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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the polar coordinate system is a
two-dimensional 300px, Bi-dimensional Cartesian coordinate system Two-dimensional space (also known as 2D space, 2-space, or bi-dimensional space) is a geometric setting in which two values (called parameter A parameter (), generally, is any characteristic ...

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

coordinate system
in which each
point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...
on a
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
is determined by a
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

distance
from a reference point and an
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

angle
from a reference direction. The reference point (analogous to the origin of a
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early fly ...
) is called the ''pole'', and the
ray Ray may refer to: Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (graph theory), an infinite sequence of vertices such that each vertex appears at most once in the sequence and each two consecutive ...
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 An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandria ) , name = Al ...

azimuth
''. Angles in polar notation are generally expressed in either
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
s or
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

radian
s (2

rad being equal to 360°).
Grégoire de Saint-Vincent Grégoire de Saint-Vincent - in latin : Gregorius a Sancto Vincentio, in dutch : Gregorius van St-Vincent - (8 September 1584 Bruges Bruges ( , nl, Brugge ; ; german: Brügge ) is the capital and largest city of the province A province is a ...
and
Bonaventura Cavalieri Bonaventura Francesco Cavalieri ( la, Bonaventura Cavalerius; 1598 – 30 November 1647) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) ...

Bonaventura Cavalieri
independently introduced the concepts in the mid-17th century, though the actual term ''polar coordinates'' has been attributed to
Gregorio Fontana Gregorio Fontana, born Giovanni Battista Lorenzo Fontana (7 December 1735 – 24 August 1803) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) ...

Gregorio Fontana
in the 18th century. The initial motivation for the introduction of the polar system was the study of
circular Circular may refer to: * The shape of a circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre; equivalently it is ...

circular
and
orbital motion In celestial mechanics, an orbit is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or pos ...

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 in a plane, such as
spiral In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:
s. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates. The polar coordinate system is extended to three dimensions in two ways: the
cylindrical A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base. This traditi ...

cylindrical
and
spherical of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a geometrical object in three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values ...

spherical
coordinate systems.


History

The concepts of angle and radius were already used by ancient peoples of the first millennium BC. The Greek astronomer and
astrologer Astrology is a pseudoscience Pseudoscience consists of statements, beliefs, or practices that claim to be both scientific and factual but are incompatible with the scientific method The scientific method is an Empirical evidenc ...

astrologer
Hipparchus Hipparchus of Nicaea (; el, Ἵππαρχος, ''Hipparkhos'';  BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of precession of the ...
(190–120 BC) created a table of
chord Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord (ast ...
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 Spirals ''On Spirals'' ( el, Περὶ ἑλίκων) is a treatise by Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Anc ...
'',
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

Archimedes
describes the
Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral In , a spiral is a which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the are:
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 Mecca, officially Makkah al-Mukarramah ( ) and commonly shortened to Makkah ( ),Quran 48:22 ' () is a city and administrative center of the Mecca Province of Saudi Arabia, and the Holiest sites in Islam, holiest city in Islam. It is inland ...

Mecca
(
qibla The qibla ( ar, قِبْلَة, links=no, lit=direction, translit=qiblah) is the direction towards the Kaaba The Kaaba (, ), also spelled Ka'bah or Kabah, sometimes referred to as al-Kaʿbah al-Musharrafah ( ar, ٱلْكَعْبَة ٱل ...

qibla
)—and its distance—from any location on the Earth. From the 9th century onward they were using
spherical trigonometry Spherical trigonometry is the branch of spherical geometry Image:Triangles (spherical geometry).jpg, 300px, The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (plan ...
and
map projection In cartography, a map projection is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the Surface (mathematics), surface of the globe ...
methods to determine these quantities accurately. The calculation is essentially the conversion of the
equatorial polar coordinates
equatorial polar coordinates
of Mecca (i.e. its
longitude Longitude (, ) is a geographic coordinate A geographic coordinate system (GCS) is a coordinate system associated with position (geometry), positions on Earth (geographic position). A GCS can give positions: *as Geodetic coordinates, ...

longitude
and
latitude In geography Geography (from Greek: , ''geographia'', literally "earth description") is a field of science devoted to the study of the lands, features, inhabitants, and phenomena of the Earth and planets. The first person to use the ...

latitude
) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the
great circle A great circle, also known as an orthodrome, of a sphere of a sphere A sphere (from Greek language, Greek —, "globe, ball") is a Geometry, geometrical object in solid geometry, three-dimensional space that is the surface of a Ball (mathem ...

