiable (i.e., the derivative is a continuous function) function. The length of the curve defined by {\displaystyle f}f can be defined as the limit of the sum of linear segment lengths for a regular partition of {\displaystyle [a,b]}[a,b] as the number of segments approaches infinity. This means

{\displaystyle L(f)=\lim {N\to \infty }\sum {i=1}^{N}{\bigg |}f(t{i})-f(t{i-1}){\bigg |}}{\displaystyle L(f)=\lim {N\to \infty }\sum {i=1}^{N}{\bigg |}f(t{i})-f(t{i-1}){\bigg |}}
where {\displaystyle t_{i}=a+i(b-a)/N=a+i\Delta t}{\displaystyle t_{i}=a+i(b-a)/N=a+i\Delta t} with {\displaystyle \Delta t={\frac {b-a}{N}}=t_{i}-t_{i-1}}{\displaystyle \Delta t={\frac {b-a}{N}}=t_{i}-t_{i-1}} for {\displaystyle i=0,1,\dotsc ,N.}{\displaystyle i=0,1,\dotsc ,N.} This definition is equivalent to the standard definition of arc length as an integral:
{\displaystyle L(f)=\lim {N\to \infty }\sum {i=1}^{N}{\bigg |}f(t{i})-f(t{i-1}){\bigg |}=\lim {N\to \infty }\sum {i=1}^{N}\left|{\frac {f(t{i})-f(t{i-1})}{\Delta t}}\right|\Delta t=\int _{a}^{b}{\Big |}f’(t){\Big |}\ dt.}{\displaystyle L(f)=\lim {N\to \infty }\sum {i=1}^{N}{\bigg |}f(t{i})-f(t{i-1}){\bigg |}=\lim {N\to \infty }\sum {i=1}^{N}\left|{\frac {f(t{i})-f(t{i-1})}{\Delta t}}\right|\Delta t=\int _{a}^{b}{\Big |}f’(t){\Big |}\ dt.}
The last equality is proved by the following steps:

The second fundamental theorem of calculus shows {\displaystyle f(t_{i})-f(t_{i-1})=\int {t{i-1}}^{t_{i}}f’(t)\ dt=\Delta t\int {0}^{1}f’(t{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta }{\displaystyle f(t_{i})-f(t_{i-1})=\int {t{i-1}}^{t_{i}}f’(t)\ dt=\Delta t\int {0}^{1}f’(t{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta } where {\displaystyle t=t_{i-1}+\theta (t_{i}-t_{i-1})}{\displaystyle t=t_{i-1}+\theta (t_{i}-t_{i-1})} over {\displaystyle \theta \in [0,1]}{\displaystyle \theta \in [0,1]} maps to {\displaystyle [t_{i-1},t_{i}]}{\displaystyle [t_{i-1},t_{i}]} and {\displaystyle dt=(t_{i}-t_{i-1})d\theta =\Delta td\theta }{\displaystyle dt=(t_{i}-t_{i-1})d\theta =\Delta td\theta }. In the below step, the following equivalent expression is used.
{\displaystyle {\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}=\int {0}^{1}f’(t{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta .}{\displaystyle {\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}=\int {0}^{1}f’(t{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta .}
The function {\displaystyle \left|f’\right|}{\displaystyle \left|f’\right|} is a continuous function from a closed interval {\displaystyle [a,b]}[a,b] to the set of real numbers, thus it is uniformly continuous according to the Heine–Cantor theorem, so there is a positive real and monotonically non-decreasing function {\displaystyle \delta (\varepsilon )}\delta(\varepsilon) of positive real numbers {\displaystyle \varepsilon }\varepsilon such that {\displaystyle \Delta t<\delta (\varepsilon )}{\displaystyle \Delta t<\delta (\varepsilon )} implies {\displaystyle \left|\left|f’(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|-\left|f’(t_{i})\right|\right|<\varepsilon }{\displaystyle \left|\left|f’(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|-\left|f’(t_{i})\right|\right|<\varepsilon } where {\displaystyle \Delta t=t_{i}-t_{i-1}}{\displaystyle \Delta t=t_{i}-t_{i-1}} and {\displaystyle \theta \in [0,1]}{\displaystyle \theta \in [0,1]}. Let’s consider the limit {\displaystyle {\ce {N\to \infty }}}{\displaystyle {\ce {N\to \infty }}} of the following formula,
{\displaystyle \sum {i=1}^{N}\left|{\frac {f(t{i})-f(t_{i-1})}{\Delta t}}\right|\Delta t-\sum {i=1}^{N}\left|f’(t{i})\right|\Delta t.}{\displaystyle \sum {i=1}^{N}\left|{\frac {f(t{i})-f(t_{i-1})}{\Delta t}}\right|\Delta t-\sum {i=1}^{N}\left|f’(t{i})\right|\Delta t.}
With the above step result, it becomes

