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

*

Arc Length

by Ed Pegg Jr., The Wolfram Demonstrations Project, 2007.

Calculus Study Guide – Arc Length (Rectification)

''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. {{Authority control Integral calculus Curves Length One-dimensional coordinate systems

curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

.
Determining the length of an irregular arc segment is also called of a curve. The advent of infinitesimal calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

led to a general formula that provides closed-form solutions in some cases.
General approach

Acurve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

in the plane
Plane or planes may refer to:
* Airplane
An airplane or aeroplane (informally plane) is a fixed-wing aircraft
A fixed-wing aircraft is a heavier-than-air flying machine
Early flying machines include all forms of aircraft studied ...

can be approximated by connecting a finite
Finite is the opposite of Infinity, infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected ...

number of points
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Scotland, Lismore, Inner Hebrides, ...

on the curve using line segment
250px, The geometric definition of a closed line segment: the intersection of all points at or to the right of ''A'' with all points at or to the left of ''B''
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' ...

s to create a polygonal path. Since it is straightforward to calculate the length
Length is a measure of distance
Distance is a numerical measurement
'
Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...

of each linear segment (using the Pythagorean theorem
In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

in Euclidean space, for example), the total length of the approximation can be found by summing
Summing (April 16, 1978 – October 10, 2008) was an United States, American thoroughbred horse racing, racehorse and Horse breeding#Terminology, sire.
Background
Summing was a bay horse bred in Kentucky by his owner Charles T. Wilson Jr. He was ...

the lengths of each linear segment; that approximation is known as the ''(cumulative) chordal
In the mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...

distance''.
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 smallIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

.
For some curves there is a smallest number $L$ that is an upper bound on the length of any polygonal approximation. These curves are called and the number $L$ is defined as the .
Definition for a smooth curve

Let $f\backslash colon;\; href="/html/ALL/s/,b.html"\; ;"title=",b">,b$ be aninjective
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

and continuously differentiable
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

function. The length of the curve defined by $f$ can be defined as the limit
Limit or Limits may refer to:
Arts and media
* Limit (music), a way to characterize harmony
* Limit (song), "Limit" (song), a 2016 single by Luna Sea
* Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

of the sum of line segment lengths for a regular partition of $;\; href="/html/ALL/s/,b.html"\; ;"title=",b">,b$Riemann integral
The partition does not need to be regular, as shown here. The approximation works as long as the width of each subdivision tends to zero.
In the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such to ...

of $,\; f\text{'}(t),$ on $;\; href="/html/ALL/s/,b.html"\; ;"title=",b">,b$ This definition of arc length shows that the length of a curve $f:;\; href="/html/ALL/s/,b.html"\; ;"title=",b">,b$ continuously differentiable on $;\; href="/html/ALL/s/,b.html"\; ;"title=",b">,b$supremum
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

is taken over all possible partitions $a=t\_0\backslash dots="b$bijection
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. Then $g=f\backslash circ\backslash varphi^:;\; href="/html/ALL/s/,d.html"\; ;"title=",d">,d$ is another continuously differentiable parameterization of the curve originally defined by $f.$ The arc length of the curve is the same regardless of the parameterization used to define the curve:
:$\backslash begin\; L(f)\; \&=\; \backslash int\_a^b\; \backslash Big,\; f\text{'}(t)\backslash Big,\; \backslash \; dt=\backslash int\_a^b\; \backslash Big,\; g\text{'}(\backslash varphi(t))\backslash varphi\text{'}(t)\backslash Big,\; \backslash \; dt\; \backslash \backslash \; \&=\; \backslash int\_a^b\; \backslash Big,\; g\text{'}(\backslash varphi(t))\backslash Big,\; \backslash varphi\text{'}(t)\backslash \; dt\; \backslash quad\; \backslash textrm\; \backslash \; \backslash varphi\; \backslash \; \backslash textrm\; \backslash \; \backslash textrm\; \backslash \; \backslash textrm\; \backslash \backslash \; \&=\; \backslash int\_c^d\; \backslash Big,\; g\text{'}(u)\backslash Big,\; \backslash \; du\; \backslash quad\; \backslash textrm\; \backslash \; \backslash textrm\; \backslash \; \backslash textrm\; \backslash \; \backslash textrm\backslash \backslash \; \&=\; L(g).\; \backslash end$
Finding arc lengths by integrating

If a planar curve in $\backslash mathbb^2$ is defined by the equation $y=f(x),$ where $f$ iscontinuously differentiable
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

, then it is simply a special case of a parametric equation where $x\; =\; t$ and $y\; =\; f(t).$ The arc length is then given by:
:$s=\backslash int\_a^b\; \backslash sqrtdx.$
Curves with closed-form solutions for arc length include the catenary
forming multiple Elastic deformation, elastic catenaries.
In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or wire rope, cable assumes under its own weight when supported only at its ends.
The catenary cu ...

