Points in the polar coordinate system with poleO and polar axisL. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3,60°). In blue, the point (4,210°).
Inmathematics, thepolar coordinate system specifies a givenpoint in aplane by using a distance and an angle as its twocoordinates. These are
the point's distance from a reference point called thepole, and
the point's direction from the pole relative to the direction of thepolar axis, aray drawn from the pole.
The distance from the pole is called theradial coordinate,radial distance or simplyradius, and the angle is called theangular coordinate,polar angle, orazimuth.[1] The pole is analogous to the origin in aCartesian coordinate system.
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 asspirals. 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 concepts of angle and radius were already used by ancient peoples of the first millenniumBC. TheGreek astronomer andastrologerHipparchus (190–120 BC) created a table ofchord functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions.[2] InOn Spirals,Archimedes describes theArchimedean 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 toMecca (qibla)—and its distance—from any location on the Earth.[3] From the 9th century onward they were usingspherical trigonometry andmap projection methods to determine these quantities accurately. The calculation is essentially the conversion of theequatorial polar coordinates of Mecca (i.e. itslongitude andlatitude) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is thegreat circle through the given location and the Earth's poles and whose polar axis is the line through the location and itsantipodal point.[4]
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 inHarvard professorJulian Lowell Coolidge'sOrigin of Polar Coordinates.[5] 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 anArchimedean spiral.Blaise Pascal subsequently used polar coordinates to calculate the length ofparabolic arcs.
InMethod of Fluxions (written 1671, published 1736), SirIsaac Newton examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems.[6] In the journalActa Eruditorum (1691),Jacob Bernoulli used a system with a point on a line, called thepole andpolar axis respectively. Coordinates were specified by the distance from the pole and the angle from thepolar axis. Bernoulli's work extended to finding theradius of curvature of curves expressed in these coordinates.
The actual termpolar coordinates has been attributed toGregorio Fontana and was used by 18th-century Italian writers. The term appeared inEnglish inGeorge Peacock's 1816 translation ofLacroix'sDifferential and Integral Calculus.[7][8]Alexis Clairaut was the first to think of polar coordinates in three dimensions, andLeonhard Euler was the first to actually develop them.[5]
A polar grid with several angles, increasing in counterclockwise orientation and labelled in degrees
The radial coordinate is often denoted byr orρ, and the angular coordinate byφ,θ, ort. The angular coordinate is specified asφ byISO standard31-11, now80000-2:2019. However, in mathematical literature the angle is often denoted by θ instead.
Angles in polar notation are generally expressed in eitherdegrees orradians (2π rad being equal to 360°). Degrees are traditionally used innavigation,surveying, and many applied disciplines, while radians are more common in mathematics and mathematicalphysics.[9]
The angleφ is defined to start at 0° from areference direction, and to increase for rotations in eitherclockwise (cw) or counterclockwise (ccw) orientation. For example, in mathematics, the reference direction is usually drawn as aray 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.
Adding any number of fullturns (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(r,φ +n × 360°) and(−r,φ + 180° +n × 360°) = (−r,φ + (2n + 1) × 180°), wheren is an arbitraryinteger.[10] Moreover, the pole itself can be expressed as (0, φ) for any angleφ.[11]
Where a unique representation is needed for any point besides the pole, it is usual to limitr to positive numbers (r > 0) andφ to either theinterval[0, 360°) or the interval(−180°, 180°], which in radians are[0, 2π) or(−π, π].[12] Another convention, in reference to the usualcodomain of thearctan function, is to allow for arbitrary nonzero real values of the radial component and restrict the polar angle to(−90°,90°]. In all cases a unique azimuth for the pole (r = 0) must be chosen, e.g.,φ = 0.
Converting between polar and Cartesian coordinates
A diagram illustrating the relationship between polar and Cartesian coordinates.
The polar coordinatesr andφ can be converted to the Cartesian coordinatesx andy by using thetrigonometric functions sine and cosine:
The Cartesian coordinatesx andy can be converted to polar coordinatesr andφ withr ≥ 0 andφ in the interval (−π,π] by:[13]where hypot is thePythagorean sum andatan2 is a common variation on thearctangent function defined as
Ifr is calculated first as above, then this formula forφ may be stated more simply using thearccosine function:
An illustration of a complex numberz plotted on the complex planeAn illustration of a complex number plotted on the complex plane usingEuler's formula
Everycomplex number can be represented as a point in thecomplex plane, and can therefore be expressed by specifying either the point's Cartesian coordinates (called rectangular or Cartesian form) or the point's polar coordinates (called polar form).