{\displaystyle \sum {i=1}^{N}\left|\int {0}^{1}f’(t{i-1}+\theta (t{i}-t_{i-1}))\ d\theta \right|\Delta t-\sum {i=1}^{N}\left|f’(t{i})\right|\Delta t.}{\displaystyle \sum {i=1}^{N}\left|\int {0}^{1}f’(t{i-1}+\theta (t{i}-t_{i-1}))\ d\theta \right|\Delta t-\sum {i=1}^{N}\left|f’(t{i})\right|\Delta t.}
Terms are rearranged so that it becomes
{\displaystyle \Delta t\sum {i=1}^{N}\left(\left|\int {0}^{1}f’(t{i-1}+\theta (t{i}-t_{i-1}))\ d\theta \right|-\int {0}^{1}\left|f’(t{i})\right|d\theta \right)\leqq \Delta t\sum {i=1}^{N}\left(\int {0}^{1}\left|f’(t{i-1}+\theta (t{i}-t_{i-1}))\right|\ d\theta -\int {0}^{1}\left|f’(t{i})\right|d\theta \right)=\Delta t\sum {i=1}^{N}\int {0}^{1}\left|f’(t{i-1}+\theta (t{i}-t_{i-1}))\right|-\left|f’(t_{i})\right|\ d\theta }{\displaystyle \Delta t\sum {i=1}^{N}\left(\left|\int {0}^{1}f’(t{i-1}+\theta (t{i}-t_{i-1}))\ d\theta \right|-\int {0}^{1}\left|f’(t{i})\right|d\theta \right)\leqq \Delta t\sum {i=1}^{N}\left(\int {0}^{1}\left|f’(t{i-1}+\theta (t{i}-t_{i-1}))\right|\ d\theta -\int {0}^{1}\left|f’(t{i})\right|d\theta \right)=\Delta t\sum {i=1}^{N}\int {0}^{1}\left|f’(t{i-1}+\theta (t{i}-t_{i-1}))\right|-\left|f’(t_{i})\right|\ d\theta }
where in the leftmost side {\displaystyle \left|f’(t_{i})\right|=\int {0}^{1}\left|f’(t{i})\right|d\theta }{\displaystyle \left|f’(t_{i})\right|=\int {0}^{1}\left|f’(t{i})\right|d\theta } is used. By {\displaystyle \left|\left|f’(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|-\left|f’(t_{i})\right|\right|<\varepsilon }{\displaystyle \left|\left|f’(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|-\left|f’(t_{i})\right|\right|<\varepsilon } for {\displaystyle N>(b-a)/\delta (\varepsilon )}{\displaystyle N>(b-a)/\delta (\varepsilon )} so that {\displaystyle \Delta t<\delta (\varepsilon )}{\displaystyle \Delta t<\delta (\varepsilon )}, it becomes
{\displaystyle \Delta t\sum {i=1}^{N}\left(\left|\int {0}^{1}f’(t{i-1}+\theta (t{i}-t_{i-1}))\ d\theta \right|-\left|f’(t_{i})\right|\right)<\varepsilon N\Delta t}{\displaystyle \Delta t\sum {i=1}^{N}\left(\left|\int {0}^{1}f’(t{i-1}+\theta (t{i}-t_{i-1}))\ d\theta \right|-\left|f’(t_{i})\right|\right)<\varepsilon N\Delta t}
with {\displaystyle \left|f’(t_{i})\right|=\int {0}^{1}\left|f’(t{i})\right|d\theta }{\displaystyle \left|f’(t_{i})\right|=\int {0}^{1}\left|f’(t{i})\right|d\theta }, {\displaystyle \varepsilon N\Delta t=\varepsilon (b-a)}{\displaystyle \varepsilon N\Delta t=\varepsilon (b-a)}, and {\displaystyle N>(b-a)/\delta (\varepsilon )}{\displaystyle N>(b-a)/\delta (\varepsilon )}. In the limit {\displaystyle N\to \infty ,}{\displaystyle N\to \infty ,} {\displaystyle \delta (\varepsilon )\to 0}{\displaystyle \delta (\varepsilon )\to 0} so {\displaystyle \varepsilon \to 0}\varepsilon \to 0 thus the left side of {\displaystyle <}{\displaystyle <} approaches to {\displaystyle 0}{\displaystyle 0}. In other words, {\displaystyle \sum {i=1}^{N}\left|{\frac {f(t{i})-f(t_{i-1})}{\Delta t}}\right|\Delta t=\sum {i=1}^{N}\left|f’(t{i})\right|\Delta t}{\displaystyle \sum {i=1}^{N}\left|{\frac {f(t{i})-f(t_{i-1})}{\Delta t}}\right|\Delta t=\sum {i=1}^{N}\left|f’(t{i})\right|\Delta t} in this limit, and the right side of this equality is just the Riemann integral of {\displaystyle \left|f’(t)\right|}{\displaystyle \left|f’(t)\right|} on {\displaystyle [a,b].}[a, b]. This definition of arc length shows that the length of a curve represented by a continuously differentiable function {\displaystyle f:[a,b]\to \mathbb {R} ^{n}}{\displaystyle f:[a,b]\to \mathbb {R} ^{n}} on {\displaystyle [a,b]}[a,b] is always finite, i.e., rectifiable.
The definition of arc length of a smooth curve as the integral of the norm of the derivative is equivalent to the definition