, circle
A circle is a shape
A shape or figure is the form of an object or its external boundary, outline, or external surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

, cycloid
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

, logarithmic spiral
Logarithmic spiral (pitch 10°)
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar
__NOTOC__
has an infinitely repeating self-similarity when it is magnified.
In mathematics
Mathematics (from Ancient Greek, G ...

, parabola
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

, semicubical parabola
In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form
: y^2 - a^2 x^3 = 0
(with ) in some Cartesian coordinate system.
Solving for y leads to the ''explicit form''
: y = \p ...

and straight line
290px, A representation of one line segment.
In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature
In mathematics, curvature is any of several str ...

. The lack of a closed form solution for the arc length of an elliptic
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the sp ...

and hyperbolic
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...

arc led to the development of the elliptic integral
In integral calculus
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integra ...

s.
Numerical integration

In most cases, including even simple curves, there are no closed-form solutions for arc length andnumerical integration
In analysis, numerical integration comprises a broad family of algorithm
of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and B. The algorithm procee ...

is necessary. Numerical integration of the arc length integral is usually very efficient. For example, consider the problem of finding the length of a quarter of the unit circle by numerically integrating the arc length integral. The upper half of the unit circle can be parameterized as $y=\backslash sqrt.$ The interval $x\backslash in;\; href="/html/ALL/s/\backslash sqrt/2,\_\backslash sqrt/2.html"\; ;"title="\backslash sqrt/2,\; \backslash sqrt/2">\backslash sqrt/2,\; \backslash sqrt/2$ corresponds to a quarter of the circle. Since $dy/dx=-x/\backslash sqrt$ and $1+(dy/dx)^2\; =\; 1/(1-x^2),$ the length of a quarter of the unit circle is
:$\backslash int\_^\backslash frac\; \backslash ,\; dx.$
The 15-point Gauss–Kronrod rule estimate for this integral of differs from the true length of
:$\backslash Big;\; href="/html/ALL/s/arcsin\_x\backslash Big.html"\; ;"title="arcsin\; x\backslash Big">arcsin\; x\backslash Big$
by and the 16-point Gaussian quadrature
In numerical analysis
(c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296...
Numerical analysis is the study ...

rule estimate of differs from the true length by only . This means it is possible to evaluate this integral to almost machine precision with only 16 integrand evaluations.
Curve on a surface

Let $\backslash mathbf(u,v)$ be a surface mapping and let $\backslash mathbf(t)\; =\; (u(t),\; v(t))$ be a curve on this surface. The integrand of the arc length integral is $,\; (\backslash mathbf\backslash circ\backslash mathbf)\text{'}(t),\; .$ Evaluating the derivative requires the chain rule for vector fields: : $D(\backslash mathbf\; \backslash circ\; \backslash mathbf)\; =\; (\backslash mathbf\_u\; \backslash \; \backslash mathbf\_v)\backslash binom\; =\; \backslash mathbf\_u\; u\text{'}\; +\; \backslash mathbf\_v\; v\text{'}.$ The squared norm of this vector is $(\backslash mathbf\_u\; u\text{'}\; +\; \backslash mathbf\_v\; v\text{'})\; \backslash cdot\; (\backslash mathbf\_u\; u\text{'}\; +\; \backslash mathbf\_v\; v\text{'})\; =\; g\_(u\text{'})^2\; +\; 2g\_u\text{'}v\text{'}\; +\; g\_(v\text{'})^2$ (where $g\_$ is thefirst fundamental form
In differential geometry, the first fundamental form is the inner product on the tangent space of a surface (differential geometry), surface in three-dimensional Euclidean space which is induced canonical form, canonically from the dot product of . ...

coefficient), so the integrand of the arc length integral can be written as $\backslash sqrt$ (where $u^1\; =\; u$ and $u^2\; =\; v$).
Other coordinate systems

