View source for Polar coordinate system

{{short description|Two-dimensional coordinate system where each point is determined by a distance from reference point and an angle from a reference direction}}
{{more footnotes|date=May 2019}}

{{Featured article}}
[[Image:Examples of Polar Coordinates.svg|thumb|250px|Points in the polar coordinate system with pole ''O'' and polar axis ''L''. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3,{{nbsp}}60°). In blue, the point (4,{{nbsp}}210°).]]
In [[mathematics]], the '''polar coordinate system''' is a [[dimension|two-dimensional]] [[coordinate system]] in which each [[point (mathematics)|point]] on a [[plane (mathematics)|plane]] is determined by a [[distance]] from a reference point and an [[angle]] from a reference direction. The reference point (analogous to the origin of a [[Cartesian coordinate system]]) is called the ''pole'', and the [[ray (geometry)|ray]] from the pole in the reference direction is the ''polar axis''. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''[[azimuth]]''.<ref name="brown">{{Cite book| last = Brown| first = Richard G.| editor = Andrew M. Gleason| year = 1997| title = Advanced Mathematics: Precalculus with Discrete Mathematics and Data Analysis| publisher = McDougal Littell| location = Evanston, Illinois| isbn = 0-395-77114-5| url = https://archive.org/details/advancedmathemat00rich_0}}</ref> The radial coordinate is often denoted by ''r'' or [[rho|''ρ'']], and the angular coordinate by [[phi|''φ'']], [[theta|''θ'']], or ''t''.  Angles in polar notation are generally expressed in either [[degree (angle)|degree]]s or [[radian]]s (2[[pi|{{pi}}]] rad being equal to 360°).

[[Grégoire de Saint-Vincent]] and [[Bonaventura Cavalieri]] independently introduced the concepts in the mid-seventeenth century, though the actual term ''polar coordinates'' has been attributed to [[Gregorio Fontana]] in the 18th-century. The initial motivation for the introduction of the polar system was the study of [[circular motion|circular]] and [[orbital motion]].

Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point. The Archimedean spiral for example is much simpler to express using polar forms. Moreover, many physical systems—such as those concerned with bodies moving around a central point or with phenomena originating from a central point—are simpler and more intuitive to model using polar coordinates. 

The polar coordinate system is extended into three dimensions with two different coordinate systems, the [[cylindrical coordinate system|cylindrical]] and [[spherical coordinate system]]. [[Alexis Clairaut]] was the first to think of polar coordinates in three dimensions, and [[Leonhard Euler]] was the first to actually develop them.


==History==
{{See also|History of trigonometric functions}}
[[File:Hipparchos 1.jpeg|thumb|left|upright|Hipparchus]]
The concepts of angle and radius were already used by ancient peoples of the first millennium [[Before Christ|BC]]. The [[Greek astronomy|Greek astronomer]] and [[astrologer]] [[Hipparchus]] (190–120 BC) created a table of [[chord (geometry)|chord]] functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions.<ref>{{cite web| last = Friendly| first = Michael| title = Milestones in the History of Thematic Cartography, Statistical Graphics, and Data Visualization| url = http://www.math.yorku.ca/SCS/Gallery/milestone/milestone.pdf|date = August 24, 2009}}</ref>
In ''[[On Spirals]]'', [[Archimedes]] describes the [[Archimedean spiral]], a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system.

From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to [[Mecca]] ([[qibla]])—and its distance—from any location on the Earth.<ref>{{Cite book
 | chapter=The Sacred Geography of Islam
 | last=King
 | first=David A.
 | title=Mathematics and the Divine: A Historical Study
 | editor1-last=Koetsier
 | editor1-first=Teun
 | editor2-first=Bergmans
 | editor2-last=Luc
 | publisher=Elsevier
 | location=Amsterdam
 | year=2005
 | pages=162–78
 | isbn=0-444-50328-5
 | url=https://books.google.com/books?id=AMOQZfrZq-EC&pg=PA161#v=onepage&f=false
 | ref=harv}}
</ref> From the 9th century onward they were using [[spherical trigonometry]] and [[map projection]] methods to determine these quantities accurately. The calculation is essentially the conversion of the [[Geodetic coordinates#Coordinates|equatorial polar coordinates]] of Mecca (i.e. its [[longitude]] and [[latitude]]) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the [[great circle]] through the given location and the Earth's poles, and whose polar axis is the line through the location and its [[antipodal point]].<ref>King ([[#CITEREFKing2005|2005]], [https://books.google.com/books?id=AMOQZfrZq-EC&pg=PA169#v=onepage&f=false p. 169]). The calculations were as accurate as could be achieved under the limitations imposed by their assumption that the Earth was a perfect sphere.</ref>

There are various accounts of the introduction of polar coordinates as part of a formal coordinate system. The full history of the subject is described in [[Harvard University|Harvard]] professor [[Julian Lowell Coolidge]]'s ''Origin of Polar Coordinates.''<ref name="coolidge">{{Cite journal| last = Coolidge| first = Julian| authorlink = Julian Lowell Coolidge| title = The Origin of Polar Coordinates| journal = American Mathematical Monthly| volume = 59| pages = 78–85| year = 1952| url = http://www-history.mcs.st-and.ac.uk/Extras/Coolidge_Polars.html| doi = 10.2307/2307104| issue = 2| publisher = Mathematical Association of America| jstor = 2307104}}</ref> [[Grégoire de Saint-Vincent]] and [[Bonaventura Cavalieri]] independently introduced the concepts in the mid-seventeenth century. Saint-Vincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653. Cavalieri first used polar coordinates to solve a problem relating to the area within an [[Archimedean spiral]]. [[Blaise Pascal]] subsequently used polar coordinates to calculate the length of [[parabola|parabolic arcs]].

