Arc length

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

Contents

• General approach
• Formula for a smooth curve
• Finding arc lengths by integration
• Numerical integration
• Curve on a surface
• Other coordinate systems
• Simple cases
• Arcs of circles
• Other simple cases
• Historical methods
• Antiquity
• 17th century
• Integral form
• Curves with infinite length
• Generalization to (pseudo-)Riemannian manifolds
• See also
• References
• Sources
• External links

Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is also called curve rectification. For a rectifiable curve these approximations don't get arbitrarily large (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) ${\displaystyle f\colon [a,b]\to \mathbb {R} ^{n}}$, 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.

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 ${\displaystyle L}$ 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 ${\displaystyle L}$.

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]

Formula for a smooth curve

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

$${\displaystyle L(f)=\lim _{N\to \infty }\sum _{i=1}^{N}{\bigg |}f(t_{i})-f(t_{i-1}){\bigg |}}$$

where ${\displaystyle t_{i}=a+i(b-a)/N=a+i\Delta t}$ with ${\displaystyle \Delta t={\frac {b-a}{N}}=t_{i}-t_{i-1}}$ for ${\displaystyle i=0,1,\dotsc ,N.}$ This definition is equivalent to the standard definition of arc length as an integral:

$${\displaystyle L(f)=\lim _{N\to \infty }\sum _{i=1}^{N}{\bigg |}f(t_{i})-f(t_{i-1}){\bigg |}=\lim _{N\to \infty }\sum _{i=1}^{N}\left|{\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}\right|\Delta t=\int _{a}^{b}{\Big |}f'(t){\Big |}\ dt.}$$

The last equality is proved by the following steps:

1. The second fundamental theorem of calculus shows $${\displaystyle f(t_{i})-f(t_{i-1})=\int _{t_{i-1}}^{t_{i}}f'(t)\ dt=\Delta t\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta }$$ where ${\displaystyle t=t_{i-1}+\theta (t_{i}-t_{i-1})}$ over ${\displaystyle \theta \in [0,1]}$ maps to ${\displaystyle [t_{i-1},t_{i}]}$ and ${\displaystyle dt=(t_{i}-t_{i-1})\,d\theta =\Delta t\,d\theta }$. In the below step, the following equivalent expression is used.$${\displaystyle {\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}=\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta .}$$
2. The function ${\displaystyle \left|f'\right|}$ is a continuous function from a closed interval ${\displaystyle [a,b]}$ 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 ${\displaystyle \delta (\varepsilon )}$ of positive real numbers ${\displaystyle \varepsilon }$ such that ${\displaystyle \Delta t<\delta (\varepsilon )}$ implies ${\displaystyle \left|\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|-\left|f'(t_{i})\right|\right|<\varepsilon }$ where ${\displaystyle \Delta t=t_{i}-t_{i-1}}$ and ${\displaystyle \theta \in [0,1]}$. Let's consider the limit ${\displaystyle {\ce {N\to \infty }}}$ of the following formula,$${\displaystyle \sum _{i=1}^{N}\left|{\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}\right|\Delta t-\sum _{i=1}^{N}\left|f'(t_{i})\right|\Delta t.}$$

With the above step result, it becomes

$${\displaystyle \sum _{i=1}^{N}\left|\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta \right|\Delta t-\sum _{i=1}^{N}\left|f'(t_{i})\right|\Delta t.}$$

Terms are rearranged so that it becomes

$${\displaystyle {\begin{aligned}&\Delta t\sum _{i=1}^{N}\left(\left|\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta \right|-\int _{0}^{1}\left|f'(t_{i})\right|d\theta \right)\\&\qquad \leqq \Delta t\sum _{i=1}^{N}\left(\int _{0}^{1}\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|\ d\theta -\int _{0}^{1}\left|f'(t_{i})\right|d\theta \right)\\&\qquad =\Delta t\sum _{i=1}^{N}\int _{0}^{1}\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|-\left|f'(t_{i})\right|\ d\theta \end{aligned}}}$$

where in the leftmost side ${\textstyle \left|f'(t_{i})\right|=\int _{0}^{1}\left|f'(t_{i})\right|d\theta }$ is used. By ${\textstyle \left|\left|f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\right|-\left|f'(t_{i})\right|\right|<\varepsilon }$ for ${\textstyle N>(b-a)/\delta (\varepsilon )}$ so that ${\displaystyle \Delta t<\delta (\varepsilon )}$, it becomes