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 Antipodal points on a circle are 180 degrees apart. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
. 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 Harvard University is a private Private or privates may refer to: Music * "In Private "In Private" was the third single in a row to be a charting success for United Kingdom, British singer Dusty Springfield, after an absence of nearly t ...

Harvard
professor Julian Lowell Coolidge's ''Origin of Polar Coordinates.'' 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 The Archimedean spiral (also known as the arithmetic spiral) is a spiral In , a spiral is a which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the are:
Archimedean spiral
.
Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, writer and Catholic Church, Catholic theologian. He was a child prodigy who was educated by his father, a tax collector i ...

Blaise Pascal
subsequently used polar coordinates to calculate the length of
parabolic arcs
parabolic arcs
. In ''
Method of Fluxions ''Method of Fluxions'' (latin De Methodis Serierum et Fluxionum) is a book by Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive ...
'' (written 1671, published 1736), Sir
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

Isaac Newton
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 Eruditorum ''Acta Eruditorum'' (Latin language, Latin for "Acts of the Erudite") was the first scientific journal of the German-speaking lands of Europe, published from 1682 to 1782. History ''Acta Eruditorum'' was founded in 1682 in Leipzig by Otto Mencke, ...

Acta Eruditorum
'' (1691),
Jacob Bernoulli Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) inclu ...
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 In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differentia ...
of curves expressed in these coordinates. The actual term ''polar coordinates'' has been attributed to
Gregorio Fontana Gregorio Fontana, born Giovanni Battista Lorenzo Fontana (7 December 1735 – 24 August 1803) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) ...

Gregorio Fontana
and was used by 18th-century Italian writers. The term appeared in
English English usually refers to: * English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, which has eventually become the World language, leading lan ...

English
in
George Peacock George Peacock FRS (9 April 1791 – 8 November 1858) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

George Peacock
's 1816 translation of Lacroix's ''Differential and Integral Calculus''.
Alexis Clairaut Alexis Claude Clairaut (; 13 May 1713 – 17 May 1765) was a French mathematician, astronomer, and geophysicist. He was a prominent Newtonian whose work helped to establish the validity of the principles and results that Sir Isaac Newton ...
was the first to think of polar coordinates in three dimensions, and
Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

Leonhard Euler
was the first to actually develop them.


Conventions

The radial coordinate is often denoted by ''r'' or ''ρ'', and the angular coordinate by , , or ''t''. The angular coordinate is specified as ''φ'' by
ISO The International Organization for Standardization (ISO ) is an international standard An international standard is a technical standard A technical standard is an established norm (social), norm or requirement for a repeatable technical task w ...
standard 31-11. However, in mathematical literature the angle is often denoted by θ instead. Angles in polar notation are generally expressed in either
degree Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
s or
radian The radian, denoted by the symbol \text, is the SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric sy ...

radian
s (2

rad being equal to 360°). Degrees are traditionally used in
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...

navigation
,
surveying Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial or three-dimensional positions of points and the distances and angles between them. A land surveying professional is called a land survey ...

surveying
, and many applied disciplines, while radians are more common in mathematics and mathematical
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
. The angle ''φ'' is defined to start at 0° from a ''reference direction'', and to increase for rotations in either or counterclockwise (ccw) orientation. For example, in mathematics, the reference direction is usually drawn as a
ray Ray may refer to: Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (graph theory), an infinite sequence of vertices such that each vertex appears at most once in the sequence and each two consecutive ...
from the pole horizontally to the right, and the polar angle increases to positive angles for ccw rotations, whereas in navigation (
bearing Bearing may refer to: * Bearing (angle), a term for direction * Bearing (mechanical), a component that separates moving parts and takes a load * Bridge bearing, a component separating a bridge pier and deck * Bearing BTS Station in Bangkok See also ...
, 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 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 and , where ''n'' is an arbitrary
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
. Moreover, the pole itself can be expressed as (0, ''φ'') for any angle ''φ''. Where a unique representation is needed for any point besides the pole, it is usual to limit ''r'' to positive numbers () and ''φ'' to either the interval or the interval , which in radians are or . Another convention, in reference to the usual codomain of the arctan function, is to allow for arbitrary nonzero real values of the radial component and restrict the polar angle to . In all cases a unique azimuth for the pole (''r'' = 0) must be chosen, e.g., ''φ'' = 0.