In ''[[Method of Fluxions]]'' (written 1671, published 1736), Sir [[Isaac Newton]] examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems.<ref>{{Cite journal| last = Boyer| first = C. B.| title = Newton as an Originator of Polar Coordinates| journal = American Mathematical Monthly| volume = 56| pages = 73–78| year = 1949| doi = 10.2307/2306162| issue = 2| publisher = Mathematical Association of America| jstor = 2306162}}</ref> In the journal ''[[Acta Eruditorum]]'' (1691), [[Jacob Bernoulli]] used a system with a point on a line, called the ''pole'' and ''polar axis'' respectively. Coordinates were specified by the distance from the pole and the angle from the ''polar axis''. Bernoulli's work extended to finding the [[Radius of curvature (mathematics)|radius of curvature]] of curves expressed in these coordinates.

The actual term ''polar coordinates'' has been attributed to [[Gregorio Fontana]] and was used by 18th-century Italian writers. The term appeared in [[English language|English]] in [[George Peacock]]'s 1816 translation of [[Sylvestre François Lacroix|Lacroix]]'s ''Differential and Integral Calculus''.<ref>{{cite web| last = Miller| first = Jeff| title = Earliest Known Uses of Some of the Words of Mathematics| url = http://jeff560.tripod.com/p.html| accessdate = 2006-09-10}}</ref><ref>{{Cite book| last = Smith| first = David Eugene| title = History of Mathematics, Vol II| publisher = Ginn and Co.| year = 1925| location = Boston| page = 324}}</ref> [[Alexis Clairaut]] was the first to think of polar coordinates in three dimensions, and [[Leonhard Euler]] was the first to actually develop them.<ref name="coolidge" />

==Conventions==
[[Image:Polar graph paper.svg|thumb|right|300px|A polar grid with several angles, increasing in ccw orientation and labelled in degrees]]
The radial coordinate is often denoted by ''r'' or [[rho|''ρ'']], and the angular coordinate by [[phi|''φ'']], [[theta|''θ'']], or ''t''.  The angular coordinate is specified as ''φ'' by [[International Organisation for Standardisation|ISO]] standard [[ISO 31-11|31-11]]. However, in mathematical literature the angle is often denoted by &theta; instead of ''φ''.

Angles in polar notation are generally expressed in either [[degree (angle)|degree]]s or [[radian]]s (2[[pi|{{pi}}]] rad being equal to 360°). Degrees are traditionally used in [[navigation]], [[surveying]], and many applied disciplines, while radians are more common in mathematics and mathematical [[physics]].<ref>{{Cite book| last = Serway| first = Raymond A.| author2 = Jewett, Jr., John W.| title = Principles of Physics| publisher = Brooks/Cole—Thomson Learning| year = 2005| isbn = 0-534-49143-X}}</ref>

The angle ''φ'' is defined to start at 0° from a ''reference direction'', and to increase for rotations in either [[clockwise|counterclockwise (ccw)]] or [[clockwise|clockwise (cw)]] orientation. In mathematics, e.g., the reference direction is usually drawn as a [[ray (geometry)|ray]] from the pole horizontally to the right, and the polar angle increases to positive angles for ccw rotations, whereas in navigation ([[bearing (navigation)|bearing]], [[heading (navigation)|heading]]) the 0°-heading is drawn vertically upwards and the angle increases for cw rotations. The polar angles decrease towards negative values for rotations in the respectively opposite orientations.

===Uniqueness of polar coordinates===
Adding any number of full [[turn (geometry)|turns]] (360°) to the angular coordinate does not change the corresponding direction.  Similarly, any polar coordinate is identical to the coordinate with the negative radial component and the opposite direction (adding 180° to the polar angle). Therefore, the same point (''r'', ''φ'') can be expressed with an infinite number of different polar coordinates {{nowrap|(''r'', ''φ'' + ''n'' × 360°)}} and {{nowrap|(−''r'', ''φ'' + 180° + ''n'' × 360°) {{=}} (−''r'', ''φ'' + (2''n'' + 1) × 180°)}}, where ''n'' is an arbitrary [[integer]].<ref>{{cite web| url = http://campuses.fortbendisd.com/campuses/documents/Teacher/2012%5Cteacher_20120507_1147.pdf| title = Polar Coordinates and Graphing| accessdate = 2006-09-22| date = 2006-04-13| format = PDF}}{{dead link|date=September 2017 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>  Moreover, the pole itself can be expressed as (0,&nbsp;''φ'') for any angle ''φ''.<ref>{{Cite book|title=Precalculus: With Unit-Circle Trigonometry|last=Lee|first=Theodore|author2=David Cohen |author3=David Sklar |year=2005|publisher=Thomson Brooks/Cole|edition=Fourth|isbn=0-534-40230-5}}</ref>

Where a unique representation is needed for any point besides the pole, it is usual to limit ''r'' to positive numbers ({{nowrap|''r'' > 0}}) and ''φ'' to the [[interval (mathematics)|interval]] [0,&nbsp;360°) or (−180°,&nbsp;180°] (in radians,  [0,&nbsp;2{{pi}}) or (−{{pi}},&nbsp;{{pi}}]).<ref>{{Cite book|title=Complex Analysis (the Hitchhiker's Guide to the Plane)|first=Ian|last=Stewart|author2=David Tall|year=1983|publisher=Cambridge University Press|isbn=0-521-28763-4}}</ref> Another convention, in reverence to the usual codomain of the [[inverse trigonometric functions|arctan-function]], is to allow for arbitrary nonzero real values of the radial component and restrict the polar angle to {{open-closed|−90°,{{nbsp}}90°}}. In all cases a unique azimuth for the pole (''r'' = 0) must be chosen, e.g., ''φ''&nbsp;=&nbsp;0.

==Converting between polar and Cartesian coordinates==
[[Image:Polar to cartesian.svg|right|thumb|250px|A diagram illustrating the relationship between polar and Cartesian coordinates.]]
[[File:Cartesian to polar.gif|right|thumb|251px|A curve on the Cartesian plane can be mapped into polar coordinates. In this animation, <math>y = \sin (6x) + 2</math> is mapped onto <math>r = \sin (6 \theta) + 2</math>. Click on image for details.]]

