In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the
polar coordinate system is a
two-dimensionalIn physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
coordinate systemIn geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
in which each point on a
planeIn mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
is determined by a
distanceDistance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...
from a fixed point and an
angleIn geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
from a fixed direction.
The fixed point (analogous to the origin of a
Cartesian systemA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
) is called the
pole, and the ray from the pole in the fixed direction is the
polar axis. The distance from the pole is called the
radial coordinate or
radius, and the angle is the
angular coordinate,
polar angle, or
azimuth.
History
The concepts of angle and radius were already used by ancient peoples of the 1st millennium BCE. The
Greek astronomerGreek astronomy is astronomy written in the Greek language in classical antiquity. Greek astronomy is understood to include the ancient Greek, Hellenistic, Greco-Roman, and Late Antiquity eras. It is not limited geographically to Greece or to ethnic Greeks, as the Greek language had become the...
and
astrologerAn astrologer practices one or more forms of astrology. Typically an astrologer draws a horoscope for the time of an event, such as a person's birth, and interprets celestial points and their placements at the time of the event to better understand someone, determine the auspiciousness of an...
HipparchusHipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created** Hipparchus , a lunar crater named in his honour...
(190-120 BCE) created a table of
chordA chord of a circle is a geometric line segment whose endpoints both lie on the circumference of the circle.A secant or a secant line is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse...
functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions.
In
On SpiralsOn Spirals is a treatise by Archimedes in 225 BC. Although Archimedes did not discover the Archimedean spiral, he employed it in this book to square the circle and trisect an angle.-Preface:...
,
ArchimedesArchimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
describes the
Archimedean spiralThe Archimedean spiral is a spiral named after the 3rd century BC Greek mathematician Archimedes. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity...
, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system.
From the 8th century CE onward, astronomers developed methods for approximating and calculating the direction to Makkah (
qiblaThe Qiblah , also transliterated as Qibla, Kiblah or Kibla, is the direction that should be faced when a Muslim prays during salah...
)—and its distance—from any location on the Earth. From the 9th century onward they were using
spherical trigonometrySpherical trigonometry is a branch of spherical geometry which deals with polygons on the sphere and the relationships between the sides and the angles...
and
map projectionA map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...
methods to determine these quantities accurately. The calculation is essentially the conversion of the equatorial polar coordinates of Mecca (i.e. its
longitudeLongitude is a geographic coordinate that specifies the east-west position of a point on the Earth's surface. It is an angular measurement, usually expressed in degrees, minutes and seconds, and denoted by the Greek letter lambda ....
and
latitudeIn geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
) 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 pointIn mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite to it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter....
.
The Persian geographer, Abū Rayhān Bīrūnī (973-1048), developed ideas which are seen as an anticipation of the polar coordinate system. Around 1025 CE, he was the first to describe a polar equi-
azimuthal equidistant projectionThe azimuthal equidistant projection is a type of map projection.A useful application for this type of projection is a polar projection in which all distances measured from the center of the map along any longitudinal line are accurate; an example of a polar azimuthal equidistant projection can be...
of the
celestial sphereIn astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the...
.
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
HarvardHarvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
professor Julian Lowell Coolidge's
Origin of Polar Coordinates. Grégoire de Saint-VincentGrégoire de Saint-Vincent , a Jesuit, was a mathematician who discovered that the area under a rectangular hyperbola is the same over [a,b] as over [c,d] when a/b = c/d...
and
Bonaventura CavalieriBonaventura Francesco Cavalieri was an Italian mathematician. He is known for his work on the problems of optics and motion, work on the precursors of infinitesimal calculus, and the introduction of logarithms to Italy...
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 spiralThe Archimedean spiral is a spiral named after the 3rd century BC Greek mathematician Archimedes. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity...
.
Blaise Pascal Blaise Pascal , was a French mathematician, physicist, inventor, writer and Catholic philosopher. He was a child prodigy who was educated by his father, a tax collector in Rouen...
subsequently used polar coordinates to calculate the length of
parabolic arcsIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
.
In
Method of FluxionsMethod of Fluxions is a book by Isaac Newton. The book was completed in 1671, and published in 1736. Fluxions is Newton's term for differential calculus...
(written 1671, published 1736), Sir
Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems. In the journal
Acta EruditorumActa Eruditorum was the first scientific journal of the German lands, published from 1682 to 1782....