{\displaystyle L(f)=\sup \sum {i=1}^{N}{\bigg |}f(t{i})-f(t_{i-1}){\bigg |}}{\displaystyle L(f)=\sup \sum {i=1}^{N}{\bigg |}f(t{i})-f(t_{i-1}){\bigg |}}
where the supremum is taken over all possible partitions {\displaystyle a=t_{0} A curve can be parameterized in infinitely many ways. Let {\displaystyle \varphi :[a,b]\to [c,d]}{\displaystyle \varphi :[a,b]\to [c,d]} be any continuously differentiable bijection. Then {\displaystyle g=f\circ \varphi ^{-1}:[c,d]\to \mathbb {R} ^{n}}{\displaystyle g=f\circ \varphi ^{-1}:[c,d]\to \mathbb {R} ^{n}} is another continuously differentiable parameterization of the curve originally defined by {\displaystyle f.}f. The arc length of the curve is the same regardless of the parameterization used to define the curve:

{\displaystyle {

}}{\displaystyle {}}

3 Finding arc lengths by integration

3.1 Numerical integration

3.2 Curve on a surface

3.3 Other coordinate systems

4 Simple cases

4.1 Arcs of circles

4.1.1 Great circles on Earth

4.2 Other simple cases

5 Historical methods

5.1 Antiquity

5.2 17th century

5.3 Integral form

6 Curves with infinite length

7 Generalization to (pseudo-)Riemannian manifolds

8 See also