The polar coordinates ''r'' and ''φ'' can be converted to the [[Cartesian coordinate system|Cartesian coordinates]] ''x'' and ''y'' by using the [[trigonometric function]]s sine and cosine:
:<math>\begin{align}
  x &= r \cos \varphi, \\
  y &= r \sin \varphi.
\end{align}</math>

The Cartesian coordinates ''x'' and ''y'' can be converted to polar coordinates ''r'' and ''φ'' with ''r''&nbsp;≥&nbsp;0 and ''φ'' in the interval (−{{pi}}, {{pi}}] by:<ref>{{Cite book|first=Bruce Follett|last=Torrence|author2=Eve Torrence|title=The Student's Introduction to Mathematica|year=1999|publisher=Cambridge University Press|isbn=0-521-59461-8}}</ref>
:<math>r = \sqrt{x^2 + y^2} \quad</math> (as in the [[Pythagorean theorem]] or the [[Euclidean norm]]), and
:<math>\varphi = \operatorname{atan2}(y, x),</math>

where [[atan2]] is a common variation on the [[arctangent]] function defined as
:<math>\operatorname{atan2}(y, x) = \begin{cases}
   \arctan\left(\frac{y}{x}\right)       & \mbox{if } x > 0\\
   \arctan\left(\frac{y}{x}\right) + \pi & \mbox{if } x < 0 \mbox{ and } y \ge 0\\
   \arctan\left(\frac{y}{x}\right) - \pi & \mbox{if } x < 0 \mbox{ and } y < 0\\
   \frac{\pi}{2}                         & \mbox{if } x = 0 \mbox{ and } y > 0\\
  -\frac{\pi}{2}                         & \mbox{if } x = 0 \mbox{ and } y < 0\\
   \text{undefined}                      & \mbox{if } x = 0 \mbox{ and } y = 0.
\end{cases}</math>

If ''r'' is calculated first as above, then this formula for φ may be stated a little more simply using the standard [[arccosine]] function:
:<math>\varphi = \begin{cases}
   \arccos\left(\frac{x}{r}\right) & \mbox{if } y \ge 0 \mbox{ and } r \neq 0 \\
  -\arccos\left(\frac{x}{r}\right) & \mbox{if } y < 0 \\
   \text{undefined}                & \mbox{if } r = 0.
\end{cases}</math>

The value of ''φ'' above is the [[principal value]] of the complex number function [[argument (complex analysis)|arg]] applied to ''x'' + ''iy''. An angle in the range [0, 2{{pi}}) may be obtained by adding 2{{pi}} to the value in case it is negative (in other words when ''y'' is negative).

==Polar equation of a curve==
The equation defining an [[algebraic curve]] expressed in polar coordinates is known as a ''polar equation''. In many cases, such an equation can simply be specified by defining ''r'' as a [[function (mathematics)|function]] of ''φ''. The resulting curve then consists of points of the form (''r''(''φ''),&nbsp;''φ'') and can be regarded as the [[graph of a function|graph]] of the polar function ''r''. Note that, in contrast to Cartesian coordinates, the independent variable ''φ'' is the ''second'' entry in the ordered pair.

Different forms of [[symmetry]] can be deduced from the equation of a polar function ''r''. If {{nowrap|''r''(−''φ'') {{=}} ''r''(''φ'')}} the curve will be symmetrical about the horizontal (0°/180°) ray, if {{nowrap|''r''({{pi}} − ''φ'') {{=}} ''r''(''φ'')}} it will be symmetric about the vertical (90°/270°) ray, and if {{nowrap|''r''(''φ'' − α) {{=}} ''r''(''φ'')}} it will be [[rotational symmetry|rotationally symmetric]] by α [[clockwise]] and [[clockwise|counterclockwise]] about the pole.

Because of the circular nature of the polar coordinate system, many curves can be described by a rather simple polar equation, whereas their Cartesian form is much more intricate. Among the best known of these curves are the [[Rose (mathematics)|polar rose]], [[Archimedean spiral]], [[Lemniscate of Bernoulli|lemniscate]], [[limaçon]], and [[cardioid]].

For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.

===Circle===
[[Image:circle r=1.svg|thumb|right|A circle with equation {{nowrap|''r''(''φ'') {{=}} 1}}]]
The general equation for a circle with a center at {{nowrap|(''r''<sub>0</sub>, <math>\gamma</math>)}} and radius ''a'' is
:<math>r^2 - 2 r r_0 \cos(\varphi - \gamma) + r_0^2 = a^2.</math>

This can be simplified in various ways, to conform to more specific cases, such as the equation
:<math>r(\varphi)=a </math>
for a circle with a center at the pole and radius ''a''.<ref name="ping">{{cite web| first=Johan| last=Claeys| url=http://www.ping.be/~ping1339/polar.htm| title=Polar coordinates| accessdate=2006-05-25| url-status=dead| archiveurl=https://web.archive.org/web/20060427230725/http://www.ping.be/~ping1339/polar.htm| archivedate=2006-04-27}}</ref>

When {{math|''r''}}<sub>0</sub> = {{math|a}}, or when the origin lies on the circle, the equation becomes
:<math>r = 2 a\cos(\varphi - \gamma).</math>

In the general case, the equation can be solved for {{math|''r''}}, giving
:<math>r = r_0 \cos(\varphi - \gamma) + \sqrt{a^2 - r_0^2 \sin^2(\varphi - \gamma)},</math>
the solution with a minus sign in front of the square root gives the same curve.

===Line===
''Radial'' lines (those running through the pole) are represented by the equation
:<math>\varphi = \gamma ,</math>
where ɣ is the angle of elevation of the line; that is, {{nowrap|ɣ {{=}} arctan ''m''}} where ''m'' is the [[slope]] of the line in the Cartesian coordinate system. The non-radial line  that crosses the radial line {{nowrap|''φ'' {{=}} ɣ}} [[perpendicular]]ly at the point (''r<sub>0</sub>'', ɣ) has the equation
:<math>r(\varphi) = {r_0}\sec(\varphi-\gamma). </math>

Otherwise stated (''r<sub>0</sub>'', ɣ) is the point in which the tangent intersects the imaginary circle of radius ''r<sub>0</sub>''.

