TheInfoList

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

A
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 ...

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 be an
injective 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
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 $, f\text{'}\left(t\right),$ on This definition of arc length shows that the length of a curve continuously differentiable on
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
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 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: :$\begin L\left(f\right) &= \int_a^b \Big, f\text{'}\left(t\right)\Big, \ dt=\int_a^b \Big, g\text{'}\left(\varphi\left(t\right)\right)\varphi\text{'}\left(t\right)\Big, \ dt \\ &= \int_a^b \Big, g\text{'}\left(\varphi\left(t\right)\right)\Big, \varphi\text{'}\left(t\right)\ dt \quad \textrm \ \varphi \ \textrm \ \textrm \ \textrm \\ &= \int_c^d \Big, g\text{'}\left(u\right)\Big, \ du \quad \textrm \ \textrm \ \textrm \ \textrm\\ &= L\left(g\right). \end$

# Finding arc lengths by integrating

If a planar curve in $\mathbb^2$ is defined by the equation $y=f\left(x\right),$ where $f$ is
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 ...
, then it is simply a special case of a parametric equation where $x = t$ and $y = f\left(t\right).$ The arc length is then given by: :$s=\int_a^b \sqrtdx.$ 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 and
numerical 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 $y=\sqrt.$ The interval corresponds to a quarter of the circle. Since $dy/dx=-x/\sqrt$ and $1+\left(dy/dx\right)^2 = 1/\left(1-x^2\right),$ the length of a quarter of the unit circle is :$\int_^\frac \, dx.$ The 15-point Gauss–Kronrod rule estimate for this integral of differs from the true length of : 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 $\mathbf\left(u,v\right)$ be a surface mapping and let $\mathbf\left(t\right) = \left(u\left(t\right), v\left(t\right)\right)$ be a curve on this surface. The integrand of the arc length integral is $, \left(\mathbf\circ\mathbf\right)\text{'}\left(t\right), .$ Evaluating the derivative requires the chain rule for vector fields: : $D\left(\mathbf \circ \mathbf\right) = \left(\mathbf_u \ \mathbf_v\right)\binom = \mathbf_u u\text{'} + \mathbf_v v\text{'}.$ The squared norm of this vector is $\left(\mathbf_u u\text{'} + \mathbf_v v\text{'}\right) \cdot \left(\mathbf_u u\text{'} + \mathbf_v v\text{'}\right) = g_\left(u\text{'}\right)^2 + 2g_u\text{'}v\text{'} + g_\left(v\text{'}\right)^2$ (where $g_$ is the
first 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 $\sqrt$ (where $u^1 = u$ and $u^2 = v$).

## Other coordinate systems

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

# 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 the
radius 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 $\theta$ 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. * $C=2\pi r,$ which is the same as $C=\pi d.$ 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 $s=\pi r.$ * For an arbitrary circular arc: ** If $\theta$ 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 $s =r\theta.$ This is a definition of the radian. ** If $\theta$ 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 $s=\frac,$ which is the same as $s=\frac.$ ** If $\theta$ 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 $s=\frac,$ which is the same as $s=\frac.$ ** If $\theta$ is in turns (one turn is a complete rotation, or 360°, or 400 grads, or $2\pi$ radians), then $s=C \theta/\text$.

### Arcs of great circles on the Earth

Two units of length, the
nautical 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 $s=\theta$ applies in the following circumstances: :* if $s$ is in nautical miles, and $\theta$ 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 $\theta$ 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.

# Historical methods

## Antiquity

For much of the
history 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: the
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 ...

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 and
Pierre 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 :$y = x^ \,$ 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 :$\textstyle a^$ so the tangent line would have the equation :$y = \textstyle \left(x - a\right) + f\left(a\right).$ 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 ...

: : $\begin AC^2 &= AB^2 + BC^2 \\ & = \textstyle \varepsilon^2 + a \varepsilon^2 \\ &= \textstyle \varepsilon^2 \left \left(1 + a \right \right) \end$ which, when solved, yields :$AC = \textstyle \varepsilon \sqrt .$ 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 made
arbitrarily 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, 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 $\gamma$ is defined to be :$\ell\left(\gamma\right)=\int_^ \sqrt \, dt,$ where $\gamma\text{'}\left(t\right)\in T_M$ is the tangent vector of $\gamma$ 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.

*
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 ...
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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 ...

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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 ...
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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 ...
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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 ...
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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 ...
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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 ...

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