Converting between polar and Cartesian coordinates

The polar coordinates ''r'' and ''φ'' can be converted to the Cartesian coordinates ''x'' and ''y'' by using the
trigonometric function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

trigonometric function
s sine and cosine: :\begin x &= r \cos \varphi, \\ y &= r \sin \varphi. \end The Cartesian coordinates ''x'' and ''y'' can be converted to polar coordinates ''r'' and ''φ'' with ''r'' ≥ 0 and ''φ'' in the interval (−, ] by: :r = \sqrt \quad (as in the
Pythagorean theorem In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...

Pythagorean theorem
or the
Euclidean norm Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...
), and :\varphi = \operatorname(y, x), where
atan2 The function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automati ...

atan2
is a common variation on the
arctangent In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

arctangent
function defined as :\operatorname(y, x) = \begin \arctan\left(\frac\right) & \mbox x > 0\\ \arctan\left(\frac\right) + \pi & \mbox x < 0 \mbox y \ge 0\\ \arctan\left(\frac\right) - \pi & \mbox x < 0 \mbox y < 0\\ \frac & \mbox x = 0 \mbox y > 0\\ -\frac & \mbox x = 0 \mbox y < 0\\ \text & \mbox x = 0 \mbox y = 0. \end If ''r'' is calculated first as above, then this formula for ''φ'' may be stated more simply using the
arccosine In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...

arccosine
function: :\varphi = \begin \arccos\left(\frac\right) & \mbox y \ge 0 \mbox r \neq 0 \\ -\arccos\left(\frac\right) & \mbox y < 0 \\ \text & \mbox r = 0. \end


Complex numbers

Every
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
can be represented as a point in the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, 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 : z = x + iy where ''i'' is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad are ...
, or can alternatively be written in polar form as :z = r(\cos\varphi + i\sin\varphi) and from there, by
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler's ...

Euler's formula
, as : z = re^ = r \exp i \varphi. where ''e'' is
Euler's number The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828, and can be characterized in many ways. It is the base of a logarithm, base of the natural logarithm. It is the Limit of a sequence, limit of ...
, and ''φ'', expressed in radians, is the
principal value In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of the complex number function
arg Arg or ARG may refer to: Places *''Arg'' () means "citadel" in Persian, and may refer to: **Arg, Iran, a village in Fars Province, Iran **Arg (Kabul), presidential palace in Kabul, Afghanistan **Arg, South Khorasan, a village in South Khorasan Pr ...
applied to ''x'' + ''iy''. To convert between the rectangular and polar forms of a complex number, the conversion formulae given above can be used. Equivalent are the and angle notations: : z = r \operatorname\mathrm \varphi = r \angle \varphi . For the operations of
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
,
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
,
exponentiation Exponentiation is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
, and
root extraction In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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: r_0 e^\, r_1 e^ = r_0 r_1 e^ ; Division: \frac = \frace^ ; Exponentiation (
De Moivre's formula In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
): \left(re^\right)^n = r^n e^ ; Root Extraction (Principal root): \sqrt = \sqrt e^


Polar equation of a curve

The equation defining an
algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
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 Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
of ''φ''. The resulting curve then consists of points of the form (''r''(''φ''), ''φ'') and can be regarded as the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...

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 Symmetry (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

symmetry
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: * If it will be rotationally symmetric by α clockwise and 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 polar rose,
Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral In , a spiral is a which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the are:
Archimedean spiral
,
lemniscate 400px, The lemniscate of Bernoulli and its two foci In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the us ...

lemniscate
,
limaçon In geometry, a limaçon or limacon , also known as a limaçon of Pascal, is defined as a roulette (curve), roulette formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. It can also ...
, and
cardioid A cardioid (from the Greek language, Greek καρδία "heart") 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 can also be defined as an epicycloid having a single Cusp ...

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

The general equation for a circle with a center at and radius ''a'' is :r^2 - 2 r r_0 \cos(\varphi - \gamma) + r_0^2 = a^2. This can be simplified in various ways, to conform to more specific cases, such as the equation :r(\varphi)=a 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 :r = 2 a\cos(\varphi - \gamma). In the general case, the equation can be solved for , giving :r = r_0 \cos(\varphi - \gamma) + \sqrt 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 : \varphi = \gamma, where \gamma is the angle of elevation of the line; that is, \varphi = \arctan m, where m is the
slope In mathematics, the slope or gradient of a line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', ...

slope
of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line \varphi = \gamma
perpendicular In elementary geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relativ ...

perpendicular
ly at the point (r_0, \gamma) has the equation : r(\varphi) = r_0 \sec(\varphi - \gamma). Otherwise stated (r_0, \gamma) is the point in which the tangent intersects the imaginary circle of radius r_0


Polar rose

A polar rose is a mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation, :r(\varphi) = a\cos\left(k\varphi + \gamma_0\right) for any constant γ0 (including 0). If ''k'' is an integer, these equations will produce a ''k''-petaled rose if ''k'' is 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 ''a'' directly represents the length or amplitude of the petals of the rose, while ''k'' relates to their spatial frequency. The constant γ0 can be regarded as a phase angle.