===Polar rose===
[[Image:Rose 2sin(4theta).svg|thumb|right|A polar rose with equation {{nowrap|''r''(''φ'') {{=}} 2 sin 4''φ''}}]]
A [[rose (mathematics)|polar rose]] is a mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,
:<math>r(\varphi) = a\cos\left(k\varphi + \gamma_0\right)</math>

for any constant ɣ<sub>0</sub> (including 0). If ''k'' is an integer, these equations will produce a ''k''-petaled rose if ''k'' is [[even and odd numbers|odd]], or a 2''k''-petaled rose if ''k'' is even. If ''k'' is rational but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. petals. The [[variable (math)|variable]] ''a'' directly represents the length or amplitude of the petals of the rose, while ''k'' relates to their spatial frequency.  The constant ɣ<sub>0</sub> can be regarded as a phase angle.
{{Clear}}

===Archimedean spiral===
[[File:Spiral of Archimedes.svg|thumb|right|One arm of an Archimedean spiral with equation {{nowrap|''r''(''φ'') {{=}} ''φ'' / 2{{pi}} }} for {{nowrap|0 < ''φ'' < 6{{pi}}}}]]
The [[Archimedean spiral]] is a spiral that was discovered by [[Archimedes]], which can also be expressed as a simple polar equation. It is represented by the equation
:<math>r(\varphi) = a+b\varphi. </math>
Changing the parameter ''a'' will turn the spiral, while ''b'' controls the distance between the arms, which for a given spiral is always constant. The Archimedean spiral has two arms, one for {{nowrap|''φ'' > 0}} and one for {{nowrap|''φ'' < 0}}. The two arms are smoothly connected at the pole. Taking the mirror image of one arm across the 90°/270° line will yield the other arm. This curve is notable as one of the first curves, after the [[conic section]]s, to be described in a mathematical treatise, and as being a prime example of a curve that is best defined by a polar equation.
{{Clear}}

===Conic sections===
[[Image:Elps-slr.svg|thumb|right|250px|Ellipse, showing semi-latus rectum]]
A [[conic section]] with one focus on the pole and the other somewhere on the 0° ray (so that the conic's [[semi-major axis|major axis]] lies along the polar axis) is given by:

: <math>r = { \ell\over {1 - e \cos \varphi}}</math>

where ''e'' is the [[eccentricity (mathematics)|eccentricity]] and <math>\ell</math> is the [[semi-latus rectum]] (the perpendicular distance at a focus from the major axis to the curve). If {{nowrap|''e'' > 1}}, this equation defines a [[hyperbola]]; if {{nowrap|''e'' {{=}} 1}}, it defines a [[parabola]]; and if {{nowrap|''e'' < 1}}, it defines an [[ellipse]]. The special case {{nowrap|''e'' {{=}} 0}} of the latter results in a circle of the radius <math>\ell</math>.
{{Clear}}

==Intersection of two polar curves==
The graphs of two polar functions <math>r=f(\theta)</math> and <math>r=g(\theta)</math> have possible intersections of three types:
# In the origin if the equations <math>f(\theta)=0</math> and <math>g(\theta)=0</math> have at least one solution each.
# All the points <math>[g(\theta_i),\theta_i]</math> where <math>\theta_i</math> are the solutions to the equation <math>f(\theta+2k\pi)=g(\theta)</math> where <math>k</math> is an integer.
# All the points <math>[g(\theta_i),\theta_i]</math> where <math>\theta_i</math> are the solutions to the equation <math>f(\theta+(2k+1)\pi)=-g(\theta)</math> where <math>k</math> is an integer.
{{Clear}}

==Complex numbers==
[[Image:Imaginarynumber2.svg|thumb|right|265px|An illustration of a complex number ''z'' plotted on the complex plane]]
[[Image:Euler's formula.svg|thumb|right|265px|An illustration of a complex number plotted on the complex plane using [[Euler's formula]]]]
Every [[complex number]] can be represented as a point in the [[complex plane]], and can therefore be expressed by specifying either the point's Cartesian coordinates (called rectangular or Cartesian form) or the point's polar coordinates (called polar form). The complex number ''z'' can be represented in rectangular form as
: <math>z = x + iy</math>
where ''i'' is the [[imaginary unit]], or can alternatively be written in polar form (via the conversion formulae given [[#Converting between polar and Cartesian coordinates|above]]) as
:<math>z = r(\cos\varphi + i\sin\varphi)</math>
and from there as
: <math>z = re^{i\varphi} </math>
where ''e'' is [[e (mathematical constant)|Euler's number]], which are equivalent as shown by [[Euler's formula]].<ref>
{{Cite book| last = Smith| first = Julius O.| title = Mathematics of the Discrete Fourier Transform (DFT)| accessdate = 2006-09-22| year = 2003| publisher = W3K Publishing| isbn = 0-9745607-0-7| chapter = Euler's Identity| chapterurl = http://ccrma-www.stanford.edu/~jos/mdft/Euler_s_Identity.html| url-status = dead| archiveurl = https://web.archive.org/web/20060915004724/http://ccrma-www.stanford.edu/~jos/mdft/Euler_s_Identity.html| archivedate = 2006-09-15
}}</ref> (Note that this formula, like all those involving exponentials of angles, assumes that the angle ''φ'' is expressed in [[radian]]s.) To convert between the rectangular and polar forms of a complex number, the conversion formulae given [[#Converting between polar and Cartesian coordinates|above]] can be used.

For the operations of [[multiplication]], [[division (mathematics)|division]], and [[exponentiation]] of complex numbers, it is generally much simpler to work with complex numbers expressed in polar form rather than rectangular form. From the laws of exponentiation:

; Multiplication: <math>r_0 e^{i\varphi_0}\, r_1 e^{i\varphi_1} = r_0 r_1 e^{i\left(\varphi_0 + \varphi_1\right)} </math>
; Division: <math>\frac{r_0 e^{i\varphi_0}}{r_1 e^{i\varphi_1}} = \frac{r_0}{r_1}e^{i(\varphi_0 - \varphi_1)} </math>
; Exponentiation ([[De Moivre's formula]]): <math>\left(re^{i\varphi}\right)^n = r^n e^{in\varphi} </math>

==Calculus==
[[Calculus]] can be applied to equations expressed in polar coordinates.<ref>{{cite web|url=http://archives.math.utk.edu/visual.calculus/5/polar.1/index.html|title=Areas Bounded by Polar Curves|author=Husch, Lawrence S.|accessdate=2006-11-25}}</ref><ref>{{cite web|url=http://archives.math.utk.edu/visual.calculus/3/polar.1/index.html|title=Tangent Lines to Polar Graphs|author=Lawrence S. Husch|accessdate=2006-11-25}}</ref>

The angular coordinate ''φ'' is expressed in radians throughout this section, which is the conventional choice when doing calculus.