Let $\backslash mathbf(t)\; =\; (r(t),\; \backslash theta(t))$ be a curve expressed in polar coordinates. The mapping that transforms from polar coordinates to rectangular coordinates is :$\backslash mathbf(r,\backslash theta)\; =\; (r\backslash cos\backslash theta,\; r\backslash sin\backslash theta\; ).$ The integrand of the arc length integral is $,\; (\backslash mathbf\backslash circ\backslash mathbf)\text{'}(t),\; .$ The chain rule for vector fields shows that $D(\backslash mathbf\backslash circ\; \backslash mathbf)\; =\; \backslash mathbf\_r\; r\text{'}\; +\; \backslash mathbf\_\; \backslash theta\text{'}.$ So the squared integrand of the arc length integral is :$(\backslash mathbf\backslash cdot\backslash mathbf)(r\text{'})^2\; +\; 2(\backslash mathbf\_r\backslash cdot\backslash mathbf\_)r\text{'}\backslash theta\text{'}\; +\; (\backslash mathbf\_\backslash cdot\backslash mathbf\_)(\backslash theta\text{'})^2\; =\; (r\text{'})^2\; +\; r^2(\backslash theta\text{'})^2.$ So for a curve expressed in polar coordinates, the arc length is :$\backslash int\_^\; \backslash sqrtdt\; =\; \backslash int\_^\; \backslash sqrtd\backslash theta.$ Now let $\backslash mathbf(t)\; =\; (r(t),\; \backslash theta(t),\; \backslash phi(t))$ be a curve expressed in spherical coordinates where $\backslash theta$ is the polar angle measured from the positive $z$-axis and $\backslash phi$ is the azimuthal angle. The mapping that transforms from spherical coordinates to rectangular coordinates is :$\backslash mathbf(r,\backslash theta,\backslash phi)\; =\; (r\backslash sin\backslash theta\backslash cos\backslash phi,\; r\backslash sin\backslash theta\backslash sin\backslash phi,\; r\backslash cos\backslash theta).$ Using the chain rule again shows that $D(\backslash mathbf\backslash circ\backslash mathbf)\; =\; \backslash mathbf\_r\; r\text{'}\; +\; \backslash mathbf\_\backslash theta\text{'}\; +\; \backslash mathbf\_\backslash phi\text{'}.$ All dot products $\backslash mathbf\_i\; \backslash cdot\; \backslash mathbf\_j$ where $i$ and $j$ differ are zero, so the squared norm of this vector is :$(\backslash mathbf\_r\backslash cdot\; \backslash mathbf\_r\; )(r\text{'}^2)\; +\; (\backslash mathbf\_\; \backslash cdot\; \backslash mathbf\_)(\backslash theta\text{'})^2\; +\; (\backslash mathbf\_\backslash cdot\; \backslash mathbf\_)(\backslash phi\text{'})^2\; =\; (r\text{'})^2\; +\; r^2(\backslash theta\text{'})^2\; +\; r^2\; \backslash sin^2\backslash theta\; (\backslash phi\text{'})^2.$ So for a curve expressed in spherical coordinates, the arc length is :$\backslash int\_^\; \backslash sqrtdt.$ A very similar calculation shows that the arc length of a curve expressed in cylindrical coordinates is :$\backslash int\_^\; \backslash sqrtdt.$Simple cases

Arcs of circles

Arc lengths are denoted by ''s'', since the Latin word for length (or size) is ''spatium''. In the following lines, $r$ represents theradius
In classical geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative ...

of a circle
A circle is a shape
A shape or figure is the form of an object or its external boundary, outline, or external surface
File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to preven ...

, $d$ is its diameter
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

, $C$ is its circumference
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

, $s$ is the length of an arc of the circle, and $\backslash theta$ is the angle which the arc subtends at the centre
Center or centre may refer to:
Mathematics
*Center (geometry)
In geometry, a centre (or center) (from Ancient Greek language, Greek ''κέντρον'') of an object is a point in some sense in the middle of the object. According to the spe ...

of the circle. The distances $r,\; d,\; C,$ and $s$ are expressed in the same units.
* $C=2\backslash pi\; r,$ which is the same as $C=\backslash pi\; d.$ This equation is a definition of
* If the arc is a semicircle
In (and more specifically ), a semicircle is a one-dimensional of points that forms half of a . The full of a semicircle always measures 180° (equivalently, , or a ). It has only one line of symmetry (). In non-technical usage, the term "semi ...