(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 curvatureIn geometry, the radius of curvature, R, of a curve at a point is a measure of the radius of the circular arc which best approximates the curve at that point. If this value taken to be positive when the curve turns anticlockwise and negative when the curve turns clockwise...
of curves expressed in these coordinates.
The actual term
polar coordinates has been attributed to
Gregorio FontanaGregorio Fontana was an Italian mathematician. He was chair of mathematics at the university of Pavia succeeding Roger Joseph Boscovich. He has been credited with the introduction of polar coordinates....
and was used by 18th-century Italian writers. The term appeared in
EnglishEnglish is a West Germanic language that arose in the Anglo-Saxon kingdoms of England and spread into what was to become south-east Scotland under the influence of the Anglian medieval kingdom of Northumbria...
in
George PeacockGeorge Peacock was an English mathematician.-Life:Peacock was born on 9 April 1791 at Thornton Hall, Denton, near Darlington, County Durham. His father, the Rev. Thomas Peacock, was a clergyman of the Church of England, incumbent and for 50 years curate of the parish of Denton, where he also kept...
's 1816 translation of
LacroixSylvestre François Lacroix was a French mathematician.He was born in Paris, and was raised in a poor family who still managed to obtain a good education for their son. He displayed a particular talent for mathematics, calculating the motions of theplanets by the age of 14...
's
Differential and Integral Calculus. Alexis Clairaut was the first to think of polar coordinates in three dimensions, and
Leonhard EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
was the first to actually develop them.
Common conventions
The radial coordinate is often denoted by
r, and the angular coordinate by
θ or
t.
Angles in polar notation are generally expressed in either
degreeA degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
s or
radianRadian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s (2
π' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
rad being equal to 360°). Degrees are traditionally used in
navigationNavigation is the process of monitoring and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks...
,
surveyingSee Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them...
, and many applied disciplines, while radians are more common in mathematics and mathematical
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
.
In many contexts, a positive angular coordinate means that the angle
θ is measured
counterclockwiseCircular motion can occur in two possible directions. A clockwise motion is one that proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back to the top...
from the axis.
In mathematical literature, the polar axis is often drawn horizontal and pointing to the right.
Uniqueness of polar coordinates
Adding any number of full turns (360°) to the angular coordinate does not change the corresponding direction. Also, a negative radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Therefore, the same point can be expressed with an infinite number of different polar coordinates or , where
n is any
integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
. Moreover, the pole itself can be expressed as (0,
θ) for any angle
θ.
Where a unique representation is needed for any point, it is usual to limit
r to non-negative numbers and
θ to the
intervalIn mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
[0, 360°) or (−180°, 180°] (in radians, [0, 2π) or (−π, π]). One must also choose a unique azimuth for the pole, e.g.,
θ = 0.
Converting between polar and Cartesian coordinates
The two polar coordinates
r and
θ can be converted to the two Cartesian coordinates
x and
y by using the
trigonometric functionIn mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
s sine and cosine:
while the
Cartesian coordinatesA Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length...
x and
y can be converted to the polar coordinates
r and
θ by:
(as in the
Pythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
), and
(where
atan2In trigonometry, the two-argument function atan2 is a variation of the arctangent function. For any real arguments and not both equal to zero, is the angle in radians between the positive -axis of a plane and the point given by the coordinates on it...
is a common variation on the arctangent function that takes into account the quadrant)
or
This gives the
θ in radians in the interval (−π, π]. In degrees this would be from -180° to 180°. These formulae assume that the pole is the Cartesian origin (0,0), that the polar axis is the Cartesian
x axis, and that the direction of the Cartesian
y axis has azimuth +π/2 radians = +90°.
Many programming languages have a function that will compute the correct angular coordinate
θ given
x and
y. For example, this function is called by
atan2In trigonometry, the two-argument function atan2 is a variation of the arctangent function. For any real arguments and not both equal to zero, is the angle in radians between the positive -axis of a plane and the point given by the coordinates on it...
(
y,
x) in the
C programming languageC is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system....
, and (
atanIn trigonometry, the two-argument function atan2 is a variation of the arctangent function. For any real arguments and not both equal to zero, is the angle in radians between the positive -axis of a plane and the point given by the coordinates on it...
y x) in
Common LispCommon Lisp, commonly abbreviated CL, is a dialect of the Lisp programming language, published in ANSI standard document ANSI INCITS 226-1994 , . From the ANSI Common Lisp standard the Common Lisp HyperSpec has been derived for use with web browsers...