===Differential calculus===
Using {{nowrap|''x'' {{=}} ''r'' cos ''φ'' }} and {{nowrap|''y'' {{=}} ''r'' sin ''φ'' }}, one can derive a relationship between derivatives in Cartesian and polar coordinates. For a given function, ''u''(''x'',''y''), it follows that (by computing its [[total derivative]]s)
:<math>\begin{align}
  r \frac{\partial u}{\partial r}
    &= r \frac{\partial u}{\partial x}\frac{\partial x}{\partial r} +
       r \frac{\partial u}{\partial y}\frac{\partial y}{\partial r}, \\[2pt]
  \frac{\partial u}{\partial \varphi}
    &= \frac{\partial u}{\partial x}\frac{\partial x}{\partial \varphi} +
       \frac{\partial u}{\partial y}\frac{\partial y}{\partial \varphi},
\end{align}</math>

or
:<math>\begin{align}
  r \frac{\partial u}{\partial r}
    &= r \frac{\partial u}{\partial x} \cos\varphi  +
       r \frac{\partial u}{\partial y} \sin\varphi
     = x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y}, \\[2pt]
  \frac{\partial u}{\partial \varphi}
    &= - \frac{\partial u}{\partial x} r \sin\varphi +
         \frac{\partial u}{\partial y} r \cos\varphi
     = -y \frac{\partial u}{\partial x} + x \frac{\partial u}{\partial y}.
\end{align}</math>

Hence, we have the following formulae:

:<math>\begin{align}
      r \frac{\partial}{\partial r} &=  x \frac{\partial}{\partial x} + y \frac{\partial}{\partial y} \\[2pt]
  \frac{\partial}{\partial \varphi} &= -y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}.
\end{align}</math>

Using the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a function ''u''(''r'',''φ''), it follows that
:<math>\begin{align}
  \frac{\partial u}{\partial x}
    &= \frac{\partial u}{\partial r}\frac{\partial r}{\partial x} +
       \frac{\partial u}{\partial \varphi}\frac{\partial \varphi}{\partial x}, \\[2pt]
  \frac{\partial u}{\partial y}
    &= \frac{\partial u}{\partial r}\frac{\partial r}{\partial y} +
       \frac{\partial u}{\partial \varphi}\frac{\partial \varphi}{\partial y},
\end{align}</math>

or
:<math>\begin{align}
  \frac{\partial u}{\partial x}
    &= \frac{\partial u}{\partial r}\frac{x}{\sqrt{x^2+y^2}} -
       \frac{\partial u}{\partial \varphi}\frac{y}{x^2+y^2} \\[2pt]
    &= \cos \varphi \frac{\partial u}{\partial r} -
       \frac{1}{r} \sin\varphi \frac{\partial u}{\partial \varphi}, \\[2pt]
  \frac{\partial u}{\partial y}
    &= \frac{\partial u}{\partial r}\frac{y}{\sqrt{x^2+y^2}} +
       \frac{\partial u}{\partial \varphi}\frac{x}{x^2+y^2} \\[2pt]
    &= \sin\varphi \frac{\partial u}{\partial r} +
       \frac{1}{r} \cos\varphi \frac{\partial u}{\partial \varphi}.
\end{align}</math>

Hence, we have the following formulae:
:<math>\begin{align}
  \frac{\partial}{\partial x}
    &= \cos \varphi \frac{\partial}{\partial r} -
       \frac{1}{r} \sin\varphi \frac{\partial}{\partial \varphi} \\[2pt]
  \frac{\partial}{\partial y}
    &= \sin \varphi \frac{\partial}{\partial r} +
       \frac{1}{r} \cos\varphi \frac{\partial}{\partial \varphi}.
\end{align}</math>

To find the Cartesian slope of the tangent line to a polar curve ''r''(''φ'') at any given point, the curve is first expressed as a system of [[parametric equations]].
:<math>\begin{align}
  x &= r(\varphi)\cos\varphi \\
  y &= r(\varphi)\sin\varphi
\end{align}</math>

[[Derivative|Differentiating]] both equations with respect to ''φ'' yields
:<math>\begin{align}
  \frac{dx}{d\varphi} &= r'(\varphi)\cos\varphi - r(\varphi)\sin\varphi \\[2pt]
  \frac{dy}{d\varphi} &= r'(\varphi)\sin\varphi + r(\varphi)\cos\varphi.
\end{align}</math>

Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point {{nowrap|(''r''(''φ''),&nbsp;''φ'')}}:
:<math>\frac{dy}{dx} = \frac{r'(\varphi)\sin\varphi + r(\varphi)\cos\varphi}{r'(\varphi)\cos\varphi-r(\varphi)\sin\varphi}.</math>

For other useful formulas including divergence, gradient, and Laplacian in polar coordinates, see [[curvilinear coordinates]].