, then $s=\backslash pi\; r.$
* For an arbitrary circular arc:
** If $\backslash theta$ is in radian
The radian, denoted by the symbol \text, is the SI unit
The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

s then $s\; =r\backslash theta.$ This is a definition of the radian.
** If $\backslash theta$ is in degrees
Degree may refer to:
As a unit of measurement
* Degree symbol (°), a notation used in science, engineering, and mathematics
* Degree (angle), a unit of angle measurement
* Degree (temperature), any of various units of temperature measurement ...

, then $s=\backslash frac,$ which is the same as $s=\backslash frac.$
** If $\backslash theta$ is in grads (100 grads, or grades, or gradians are one right-angle
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

), then $s=\backslash frac,$ which is the same as $s=\backslash frac.$
** If $\backslash theta$ is in turns (one turn is a complete rotation, or 360°, or 400 grads, or $2\backslash pi$ radians), then $s=C\; \backslash theta/\backslash text$.
Arcs of great circles on the Earth

Two units of length, thenautical mile
A nautical mile is a unit of length
A unit of length refers to any arbitrarily chosen and accepted reference standard for measurement of length. The most common units in modern use are the metric system, metric units, used in every country gl ...

and the metre
The metre ( Commonwealth spelling) or meter (American spelling
Despite the various English dialects spoken from country to country and within different regions of the same country, there are only slight regional variations in English ...

(or kilometre), were originally defined so the lengths of arcs of great circle
A great circle, also known as an orthodrome, of a sphere
of a sphere
A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathem ...

s on the Earth's surface would be simply numerically related to the angles they subtend at its centre. The simple equation $s=\backslash theta$ applies in the following circumstances:
:* if $s$ is in nautical miles, and $\backslash theta$ is in arcminute
A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol ', is a unit of angular measurement equal to of one degree. Since one degree is of a turn (or complete rotation), one minute of arc is of a turn. The n ...

s ( degree), or
:* if $s$ is in kilometres, and $\backslash theta$ is in centigrades ( grad).
The lengths of the distance units were chosen to make the circumference of the Earth equal kilometres, or nautical miles. Those are the numbers of the corresponding angle units in one complete turn.
Those definitions of the metre and the nautical mile have been superseded by more precise ones, but the original definitions are still accurate enough for conceptual purposes and some calculations. For example, they imply that one kilometre is exactly 0.54 nautical miles. Using official modern definitions, one nautical mile is exactly 1.852 kilometres, which implies that 1 kilometre is about nautical miles. This modern ratio differs from the one calculated from the original definitions by less than one part in 10,000.
Length of an arc of a parabola

Historical methods

Antiquity

For much of thehistory of mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the History of mathematical notation, mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples ...

, even the greatest thinkers considered it impossible to compute the length of an irregular arc. Although Archimedes
Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

had pioneered a way of finding the area beneath a curve with his "method of exhaustion
The method of exhaustion (; ) is a method of finding the area
Area is the quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of ...

", few believed it was even possible for curves to have definite lengths, as do straight lines. The first ground was broken in this field, as it often has been in calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

, by approximation
An approximation is anything that is intentionally similar but not exactly equal
Equal or equals may refer to:
Arts and entertainment
* Equals (film), ''Equals'' (film), a 2015 American science fiction film
* Equals (game), ''Equals'' (game), a ...

. People began to inscribe polygon
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

s within the curves and compute the length of the sides for a somewhat accurate measurement of the length. By using more segments, and by decreasing the length of each segment, they were able to obtain a more and more accurate approximation. In particular, by inscribing a polygon of many sides in a circle, they were able to find approximate values of π.
17th century

In the 17th century, the method of exhaustion led to the rectification by geometrical methods of several transcendental curves: thelogarithmic spiral
Logarithmic spiral (pitch 10°)
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar
__NOTOC__
has an infinitely repeating self-similarity when it is magnified.
In mathematics
Mathematics (from Ancient Greek, G ...

by Evangelista Torricelli
Evangelista Torricelli ( , also , ; 15 October 160825 October 1647) was an Italian
Italian may refer to:
* Anything of, from, or related to the country and nation of Italy
** Italians, an ethnic group or simply a citizen of the Italian Republic
...

in 1645 (some sources say John Wallis
John Wallis (; la, Wallisius; ) was an English clergyman and mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), f ...

in the 1650s), the cycloid
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

by Christopher Wren
Sir Christopher Wren PRS FRS
FRS may also refer to:
Government and politics
* Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States
* Fa ...

in 1658, and the catenary
forming multiple Elastic deformation, elastic catenaries.
In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or wire rope, cable assumes under its own weight when supported only at its ends.
The catenary cu ...

by Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the " 1666–1676" section. ( – 14 November 1716) was a German polymath
A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, " ...

in 1691.
In 1659, Wallis credited William Neile's discovery of the first rectification of a nontrivial algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

, the semicubical parabola
In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form
: y^2 - a^2 x^3 = 0
(with ) in some Cartesian coordinate system.
Solving for y leads to the ''explicit form''
: y = \p ...