. In both cases, the result is an angle in radians in the range (−π, π]. If desired an angle in the range [0, 2π) may be obtained by adding 2π to the value if it is negative.
The value of
θ above is the
principal valueIn considering complex multiple-valued functions in complex analysis, the principal values of a function are the values along one chosen branch of that function, so it is single-valued.-Motivation:...
of the complex number function arg except that arg does not define a value at the origin when
x and
y are zero. The value zero above is just a convenient value that is often chosen.
Polar equation of a curve
The equation defining an
algebraic curveIn algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
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
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
of
θ. The resulting curve then consists of points of the form (
r(
θ),
θ) and can be regarded as the
graphIn mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...
of the polar function
r.
Different forms of
symmetrySymmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
can be deduced from the equation of a polar function
r. If the curve will be symmetrical about the horizontal (0°/180°) ray, if it will be symmetric about the vertical (90°/270°) ray, and if it will be
rotationally symmetricGenerally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...
α
counterclockwiseCircular motion can occur in two possible directions. A clockwise motion is one that proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back to the top...
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
polar roseIn mathematics, a rose or rhodonea curve is a sinusoid plotted in polar coordinates. Up to similarity, thesecurves can all be expressed by a polar equation of the form\!\,r=\cos.If k is an integer, the curve will be rose shaped with...
,
Archimedean spiralThe Archimedean spiral is a spiral named after the 3rd century BC Greek mathematician Archimedes. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity...
,
lemniscateIn geometry, the lemniscate of Bernoulli is a plane curve defined from two given points F1 and F2, known as foci, at distance 2a from each other as the locus of points P so that PF1·PF2 = a2. The curve has a shape similar to the numeral 8 and to the ∞ symbol. Its name is from lemniscus, which is...
,
limaçonIn geometry, a limaçon or limacon , also known as a limaçon of Pascal, is defined as a roulette formed when a circle rolls around the outside of a circle of equal radius. It can also be defined as the roulette formed when a circle rolls around a circle with half its radius so that the smaller...
, and
cardioidA cardioid is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It is therefore a type of limaçon and can also be defined as an epicycloid having a single cusp...
.
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
The general equation for a circle with a center at and radius
a is
This can be simplified in various ways, to conform to more specific cases, such as the equation
for a circle with a center at the pole and radius
a.
When
0 = , or when the origin lies on the circle, the equation becomes
.
In the general case, the equation can be solved for , giving
,
the solution with a minus sign in front of the square root giving the same curve.
Line
Radial lines (those running through the pole) are represented by the equation
,
where φ is the angle of elevation of the line; that is, where
m is the
slopeIn mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....
of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line
perpendicularIn geometry, two lines or planes are considered perpendicular to each other if they form congruent adjacent angles . The term may be used as a noun or adjective...
ly at the point (
r0, φ) has the equation
Otherwise stated (
r0, φ) is the point in which the tangent intersects the imaginary circle of radius
r0.
Polar rose
A
polar roseIn mathematics, a rose or rhodonea curve is a sinusoid plotted in polar coordinates. Up to similarity, thesecurves can all be expressed by a polar equation of the form\!\,r=\cos.If k is an integer, the curve will be rose shaped with...
is a famous mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,
for any constant φ
0 (including 0). If
k is an integer, these equations will produce a
k-petaled rose if
k is
oddIn mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...
, 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
a represents the length of the petals of the rose.
Archimedean spiral
The
Archimedean spiralThe Archimedean spiral is a spiral named after the 3rd century BC Greek mathematician Archimedes. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity...
is a famous spiral that was discovered by
ArchimedesArchimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
, which also can be expressed as a simple polar equation. It is represented by the equation
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 and one for . 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 sectionIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
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.
Conic sections
A conic section with one focus on the pole and the other somewhere on the 0° ray (so that the conic's
major axisThe major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...
lies along the polar axis) is given by:
-
where
e is the
eccentricityIn mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...
and
is the semi-latus rectum (the perpendicular distance at a focus from the major axis to the curve). If , this equation defines a
hyperbolaIn mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
; if , it defines a
parabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
; and if , it defines an
ellipseIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
. The special case of the latter results in a circle of radius
.