===Integral calculus (arc length)===
The arc length (length of a line segment) defined by a polar function is found by the integration over the curve ''r''(''φ''). Let ''L'' denote this length along the curve starting from points ''A'' through to point ''B'', where these points correspond to ''φ'' = ''a'' and ''φ'' = ''b'' such that {{nowrap|0 < ''b'' − ''a'' < 2{{pi}}}}. The length of ''L'' is given by the following integral

:<math>L = \int_a^b \sqrt{ \left[r(\varphi)\right]^2 + \left[ {\tfrac{dr(\varphi) }{ d\varphi }} \right] ^2 } d\varphi</math>

===Integral calculus (area)===
[[Image:Polar coordinates integration region.svg|thumb|The integration region ''R'' is bounded by the curve ''r''(''φ'') and the rays ''φ'' = ''a'' and ''φ'' = ''b''.]]
Let ''R'' denote the region enclosed by a curve ''r''(''φ'') and the rays ''φ'' = ''a'' and ''φ'' = ''b'', where {{nowrap|0 < ''b'' − ''a'' ≤ 2{{pi}}}}. Then, the area of ''R'' is

:<math>\frac12\int_a^b \left[r(\varphi)\right]^2\, d\varphi.</math>

[[Image:Polar coordinates integration Riemann sum.svg|thumb|The region ''R'' is approximated by ''n'' sectors (here, ''n'' = 5).]]
[[File:Planimeter.jpg|thumb|A [[planimeter]], which mechanically computes polar integrals]]
This result can be found as follows. First, the interval {{nowrap|[''a'', ''b'']}} is divided into ''n'' subintervals, where ''n'' is an arbitrary positive integer. Thus Δ''φ'', the length of each subinterval, is equal to {{nowrap|''b'' − ''a''}} (the total length of the interval), divided by ''n'', the number of subintervals. For each subinterval ''i'' = 1, 2, ..., ''n'', let ''φ''<sub>''i''</sub> be the midpoint of the subinterval, and construct a [[circular sector|sector]] with the center at the pole, radius ''r''(''φ''<sub>''i''</sub>), central angle Δ''φ'' and arc length ''r''(''φ''<sub>''i''</sub>)Δ''φ''. The area of each constructed sector is therefore equal to
:<math>\left[r(\varphi_i)\right]^2 \pi \cdot \frac{\Delta \varphi}{2\pi} = \frac{1}{2}\left[r(\varphi_i)\right]^2 \Delta \varphi.</math>
Hence, the total area of all of the sectors is
:<math>\sum_{i=1}^n \tfrac12r(\varphi_i)^2\,\Delta\varphi.</math>

As the number of subintervals ''n'' is increased, the approximation of the area continues to improve. In the limit as {{nowrap|''n'' → ∞}}, the sum becomes the [[Riemann sum]] for the above integral.

A mechanical device that computes area integrals is the [[planimeter]], which measures the area of plane figures by tracing them out: this replicates integration in polar coordinates by adding a joint so that the 2-element [[Linkage (mechanical)|linkage]] effects [[Green's theorem]], converting the quadratic polar integral to a linear integral.

====Generalization====
Using [[Cartesian coordinates]], an infinitesimal area element can be calculated as ''dA'' = ''dx'' ''dy''. The [[integration by substitution#Substitution for multiple variables|substitution rule for multiple integrals]] states that, when using other coordinates, the [[Jacobian matrix and determinant|Jacobian]] determinant of the coordinate conversion formula has to be considered:
: <math>J = \det\frac{\partial(x, y)}{\partial(r, \varphi)}
  = \begin{vmatrix}
      \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \varphi} \\[2pt]
      \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \varphi}
    \end{vmatrix}
  = \begin{vmatrix}
      \cos\varphi & -r\sin\varphi \\
      \sin\varphi &  r\cos\varphi
    \end{vmatrix}
  = r\cos^2\varphi + r\sin^2\varphi = r.
</math>

Hence, an area element in polar coordinates can be written as
:<math>dA = dx\,dy\ = J\,dr\,d\varphi = r\,dr\,d\varphi.</math>

Now, a function, that is given in polar coordinates, can be integrated as follows:
:<math>\iint_R f(x, y)\, dA = \int_a^b \int_0^{r(\varphi)} f(r, \varphi)\,r\,dr\,d\varphi.</math>

Here, ''R'' is the same region as above, namely, the region enclosed by a curve ''r''(''ϕ'') and the rays ''φ'' = ''a'' and ''φ'' = ''b''. The formula for the area of ''R'' mentioned above is retrieved by taking ''f'' identically equal to 1.

[[File:The Gaussian integral.svg|thumb|The Gaussian integral]]

A more surprising application of this result yields the [[Gaussian integral]], here denoted ''K'':
:<math>K=\int_{-\infty}^\infty e^{-x^2} \, dx = \sqrt\pi.</math>

===Vector calculus===
[[Vector calculus]] can also be applied to polar coordinates. For a planar motion, let <math>\mathbf{r}</math> be the position vector {{nowrap|(''r''&nbsp;cos(''φ''), ''r''&nbsp;sin(''φ''))}}, with ''r'' and ''φ'' depending on time ''t''.

We define the unit vectors
:<math>\hat{\mathbf{r}} = (\cos(\varphi), \sin(\varphi))</math>

in the direction of <math>\mathbf{r}</math> and
:<math>\hat{\boldsymbol{\varphi}} = (-\sin(\varphi), \cos(\varphi)) = \hat{\mathbf{k}} \times \hat{\mathbf{r}} \ ,</math>

in the plane of the motion perpendicular to the radial direction, where <math>\hat{\mathbf{k}}</math> is a unit vector normal to the plane of the motion.

Then
:<math>\begin{align}
         \mathbf{r} &= (x,\ y) = r(\cos\varphi,\ \sin\varphi) = r \hat{\mathbf{r}}\ , \\
   \dot{\mathbf{r}} &= \left(\dot{x},\ \dot{y}\right) = \dot{r}(\cos\varphi,\ \sin\varphi) + r\dot{\varphi}(-\sin\varphi,\ \cos\varphi) = \dot{r}\hat{\mathbf{r}} + r\dot{\varphi}\hat{\boldsymbol{\varphi}}\ ,\\
  \ddot{\mathbf{r}} &= \left(\ddot{x},\ \ddot{y}\right) \\
                    &= \ddot{r}(\cos\varphi,\ \sin\varphi) + 2\dot{r}\dot{\varphi}(-\sin\varphi,\ \cos\varphi) +  r\ddot{\varphi}(-\sin\varphi,\ \cos\varphi) - r\dot{\varphi}^2(\cos\varphi,\ \sin\varphi) \\
                    &= \left(\ddot{r} - r\dot{\varphi}^2\right) \hat{\mathbf{r}} + \left(r\ddot{\varphi} + 2\dot{r}\dot{\varphi}\right) \hat{\boldsymbol{\varphi}} \\
                    &= \left(\ddot{r} - r\dot{\varphi}^2\right) \hat{\mathbf{r}} + \frac{1}{r}\; \frac{d}{dt} \left(r^2\dot{\varphi}\right) \hat{\boldsymbol{\varphi}}.
\end{align}</math>