. The accompanying figures appear on page 145. On page 91, William Neile is mentioned as ''Gulielmus Nelius''.
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 andPierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of suc ...

.
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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

. 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).
Building on his previous work with tangents, Fermat used the curve
:$y\; =\; x^\; \backslash ,$
whose tangent
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

at ''x'' = ''a'' had a slope
In mathematics, the slope or gradient of a line
Line, lines, The Line, or LINE may refer to:
Arts, entertainment, and media Films
* ''Lines'' (film), a 2016 Greek film
* ''The Line'' (2017 film)
* ''The Line'' (2009 film)
* ''The Line'', ...

of
:$\backslash textstyle\; a^$
so the tangent line would have the equation
:$y\; =\; \backslash textstyle\; (x\; -\; a)\; +\; f(a).$
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
In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

:
: $\backslash begin\; AC^2\; \&=\; AB^2\; +\; BC^2\; \backslash \backslash \; \&\; =\; \backslash textstyle\; \backslash varepsilon^2\; +\; a\; \backslash varepsilon^2\; \backslash \backslash \; \&=\; \backslash textstyle\; \backslash varepsilon^2\; \backslash left\; (1\; +\; a\; \backslash right\; )\; \backslash end$
which, when solved, yields
:$AC\; =\; \backslash textstyle\; \backslash varepsilon\; \backslash sqrt\; .$
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 madearbitrarily largeIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

. 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
The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractal
In mathematics, a fractal is a subset of Euclidean space with a Hausdorff dimension, fractal dimension that st ...

. 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
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point (geometry), ...

and Hausdorff measure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

are used to quantify the size of such curves.
Generalization to (pseudo-)Riemannian manifolds

Let $M$ be a (pseudo-)Riemannian manifold, $\backslash gamma:;\; href="/html/ALL/s/,1.html"\; ;"title=",1">,1$ a curve in $M$ and $g$ the (pseudo-)metric tensor
In the mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...

.
The length of $\backslash gamma$ is defined to be
:$\backslash ell(\backslash gamma)=\backslash int\_^\; \backslash sqrt\; \backslash ,\; dt,$
where $\backslash gamma\text{'}(t)\backslash in\; T\_M$ is the tangent vector of $\backslash gamma$ at $t.$ 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
The theory of relativity usually encompasses two interrelated theories by Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born , widely acknowledged to be one of the greatest physicists of all time ...

, arc length of timelike curves (world line
The world line (or worldline) of an object is the path (topology), path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics.
The concept of a "world line" is disti ...

s) is the proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural ...

elapsed along the world line, and arc length of a spacelike curve the proper distance
Proper length or rest length is the length of an object in the object's rest frame.
The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on th ...

along the curve.
See also

*Arc (geometry)
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

* Circumference
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

* Crofton formula
* Elliptic integral
In integral calculus
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integra ...

* Geodesic
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

s
* Intrinsic equationIn geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that ...

* Integral approximations
* Line integral
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

* Meridian arc
In geodesy
Geodesy () is the Earth science
Earth science or geoscience includes all fields of natural science
Natural science is a branch of science
Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic ...

* Multivariable calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one Variable (mathematics), variable to calculus with function of several variables, functions of several variables: the Differential calculus, different ...

* Sinuosity
The meandering '' Rio Cauto'' at Guamo Embarcadero, 1..html" ;"title="Cuba, is not taking the shortest path downslope. Therefore, its sinuosity index is > 1.">Cuba, is not taking the shortest path downslope. Therefore, its sinuosi ...

References

Sources

*External links

**

Arc Length

by Ed Pegg Jr., The Wolfram Demonstrations Project, 2007.

Calculus Study Guide – Arc Length (Rectification)

''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. {{Authority control Integral calculus Curves Length One-dimensional coordinate systems