Archimedean spiral

The
Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral In , a spiral is a which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the are:
Archimedean spiral
is a spiral discovered by
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

Archimedes
which can also be expressed as a simple polar equation. It is represented by the equation :r(\varphi) = a+b\varphi. 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. If , 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s, to be described in a mathematical treatise, and as a prime example of a curve best defined by a polar equation.


Conic sections

A
conic section In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
with one focus on the pole and the other somewhere on the 0° ray (so that the conic's
major axis In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
lies along the polar axis) is given by: : r = where ''e'' is the
eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off- center, in geometry * Eccentricity (graph theory) of a ...
and \ell is the
semi-latus rectum In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
(the perpendicular distance at a focus from the major axis to the curve). If , this equation defines a
hyperbola In mathematics, a hyperbola () (adjective form hyperbolic, ) (plural ''hyperbolas'', or ''hyperbolae'' ()) is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it ...

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

parabola
; and if , it defines an
ellipse In , an ellipse is a surrounding two , such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a , which is the special type of ellipse in which the two focal points are t ...

ellipse
. The special case of the latter results in a circle of the radius \ell.


Intersection of two polar curves

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


Calculus

Calculus can be applied to equations expressed in polar coordinates. The angular coordinate ''φ'' 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 (by computing its total derivatives) or :\begin r \frac &= r \frac \cos\varphi + r \frac \sin\varphi = x \frac + y \frac, \\[2pt] \frac &= - \frac r \sin\varphi + \frac r \cos\varphi = -y \frac + x \frac. \end Hence, we have the following formulae: :\begin r \frac &= x \frac + y \frac \\[2pt] \frac &= -y \frac + x \frac. \end Using the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a function ''u''(''r'',''φ''), it follows that :\begin \frac &= \frac\frac + \frac\frac, \\[2pt] \frac &= \frac\frac + \frac\frac, \end or :\begin \frac &= \frac\frac - \frac\frac \\[2pt] &= \cos \varphi \frac - \frac \sin\varphi \frac, \\[2pt] \frac &= \frac\frac + \frac\frac \\[2pt] &= \sin\varphi \frac + \frac \cos\varphi \frac. \end Hence, we have the following formulae: :\begin \frac &= \cos \varphi \frac - \frac \sin\varphi \frac \\[2pt] \frac &= \sin \varphi \frac + \frac \cos\varphi \frac. \end 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. :\begin x &= r(\varphi)\cos\varphi \\ y &= r(\varphi)\sin\varphi \end Derivative, Differentiating both equations with respect to ''φ'' yields :\begin \frac &= r'(\varphi)\cos\varphi - r(\varphi)\sin\varphi \\[2pt] \frac &= r'(\varphi)\sin\varphi + r(\varphi)\cos\varphi. \end Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point : :\frac = \frac. 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 . The length of ''L'' is given by the following integral :L = \int_a^b \sqrt d\varphi


Integral calculus (area)

Let ''R'' denote the region enclosed by a curve ''r''(''φ'') and the rays ''φ'' = ''a'' and ''φ'' = ''b'', where . Then, the area of ''R'' is :\frac12\int_a^b \left[r(\varphi)\right]^2\, d\varphi. This result can be found as follows. First, the interval is divided into ''n'' subintervals, where ''n'' is some positive integer. Thus Δ''φ'', the angle measure of each subinterval, is equal to (the total angle measure of the interval), divided by ''n'', the number of subintervals. For each subinterval ''i'' = 1, 2, ..., ''n'', let ''φ''''i'' be the midpoint of the subinterval, and construct a circular sector, sector with the center at the pole, radius ''r''(''φ''''i''), central angle Δ''φ'' and arc length ''r''(''φ''''i'')Δ''φ''. The area of each constructed sector is therefore equal to :\left[r(\varphi_i)\right]^2 \pi \cdot \frac = \frac\left[r(\varphi_i)\right]^2 \Delta \varphi. Hence, the total area of all of the sectors is :\sum_^n \tfrac12r(\varphi_i)^2\,\Delta\varphi. As the number of subintervals ''n'' is increased, the approximation of the area improves. Taking , 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: : J = \det\frac = \begin \frac & \frac \\[2pt] \frac & \frac \end = \begin \cos\varphi & -r\sin\varphi \\ \sin\varphi & r\cos\varphi \end = r\cos^2\varphi + r\sin^2\varphi = r. Hence, an area element in polar coordinates can be written as :dA = dx\,dy\ = J\,dr\,d\varphi = r\,dr\,d\varphi. Now, a function, that is given in polar coordinates, can be integrated as follows: :\iint_R f(x, y)\, dA = \int_a^b \int_0^ f(r, \varphi)\,r\,dr\,d\varphi. 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'' is retrieved by taking ''f'' identically equal to 1. A more surprising application of this result yields the Gaussian integral, here denoted ''K'': :K=\int_^\infty e^ \, dx = \sqrt\pi.