Complex numbers
Every
complex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
can be represented as a point in the
complex planeIn mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
, 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
-
where
i is the
imaginary unitIn mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
, or can alternatively be written in polar form (via the conversion formulae given above) as
and from there as
-
where
e is
Euler's numberThe mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...
, which are equivalent as shown by
Euler's formulaEuler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
. (Note that this formula, like all those involving exponentials of angles, assumes that the angle
θ is expressed in
radianRadian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s.) To convert between the rectangular and polar forms of a complex number, the conversion formulae given above can be used.
For the operations of
multiplicationMultiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
,
divisionright|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
, and
exponentiationExponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
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:
-
-
-
- Exponentiation (De Moivre's formula
In mathematics, de Moivre's formula , named after Abraham de Moivre, states that for any complex number x and integer n it holds that...
):
-
Calculus
CalculusCalculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
can be applied to equations expressed in polar coordinates.
The angular coordinate
θ is expressed in radians throughout this section, which is the conventional choice when doing calculus.
Differential calculus
Using and , one can derive a relationship between derivatives in Cartesian and polar coordinates. For a given function,
u(
x,
y), it follows that
or
Hence, we have the following formulae:
To find the Cartesian slope of the tangent line to a polar curve
r(
θ) at any given point, the curve is first expressed as a system of parametric equations.
DifferentiatingIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
both equations with respect to
θ yields
Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point :
For other useful formulas including divergence, gradient, and Laplacian in polar coordinates, see
curvilinear coordinatesCurvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally invertible at each point. This means that one can convert a point given...
.
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(
θ). Let
L denote this length along the curve starting from points
A through to point
B, where these points correspond to
θ =
a and
θ =
b such that . The length of
L is given by the following integral
Integral calculus (Area)
Let
R denote the region enclosed by a curve
r(
θ) and the rays
θ =
a and
θ =
b, where . Then, the area of
R is
This result can be found as follows. First, the interval is divided into
n subintervals, where
n is an arbitrary positive integer. Thus Δ
θ, the length of each subinterval, is equal to (the total length of the interval), divided by
n, the number of subintervals. For each subinterval
i = 1, 2, …,
n, let
θi be the midpoint of the subinterval, and construct a
sectorA circular sector or circle sector, is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. In the diagram, θ is the central angle in radians, r the radius of the circle, and L is the arc length of the...
with the center at the pole, radius
r(
θi), central angle Δ
θ and arc length
r(
θi)Δ
θ. The area of each constructed sector is therefore equal to
Hence, the total area of all of the sectors is
As the number of subintervals
n is increased, the approximation of the area continues to improve. In the limit as , the sum becomes the
Riemann sumIn mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It mayalso be used to define the integration operation. The method was named after German mathematician Bernhard Riemann....
for the above integral.
A mechanical device that computes area integrals is the
planimeterA planimeter is a measuring instrument used to determine the area of an arbitrary two-dimensional shape.-Construction:There are several kinds of planimeters, but all operate in a similar way. The precise way in which they are constructed varies, with the main types of mechanical planimeter being...
, 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
linkageA mechanical linkage is an assembly of bodies connected together to manage forces and movement. The movement of a body, or link, is studied using geometry so the link is considered to be rigid. The connections between links are modeled as providing ideal movement, pure rotation or sliding for...
effects
Green's theoremIn mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C...
, 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
substitution ruleIn calculus, integration by substitution is a method for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool for mathematicians...
for multiple integrals states that, when using other coordinates, the Jacobian determinant of the coordinate conversion formula has to be considered:
-
Hence, an area element in polar coordinates can be written as
Now, a function that is given in polar coordinates can be integrated as follows:
Here, R is the same region as above, namely, the region enclosed by a curve r(θ) and the rays θ = a and θ = b.
The formula for the area of R mentioned above is retrieved by taking f identically equal to 1. A more surprising application of this result yields the Gaussian integralThe Gaussian integral, also known as the Euler-Poisson integral or Poisson integral, is the integral of the Gaussian function e−x2 over the entire real line.It is named after the German mathematician and...
Vector calculus
Vector calculus can also be applied to polar coordinates. For a planar motion, let 
be the position vector , with r and θ depending on time t.