====Centrifugal and Coriolis terms====
{{See also|Mechanics of planar particle motion|Centrifugal force (rotating reference frame)}}
{{multiple image
|align = vertical
|width1 = 100
|image1 = Position vector plane polar coords.svg
|caption1 = Position vector '''r''', always points radially from the origin.
|width2 = 150
|image2 = Velocity vector plane polar coords.svg
|caption2 = Velocity vector '''v''', always tangent to the path of motion.
|width3 = 200
|image3 = Acceleration vector plane polar coords.svg
|caption3 = Acceleration vector '''a''', not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.
|footer = Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 2d space, but a plane in any higher dimension.
}}

The term <math>r\dot\varphi^2</math> is sometimes referred to as the ''centripetal acceleration'', and the term <math>2\dot r \dot\varphi</math> as the ''Coriolis acceleration''. For example, see Shankar.<ref name=Shankar>{{Cite book|title=Principles of Quantum Mechanics|author=Ramamurti Shankar|edition=2nd|page=81|url=https://books.google.com/?id=2zypV5EbKuIC&pg=PA81&dq=Coriolis+%22polar+coordinates%22|year=1994|isbn=0-306-44790-8|publisher=Springer}}</ref>

Note: these terms, that appear when acceleration is expressed in polar coordinates, are a mathematical consequence of differentiation; they appear whenever polar coordinates are used. In planar particle dynamics these accelerations appear when setting up Newton's [[Newton's second law|second law of motion]] in a rotating frame of reference. Here these extra terms are often called [[fictitious force]]s; fictitious because they are simply a result of a change in coordinate frame. That does not mean they do not exist, rather they exist only in the rotating frame.

[[Image:Co-rotating frame vector.svg|thumb|Inertial frame of reference ''S'' and instantaneous non-inertial co-rotating frame of reference ''S′''. The co-rotating frame rotates at angular rate &Omega; equal to the rate of rotation of the particle about the origin of ''S′'' at the particular moment ''t''. Particle is located at vector position ''r''(''t'') and unit vectors are shown in the radial direction to the particle from the origin, and also in the direction of increasing angle ''ϕ'' normal to the radial direction. These unit vectors need not be related to the tangent and normal to the path. Also, the radial distance ''r'' need not be related to the radius of curvature of the path.]]

=====Co-rotating frame=====
For a particle in planar motion, one approach to attaching physical significance to these terms is based on the concept of an instantaneous ''co-rotating frame of reference''.<ref name=Taylor>For the following discussion, see {{Cite book|author=John R Taylor|title=Classical Mechanics|page=§9.10, pp. 358–359|isbn=1-891389-22-X|publisher=University Science Books|year=2005}}</ref> To define a co-rotating frame, first an origin is selected from which the distance ''r''(''t'') to the particle is defined. An axis of rotation is set up that is perpendicular to the plane of motion of the particle, and passing through this origin. Then, at the selected moment ''t'', the rate of rotation of the co-rotating frame Ω is made to match the rate of rotation of the particle about this axis, ''dφ''/''dt''. Next, the terms in the acceleration in the inertial frame are related to those in the co-rotating frame. Let the location of the particle in the inertial frame be (''r(''t''), ''φ''(''t'')), and in the co-rotating frame be (''r(t), ''φ''′(t)''). Because the co-rotating frame rotates at the same rate as the particle, ''dφ''′/''dt'' = 0. The fictitious centrifugal force in the co-rotating frame is ''mrΩ<sup>2</sup>, radially outward. The velocity of the particle in the co-rotating frame also is radially outward, because ''dφ''′/''dt'' = 0. The ''fictitious Coriolis force'' therefore has a value −2''m''(''dr''/''dt'')Ω, pointed in the direction of increasing ''φ'' only. Thus, using these forces in Newton's second law we find:
:<math>\boldsymbol{F} + \boldsymbol{F}_\text{cf} + \boldsymbol{F}_\text{Cor} = m\ddot{\boldsymbol{r}} \ , </math>

where over dots represent time differentiations, and '''F''' is the net real force (as opposed to the fictitious forces). In terms of components, this vector equation becomes:
:<math>\begin{align}
             F_r + mr\Omega^2 &= m\ddot{r} \\
  F_\varphi - 2m\dot{r}\Omega &= mr\ddot{\varphi} \ ,
\end{align}</math>

which can be compared to the equations for the inertial frame:
:<math>\begin{align}
        F_r &= m\ddot{r} - mr\dot{\varphi}^2 \\
  F_\varphi &= mr\ddot{\varphi} + 2m\dot{r}\dot{\varphi} \ .
\end{align}</math>

This comparison, plus the recognition that by the definition of the co-rotating frame at time ''t'' it has a rate of rotation Ω = ''dφ''/''dt'', shows that we can interpret the terms in the acceleration (multiplied by the mass of the particle) as found in the inertial frame as the negative of the centrifugal and Coriolis forces that would be seen in the instantaneous, non-inertial co-rotating frame.

For general motion of a particle (as opposed to simple circular motion), the centrifugal and Coriolis forces in a particle's frame of reference commonly are referred to the instantaneous [[osculating circle]] of its motion, not to a fixed center of polar coordinates. For more detail, see [[Centripetal force#Local coordinates|centripetal force]].