Vector calculus

Vector calculus can also be applied to polar coordinates. For a planar motion, let \mathbf be the position vector , with ''r'' and ''φ'' depending on time ''t''. We define the unit vectors :\hat = (\cos(\varphi), \sin(\varphi)) in the direction of \mathbf and :\hat = (-\sin(\varphi), \cos(\varphi)) = \hat \times \hat \ , in the plane of the motion perpendicular to the radial direction, where \hat is a unit vector normal to the plane of the motion. Then :\begin \mathbf &= (x,\ y) = r(\cos\varphi,\ \sin\varphi) = r \hat\ , \\ \dot &= \left(\dot,\ \dot\right) = \dot(\cos\varphi,\ \sin\varphi) + r\dot(-\sin\varphi,\ \cos\varphi) = \dot\hat + r\dot\hat\ ,\\ \ddot &= \left(\ddot,\ \ddot\right) \\ &= \ddot(\cos\varphi,\ \sin\varphi) + 2\dot\dot(-\sin\varphi,\ \cos\varphi) + r\ddot(-\sin\varphi,\ \cos\varphi) - r\dot^2(\cos\varphi,\ \sin\varphi) \\ &= \left(\ddot - r\dot^2\right) \hat + \left(r\ddot + 2\dot\dot\right) \hat \\ &= \left(\ddot - r\dot^2\right) \hat + \frac\; \frac \left(r^2\dot\right) \hat. \end


Centrifugal and Coriolis terms

The term r\dot\varphi^2 is sometimes referred to as the ''centripetal acceleration'', and the term 2\dot r \dot\varphi as the ''Coriolis acceleration''. For example, see Shankar. 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 forces; 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.


=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''.For the following discussion, see 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 ''mrΩ2, 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: :\boldsymbol + \boldsymbol_\text + \boldsymbol_\text = m\ddot \ , 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: :\begin F_r + mr\Omega^2 &= m\ddot \\ F_\varphi - 2m\dot\Omega &= mr\ddot \ , \end which can be compared to the equations for the inertial frame: :\begin F_r &= m\ddot - mr\dot^2 \\ F_\varphi &= mr\ddot + 2m\dot\dot \ . \end 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 , the plane minus the origin. In these coordinates, the Euclidean metric tensor is given byds^2 = dr^2 + r^2 d\theta^2.This can be seen via the change of variables formula for the metric tensor, or by computing the differential forms ''dx'', ''dy'' via the exterior derivative of the 0-forms , and substituting them in the Euclidean metric tensor . An Orthonormality, orthonormal Moving frame, frame with respect to this metric is given bye_r = \frac, \quad e_\theta = \frac1r \frac,with Moving frame#Coframes, dual coframee^r = dr, \quad e^\theta = r d\theta.The connection form relative to this frame and the Levi-Civita connection is given by the skew-symmetric matrix of 1-forms\omega^i_j = \begin 0 & -d\theta \\ d\theta & 0\endand hence the curvature form 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 A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism (geometry), prism with a circle as its base. This traditi ...

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 Circular may refer to: * The shape of a circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre; equivalently it is ...

circular
and
orbital motion In celestial mechanics, an orbit is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or pos ...

orbital motion
.


Position and navigation

Polar coordinates are used often in
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...

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. 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).


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 sources, 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 at its target design frequency. The pattern shifts toward omnidirectionality at lower frequencies.


See also

*Curvilinear coordinates *List of canonical coordinate transformations *Log-polar coordinates *Polar decomposition *Unit circle


References


General references

* * *


External links

* *
Coordinate Converter — converts between polar, Cartesian and spherical coordinatesPolar Coordinate System Dynamic Demo
{{DEFAULTSORT:Polar Coordinate System Two-dimensional coordinate systems Orthogonal coordinate systems