We define the unit vectors
in the direction of r and
in the plane of the motion perpendicular to the radial direction, where 
is a unit vector normal to the plane of the motion.
Then
-
where h is the specific angular momentum.
Centrifugal and Coriolis terms
The term 
is sometimes referred to as the centrifugal term, and the term 
as the Coriolis term. For example, see Shankar. Although these equations bear some resemblance in form to the centrifugalCentrifugal force can generally be any force directed outward relative to some origin. More particularly, in classical mechanics, the centrifugal force is an outward force which arises when describing the motion of objects in a rotating reference frame...
and Coriolis effectIn physics, the Coriolis effect is a deflection of moving objects when they are viewed in a rotating reference frame. In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right...
s found in rotating reference frames, nonetheless these are not the same things. For example, the physical centrifugal and Coriolis forces appear only in non-inertial frames of reference. In contrast, these terms that appear when acceleration is expressed in polar coordinates are a mathematical consequence of differentiation; these terms appear wherever polar coordinates are used. In particular, these terms appear even when polar coordinates are used in inertial frames of reference, where the physical centrifugal and Coriolis forces never appear.
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. 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, dθ/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), θ(t)), and in the co-rotating frame be (r(t), θ′(t)). Because the co-rotating frame rotates at the same rate as the particle, dθ′/dt = 0. The fictitious centrifugal force in the co-rotating frame is mrΩ2, radially outward. The velocity of the particle in the co-rotating frame also is radially outward, because dθ′/dt = 0. The fictitious Coriolis force therefore has a value −2m(dr/dt)Ω, pointed in the direction of increasing θ only. Thus, using these forces in Newton's second law we find:
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:
which can be compared to the equations for the inertial frame:
This comparison, plus the recognition that by the definition of the co-rotating frame at time t it has a rate of rotation Ω = dθ/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 circleIn differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p...
of its motion, not to a fixed center of polar coordinates. For more detail, see centripetal force.
Connection to spherical and cylindrical coordinates
The polar coordinate system is extended into three dimensions with two different coordinate systems, the cylindricalA cylindrical coordinate system is a three-dimensional coordinate systemthat specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis...
and spherical coordinate systemIn mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its inclination angle measured from a fixed zenith direction, and the azimuth angle of...
.
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 circularIn physics, circular motion is rotation along a circular path or a circular orbit. It can be uniform, that is, with constant angular rate of rotation , or non-uniform, that is, with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of...
and orbital motion.
Position and navigation
Polar coordinates are used often in navigationNavigation is the process of monitoring and controlling the movement of a craft or vehicle from one place to another. It is also the term of art used for the specialized knowledge used by navigators to perform navigation tasks...
, as the destination or direction of travel can be given as an angle and distance from the object being considered. For instance, aircraftAn aircraft is a vehicle that is able to fly by gaining support from the air, or, in general, the atmosphere of a planet. An aircraft counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines.Although...
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 clockwiseCircular motion can occur in two possible directions. A clockwise motion is one that proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back to the top...
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. Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read zero-niner-zero by air traffic controlAir traffic control is a service provided by ground-based controllers who direct aircraft on the ground and in the air. The primary purpose of ATC systems worldwide is to separate aircraft to prevent collisions, to organize and expedite the flow of traffic, and to provide information and other...
).
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 equationUsed in hydrogeology, the groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through an aquifer. The transient flow of groundwater is described by a form of the diffusion equation, similar to that used in heat transfer to describe the flow...
when applied to radially symmetric wells. Systems with a radial force are also good candidates for the use of the polar coordinate system. These systems include gravitational fieldsGravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
, which obey the inverse-square lawIn physics, an inverse-square law is any physical law stating that a specified physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity....
, as well as systems with point sourceA point source is a localised, relatively small source of something.Point source may also refer to:*Point source , a localised source of pollution**Point source water pollution, water pollution with a localized source...
s, such as radio antennasAn antenna is an electrical device which converts electric currents into radio waves, and vice versa. It is usually used with a radio transmitter or radio receiver...
.
Radially asymmetric systems may also be modeled with polar coordinates. For example, a microphoneA microphone is an acoustic-to-electric transducer or sensor that converts sound into an electrical signal. In 1877, Emile Berliner invented the first microphone used as a telephone voice transmitter...
's 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 at its target design frequency. The pattern shifts toward omnidirectionality at lower frequencies.
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