$${\displaystyle \Delta t\sum _{i=1}^{N}\left(\left|\int _{0}^{1}f'(t_{i-1}+\theta (t_{i}-t_{i-1}))\ d\theta \right|-\left|f'(t_{i})\right|\right)<\varepsilon N\Delta t}$$

with ${\displaystyle \left|f'(t_{i})\right|=\int _{0}^{1}\left|f'(t_{i})\right|d\theta }$, ${\displaystyle \varepsilon N\Delta t=\varepsilon (b-a)}$, and ${\displaystyle N>(b-a)/\delta (\varepsilon )}$. In the limit ${\displaystyle N\to \infty ,}$${\displaystyle \delta (\varepsilon )\to 0}$ so ${\displaystyle \varepsilon \to 0}$ thus the left side of ${\displaystyle <}$ approaches ${\displaystyle 0}$. In other words, ${\displaystyle \sum _{i=1}^{N}\left|{\frac {f(t_{i})-f(t_{i-1})}{\Delta t}}\right|\Delta t=\sum _{i=1}^{N}\left|f'(t_{i})\right|\Delta t}$ in this limit, and the right side of this equality is just the Riemann integral of ${\displaystyle \left|f'(t)\right|}$ on ${\displaystyle [a,b].}$ This definition of arc length shows that the length of a curve represented by a continuously differentiable function ${\displaystyle f:[a,b]\to \mathbb {R} ^{n}}$ on ${\displaystyle [a,b]}$ 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

$${\displaystyle L(f)=\sup \sum _{i=1}^{N}{\bigg |}f(t_{i})-f(t_{i-1}){\bigg |}}$$

where the supremum is taken over all possible partitions ${\displaystyle a=t_{0}<t_{1}<\dots <t_{N-1}<t_{N}=b}$ of ${\displaystyle [a,b].}$ ^[3] This definition as the supremum of the all possible partition sums is also valid if ${\displaystyle f}$ is merely continuous, not differentiable.

A curve can be parameterized in infinitely many ways. Let ${\displaystyle \varphi$ :[a,b]\to [c,d]} be any continuously differentiable bijection. Then ${\displaystyle g=f\circ \varphi ^{-1}:[c,d]\to \mathbb {R} ^{n}}$ is another continuously differentiable parameterization of the curve originally defined by ${\displaystyle f.}$ The arc length of the curve is the same regardless of the parameterization used to define the curve:

$${\displaystyle {\begin{aligned}L(f)&=\int _{a}^{b}{\Big |}f'(t){\Big |}\ dt=\int _{a}^{b}{\Big |}g'(\varphi (t))\varphi '(t){\Big |}\ dt\\&=\int _{a}^{b}{\Big |}g'(\varphi (t)){\Big |}\varphi '(t)\ dt\quad {\text{in the case }}\varphi {\text{ is non-decreasing}}\\&=\int _{c}^{d}{\Big |}g'(u){\Big |}\ du\quad {\text{using integration by substitution}}\\&=L(g).\end{aligned}}}$$

Finding arc lengths by integration

If a planar curve in ${\displaystyle \mathbb {R} ^{2}}$ is defined by the equation ${\displaystyle y=f(x),}$ where ${\displaystyle f}$ is continuously differentiable, then it is simply a special case of a parametric equation where ${\displaystyle x=t}$ and ${\displaystyle y=f(t).}$ The Euclidean distance of each infinitesimal segment of the arc can be given by:

$${\displaystyle {\sqrt {dx^{2}+dy^{2}}}={\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}\,}}dx.}$$

The arc length is then given by:

$${\displaystyle s=\int _{a}^{b}{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}\,}}dx.}$$

Curves with closed-form solutions for arc length include the catenary, circle, cycloid, logarithmic spiral, parabola, semicubical parabola and straight line. The lack of a closed form solution for the arc length of an elliptic and hyperbolic arc led to the development of the elliptic integrals.

Simple cases

Historical methods

Integral form

Before the full formal development of calculus, the basis for the modern integral form for arc length was independently discovered by Hendrik van Heuraet and Pierre de Fermat.

