In mathematics, the polar coordinate system is a

two-dimensional In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise ...

coordinate system in which each point on a plane is determined by a

distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...

from a reference point and an

angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...

from a reference direction. The reference point (analogous to the origin of a

Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured ...

) is called the ''pole'', and the

ray Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (g ...

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 a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematicall ...

''. Angles in polar notation are generally expressed in either

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...

s or radians (2 rad being equal to 360°). Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-17th century, though the actual term "polar coordinates" has been attributed to Gregorio Fontana in the 18th century. The initial motivation for the introduction of the polar system was the study of circular and

orbital motion In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...

. 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:cylindrical and 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 range of divinatory practices, recognized as pseudoscientific since the 18th century, that claim to discern information about human affairs and terrestrial events by studying the apparent positions of celestial objects. Di ...

Hipparchus Hipparchus (; 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 the precession of the equ ...

(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 ( ...

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, written around 225 BC. Notably, Archimedes employed the Archimedean spiral in this book to square the circle Squaring the circle is a problem in geometry first pro ...

'',

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scienti ...

describes the

Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a cons ...

, 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, commonly shortened to Makkah ()) is a city and administrative center of the Mecca Province of Saudi Arabia, and the holiest city in Islam. It is inland from Jeddah on the Red Sea, in a narrow val ...

( qibla)—and its distance—from any location on the Earth. From the 9th century onward they were using spherical trigonometry and

map projection In cartography, map projection is the term used to describe a broad set of transformations employed to represent the two-dimensional curved surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longit ...

methods to determine these quantities accurately. The calculation is essentially the conversion of the equatorial polar coordinates of Mecca (i.e. its

longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...

and

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north po ...

) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the great circle through the given location and the Earth's poles and whose polar axis is the line through the location and its

antipodal point In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true ...

. 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 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

Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic writer. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pascal's earlies ...

subsequently used polar coordinates to calculate the length of parabolic arcs. In ''

Method of Fluxions ''Method of Fluxions'' ( la, De Methodis Serierum et Fluxionum) is a mathematical treatise by Sir Isaac Newton which served as the earliest written formulation of modern calculus. The book was completed in 1671, and published in 1736. Fluxion ...

'' (written 1671, published 1736), Sir

Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the g ...

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'' (1691), Jacob Bernoulli used a system with a point on a line, called the ''pole'' and ''polar axis'' respectively. Coordinates were specified by the distance from the pole and the angle from the ''polar axis''. Bernoulli's work extended to finding the radius of curvature of curves expressed in these coordinates. The actual term ''polar coordinates'' has been attributed to Gregorio Fontana and was used by 18th-century Italian writers. The term appeared in English in

George Peacock George Peacock FRS (9 April 1791 – 8 November 1858) was an English mathematician and Anglican cleric. He founded what has been called the British algebra of logic. Early life Peacock was born on 9 April 1791 at Thornton Hall, Denton, nea ...

's 1816 translation of Lacroix's ''Differential and Integral Calculus''. Alexis Clairaut 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, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...

was the first to actually develop them.

Conventions

The radial coordinate is often denoted by ''r'' or ''ρ'', and the angular coordinate by ''φ'', ''θ'', or ''t''. The angular coordinate is specified as ''φ'' by ISO standard 31-11. However, in mathematical literature the angle is often denoted by θ instead. Angles in polar notation are generally expressed in either

s or radians (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, ...

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

, and many applied disciplines, while radians are more common in mathematics and mathematical

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...

. The angle ''φ'' is defined to start at 0° from a ''reference direction'', and to increase for rotations in either clockwise (cw) or counterclockwise (ccw) orientation. For example, in mathematics, the reference direction is usually drawn as a

from the pole horizontally to the right, and the polar angle increases to positive angles for ccw rotations, whereas in navigation ( bearing, 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 is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

. 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, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in ...

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:

\begin
r &= \sqrt = \operatorname(x,y)\\
\varphi &= \operatorname(y, x),
\end

where hypot is the Pythagorean addition, Pythagorean sum and atan2 is a common variation on the

arctangent In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). S ...

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 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 extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...

can be represented as a point in the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...

, 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 x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition a ...