== Differential geometry ==
In the modern terminology of [[differential geometry]], polar coordinates provide [[coordinate charts]] for the [[differentiable manifold]] ℝ<sup>2</sup> \ {(0,0)}, the plane minus the origin. In these coordinates, the Euclidean [[metric tensor]] is given by<math display="block">ds^2 = dr^2 + r^2 d\theta^2.</math>This can be seen via the change of variables formula for the metric tensor, or by computing the [[differential form]]s ''dx'', ''dy'' via the [[exterior derivative]] of the 0-forms ''x = r cos(θ), y = r sin(θ)'' and substituting them in the Euclidean metric tensor ''ds<sup>2</sup> = dx<sup>2</sup> + dy<sup>2</sup>''. An [[Orthonormality|orthonormal]] [[Moving frame|frame]] with respect to this metric is given by<math display="block">e_r = \frac{\partial}{\partial r}, \quad e_\theta = \frac1r \frac{\partial}{\partial \theta},</math>with [[Moving frame#Coframes|dual coframe]]<math display="block">e^r = dr, \quad e^\theta = r d\theta.</math>The [[connection form]] relative to this frame and the [[Levi-Civita connection]] is given by the skew-symmetric matrix of 1-forms<math display="block">\omega^i_j = \begin{pmatrix} 0 & -d\theta \\ d\theta & 0\end{pmatrix}</math>and hence the [[curvature form]] Ω = dω + ω∧ω vanishes identically. Therefore, as expected, the punctured plane is a [[flat manifold]].

==Extensions in 3D==
The polar coordinate system is extended into three dimensions with two different coordinate systems, the [[cylindrical coordinate system|cylindrical]] and [[spherical coordinate system]].

==Applications==
Polar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane. They are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point. For instance, the examples above show how elementary polar equations suffice to define curves—such as the Archimedean spiral—whose equation in the Cartesian coordinate system would be much more intricate. Moreover, many physical systems—such as those concerned with bodies moving around a central point or with phenomena originating from a central point—are simpler and more intuitive to model using polar coordinates. The initial motivation for the introduction of the polar system was the study of [[circular motion|circular]] and [[orbital motion]].

===Position and navigation===
Polar coordinates are used often in [[navigation]] as the destination or direction of travel can be given as an angle and distance from the object being considered. For instance, [[aircraft]] use a slightly modified version of the polar coordinates for navigation. In this system, the one generally used for any sort of navigation, the 0° ray is generally called heading 360, and the angles continue in a [[clockwise]] direction, rather than counterclockwise, as in the mathematical system. Heading 360 corresponds to [[magnetic north]], while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively.<ref>{{cite web|url=http://www.thaitechnics.com/nav/adf.html|title=Aircraft Navigation System|accessdate=2006-11-26|first=Sumrit|last=Santhi}}</ref> Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read [[ICAO spelling alphabet|zero-niner-zero]] by [[air traffic control]]).<ref>{{cite web|url=https://www.faa.gov/regulations_policies/handbooks_manuals/aircraft/airplane_handbook/media/faa-h-8083-3a-7of7.pdf|title=Emergency Procedures|format=PDF|accessdate=2007-01-15|url-status=dead|archiveurl=https://web.archive.org/web/20130603111635/http://www.faa.gov/regulations_policies/handbooks_manuals/aircraft/airplane_handbook/media/faa-h-8083-3a-7of7.pdf|archivedate=2013-06-03}}</ref>

===Modeling===
Systems displaying [[radial symmetry]] provide natural settings for the polar coordinate system, with the central point acting as the pole. A prime example of this usage is the [[groundwater flow equation]] when applied to radially symmetric wells. Systems with a [[central force|radial force]] are also good candidates for the use of the polar coordinate system. These systems include [[gravitation|gravitational fields]], which obey the [[inverse-square law]], as well as systems with [[point source]]s, such as [[antenna (radio)|radio antennas]].

Radially asymmetric systems may also be modeled with polar coordinates. For example, a [[microphone]]'s [[Microphone pick up patterns|pickup pattern]] illustrates its proportional response to an incoming sound from a given direction, and these patterns can be represented as polar curves. The curve for a standard cardioid microphone, the most common unidirectional microphone, can be represented as {{nowrap|''r'' {{=}} 0.5 + 0.5sin(''ϕ'')}} at its target design frequency.<ref>{{Cite book|last=Eargle|first=John|authorlink=John M. Eargle|title=Handbook of Recording Engineering|year=2005|edition=Fourth|publisher=Springer|isbn = 0-387-28470-2}}</ref> The pattern shifts toward omnidirectionality at lower frequencies.

==See also==
{{Portal|Mathematics}}
*[[Curvilinear coordinates]]
*[[List of canonical coordinate transformations]]
*[[Log-polar coordinates]]
*[[Polar decomposition]]
*[[Unit circle]]

==References==
'''Specific'''
{{Reflist|30em}}

'''General'''
{{refbegin}}
*{{Cite book|last=Adams|first=Robert|author2=Christopher Essex|title=Calculus: a complete course|edition=Eighth|year=2013|publisher=Pearson Canada Inc.|isbn=978-0-321-78107-9}}
*{{Cite book|last=Anton|first=Howard|author2=Irl Bivens |author3=Stephen Davis |title=Calculus|edition=Seventh|year=2002|publisher=Anton Textbooks, Inc.|isbn=0-471-38157-8}}
*{{Cite book|last=Finney|first=Ross|author2=George Thomas|author3=Franklin Demana|author4=Bert Waits|title=Calculus: Graphical, Numerical, Algebraic|edition=Single Variable Version|date=June 1994|publisher=Addison-Wesley Publishing Co.|isbn=0-201-55478-X|url=https://archive.org/details/calculusgraphica00ross}}
{{refend}}

==External links==
{{wikibooks|Calculus|Polar Integration}}
* {{springer|title=Polar coordinates|id=p/p073410}}
*{{dmoz|Science/Math/Software/Graphing/|Graphing Software}}
*[http://www.random-science-tools.com/maths/coordinate-converter.htm Coordinate Converter &mdash; converts between polar, Cartesian and spherical coordinates]
*[http://scratch.mit.edu/projects/nevit/691690 Polar Coordinate System Dynamic Demo]

{{Orthogonal coordinate systems}}

{{DEFAULTSORT:Polar Coordinate System}}
[[Category:Two-dimensional coordinate systems]]