In 1659 van Heuraet published a construction showing that the problem of determining arc length could be transformed into the problem of determining the area under a curve (i.e., an integral). As an example of his method, he determined the arc length of a semicubical parabola, which required finding the area under a parabola. ^[9] In 1660, Fermat published a more general theory containing the same result in his De linearum curvarum cum lineis rectis comparatione dissertatio geometrica (Geometric dissertation on curved lines in comparison with straight lines). ^[10]

Building on his previous work with tangents, Fermat used the curve

${\displaystyle y=x^{\frac {3}{2}}\,}$

whose tangent at x = a had a slope of

${\displaystyle {3 \over 2}a^{\frac {1}{2}}}$

so the tangent line would have the equation

${\displaystyle y={3 \over 2}a^{\frac {1}{2}}(x-a)+f(a).}$

Next, he increased a by a small amount to a + ε, making segment AC a relatively good approximation for the length of the curve from A to D. To find the length of the segment AC, he used the Pythagorean theorem:

${\displaystyle {\begin{aligned}AC^{2}&=AB^{2}+BC^{2}\\&=\varepsilon ^{2}+{9 \over 4}a\varepsilon ^{2}\\&=\varepsilon ^{2}\left(1+{9 \over 4}a\right)\end{aligned}}}$

which, when solved, yields

${\displaystyle AC=\varepsilon {\sqrt {1+{9 \over 4}a\,}}.}$

In order to approximate the length, Fermat would sum up a sequence of short segments.

Curves with infinite length

As mentioned above, some curves are non-rectifiable. That is, there is no upper bound on the lengths of polygonal approximations; the length can be made arbitrarily large. Informally, such curves are said to have infinite length. There are continuous curves on which every arc (other than a single-point arc) has infinite length. An example of such a curve is the Koch curve. Another example of a curve with infinite length is the graph of the function defined by f(x) = x sin(1/x) for any open set with 0 as one of its delimiters and f(0) = 0. Sometimes the Hausdorff dimension and Hausdorff measure are used to quantify the size of such curves.

Generalization to (pseudo-)Riemannian manifolds

Let ${\displaystyle M}$ be a (pseudo-)Riemannian manifold, ${\displaystyle g}$ the (pseudo-) metric tensor, ${\displaystyle \gamma$ :[0,1]\rightarrow M} a curve in ${\displaystyle M}$ defined by ${\displaystyle n}$ parametric equations

${\displaystyle \gamma (t)=[\gamma ^{1}(t),\dots ,\gamma ^{n}(t)],\quad t\in [0,1]}$

and

${\displaystyle \gamma (0)=\mathbf {x} ,\,\,\gamma (1)=\mathbf {y} }$

The length of ${\displaystyle \gamma }$, is defined to be

${\displaystyle \ell (\gamma )=\int \limits _{0}^{1}||\gamma '(t)||_{\gamma (t)}dt}$,

or, choosing local coordinates ${\displaystyle x}$,

${\displaystyle \ell (\gamma )=\int \limits _{0}^{1}{\sqrt {\pm \sum _{i,j=1}^{n}g_{ij}(x(\gamma (t))){\frac {dx^{i}(\gamma (t))}{dt}}{\frac {dx^{j}(\gamma (t))}{dt}}}}dt}$,

where

${\displaystyle \gamma '(t)\in T_{\gamma (t)}M}$

is the tangent vector of ${\displaystyle \gamma }$ at ${\displaystyle t.}$ The sign in the square root is chosen once for a given curve, to ensure that the square root is a real number. The positive sign is chosen for spacelike curves; in a pseudo-Riemannian manifold, the negative sign may be chosen for timelike curves. Thus the length of a curve is a non-negative real number. Usually no curves are considered which are partly spacelike and partly timelike.

In theory of relativity, arc length of timelike curves (world lines) is the proper time elapsed along the world line, and arc length of a spacelike curve the proper distance along the curve.

See also

• Arc (geometry)
• Circumference
• Crofton formula
• Elliptic integral
• Geodesics
• Intrinsic equation
• Integral approximations
• Line integral
• Meridian arc
• Multivariable calculus
• Sinuosity

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References

Sources

• Farouki, Rida T. (1999). "Curves from motion, motion from curves". In Laurent, P.-J.; Sablonniere, P.; Schumaker, L. L. (eds.). Curve and Surface Design: Saint-Malo 1999. Vanderbilt Univ. Press. pp. 63–90. ISBN   978-0-8265-1356-4.

External links

Wikimedia Commons has media related to Arc length .

• "Rectifiable curve", Encyclopedia of Mathematics , EMS Press, 2001 [1994]
• The History of Curvature
• Weisstein, Eric W. "Arc Length". MathWorld .
• Arc Length by Ed Pegg Jr., The Wolfram Demonstrations Project, 2007.
• Calculus Study Guide – Arc Length (Rectification)
• Famous Curves Index The MacTutor History of Mathematics archive
• Arc Length Approximation by Chad Pierson, Josh Fritz, and Angela Sharp, The Wolfram Demonstrations Project.
• Length of a Curve Experiment Illustrates numerical solution of finding length of a curve.