, or can alternatively be written in polar form as

z = r(\cos\varphi + i\sin\varphi)

and from there, by Euler's formula, as

z = re^ = r \exp i \varphi.

where ''e'' is Euler's number, and ''φ'', expressed in radians, is the principal value of the complex number function arg 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 mathematical operations of arithmetic, with the other ones being ad ...

, division,

exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...

, and root extraction 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):

\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 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 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 discre ...

of the polar function ''r''. Note that, in contrast to Cartesian coordinates, the independent variable ''φ'' is the ''second'' entry in the ordered pair. Different forms of symmetry can be deduced from the equation of a polar function ''r'': * If 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 Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...

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,

, lemniscate,

limaçon In geometry, a limaçon or limacon , also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. I ...

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

Circle

The general equation for a circle with a center at

(r_0, \gamma)

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 or 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 is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is used ...

of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line

\varphi = \gamma

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...

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

is a spiral discovered by

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, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a ...

s, to be described in a mathematical treatise, and as a prime example of a curve best defined by a polar equation.

Conic sections

with one focus on the pole and the other somewhere on the 0° ray (so that the conic's

major axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longe ...

lies along the polar axis) is given by:

r =

where ''e'' is the eccentricity and

\ell

is the semi-latus rectum (the perpendicular distance at a focus from the major axis to the curve). If , this equation defines a hyperbola; if , it defines a

parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descri ...

; and if , it defines an 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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

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 derivative In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with r ...

s) or

\frac &= - \frac r \sin\varphi + \frac r \cos\varphi = -y \frac + x \frac. \end

Hence, we have the following formulae:

\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

\frac &= \frac\frac + \frac\frac, \end

&= \sin\varphi \frac + \frac \cos\varphi \frac. \end

Hence, we have the following formulae:

\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

Differentiating both equations with respect to ''φ'' yields

\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

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 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

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 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 substitution rule for multiple integrals states that, when using other coordinates, the

Jacobian In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: *Jacobian matrix and determinant *Jacobian elliptic functions *Jacobian variety *Intermediate Jacobian In mathematics, the intermediate Jacobian of a compact Kähler m ...

determinant of the coordinate conversion formula has to be considered:

\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:

\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 \boldsymbol \varphi = (-\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

This equation can be obtain by taking derivative of the function and derivatives of the unit basis vectors. For a curve in 2D with the parameter is

\theta

the previous equation simplify to:

\begin
\vec r &= r(\theta) \hat e_r\\
\frac   &= \frac  \hat e_r+r\hat e_\theta\\
\frac   &= (\frac  -r) \hat e_r +\frac   \hat e_\theta
\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 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 ''mr''Ω², 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 In 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 ...

of its motion, not to a fixed center of polar coordinates. For more detail, see centripetal force.

Differential geometry

In the modern terminology of differential geometry, polar coordinates provide

coordinate charts In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an a ...

for the differentiable manifold , the plane minus the origin. In these coordinates, the Euclidean metric tensor is given by

ds^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 form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...

s ''dx'', ''dy'' via the exterior derivative of the 0-forms , and substituting them in the Euclidean metric tensor . An orthonormal frame with respect to this metric is given by

e_r = \frac, \quad e_\theta = \frac1r \frac,

with dual coframe

e^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

_j = \begin 0 & -d\theta \\ d\theta & 0\end

and hence the curvature form vanishes. 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 and

spherical coordinate system In 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 ''polar angle'' me ...

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 and

Position and navigation

Polar coordinates are used often in

as the destination or direction of travel can be given as an angle and distance from the object being considered. For instance,

aircraft An aircraft is a vehicle that is able to fly by gaining support from the air. It 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. ...

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 zero-niner-zero by

air traffic control Air traffic control (ATC) is a service provided by ground-based air traffic controllers who direct aircraft on the ground and through a given section of controlled airspace, and can provide advisory services to aircraft in non-controlled airsp ...

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 Used 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, simila ...

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 fields, which obey the inverse-square law, as well as systems with point sources, such as radio antennas. Radially asymmetric systems may also be modeled with polar coordinates. For example, a

microphone A microphone, colloquially called a mic or mike (), is a transducer that converts sound into an electrical signal. Microphones are used in many applications such as telephones, hearing aids, public address systems for concert halls and pub ...

'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.

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