In polar form, the distance and angle coordinates are often referred to as the number'smagnitude andargument respectively. Two complex numbers can be multiplied by adding their arguments and multiplying their magnitudes.
The complex numberz can be represented in rectangular form aswherei is theimaginary unit, or can alternatively be written in polar form asand from there, byEuler's formula,[14] aswheree isEuler's number, andφ, expressed in radians, is theprincipal value of the complex number functionarg applied tox +iy. To convert between the rectangular and polar forms of a complex number, the conversion formulae given above can be used. Equivalent are thecis andangle notations:
For the operations ofmultiplication,division,exponentiation, androot 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:
A curve on the Cartesian plane can be mapped into polar coordinates. In this animation, is mapped onto. Click on image for details.
The equation defining aplane curve expressed in polar coordinates is known as apolar equation. In many cases, such an equation can simply be specified by definingr as afunction ofφ. The resulting curve then consists of points of the form (r(φ), φ) and can be regarded as thegraph of the polar functionr. Note that, in contrast to Cartesian coordinates, the independent variableφ is thesecond entry in the ordered pair.
Different forms ofsymmetry can be deduced from the equation of a polar functionr:
Ifr(−φ) =r(φ) the curve will be symmetrical about the horizontal (0°/180°) ray;
Ifr(π −φ) =r(φ) it will be symmetric about the vertical (90°/270°) ray:
Ifr(φ − α) =r(φ) it will berotationally 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 thepolar rose,Archimedean spiral,lemniscate,limaçon, andcardioid.
For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.
Radial lines (those running through the pole) are represented by the equationwhere is the angle of elevation of the line; that is,, where is theslope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial lineperpendicularly at the point has the equation
Otherwise stated is the point in which the tangent intersects the imaginary circle of radius
Apolar rose is a mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,
for any constant γ0 (including 0). Ifk is an integer, these equations will produce ak-petaled rose ifk isodd, or a 2k-petaled rose ifk is even. Ifk 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. Thevariablea directly represents the length or amplitude of the petals of the rose, whilek relates to their spatial frequency. The constant γ0 can be regarded as a phase angle.
One arm of an Archimedean spiral with equationr(φ) =φ / 2π for0 <φ < 6π
TheArchimedean spiral is a spiral discovered byArchimedes which can also be expressed as a simple polar equation. It is represented by the equationChanging the parametera will turn the spiral, whileb controls the distance between the arms, which for a given spiral is always constant. The Archimedean spiral has two arms, one forφ > 0 and one forφ < 0. The two arms are smoothly connected at the pole. Ifa = 0, 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 theconic sections, to be described in a mathematical treatise, and as a prime example of a curve best defined by a polar equation.
Aconic section with one focus on the pole and the other somewhere on the 0° ray (so that the conic'smajor axis lies along the polar axis) is given by:wheree is theeccentricity and is thesemi-latus rectum (the perpendicular distance at a focus from the major axis to the curve). Ife > 1, this equation defines ahyperbola; ife = 1, it defines aparabola; and ife < 1, it defines anellipse. The special casee = 0 of the latter results in a circle of the radius.
A quadratrix in the first quadrant (x, y) is a curve withy = ρ sin θ equal to the fraction of the quarter circle with radiusr determined by the radius through the curve point. Since this fraction is, the curve is given by.[16]
Usingx =r cosφ andy =r sinφ, one can derive a relationship between derivatives in Cartesian and polar coordinates. For a given function,u(x,y), it follows that (by computing itstotal derivatives)or
Hence, we have the following formula:
Using the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a functionu(r,φ), it follows thator
Hence, we have the following formulae:
To find the Cartesian slope of the tangent line to a polar curver(φ) at any given point, the curve is first expressed as a system ofparametric equations.
The arc length (length of a line segment) defined by a polar function is found by the integration over the curver(φ). LetL denote this length along the curve starting from pointsA through to pointB, where these points correspond toφ =a andφ =b such that0 <b −a < 2π. The length ofL is given by the following integral
The integration regionR is bounded by the curver(φ) and the raysφ =a andφ =b.
LetR denote the region enclosed by a curver(φ) and the raysφ =a andφ =b, where0 <b −a ≤ 2π. Then, the area ofR is
The regionR is approximated byn sectors (here,n = 5).Aplanimeter, which mechanically computes polar integrals
This result can be found as follows. First, the interval[a,b] is divided inton subintervals, wheren is some positive integer. Thus Δφ, the angle measure of each subinterval, is equal tob −a (the total angle measure of the interval), divided byn, the number of subintervals. For each subintervali = 1, 2, ...,n, letφi be the midpoint of the subinterval, and construct asector with the center at the pole, radiusr(φi), central angle Δφ and arc lengthr(φi)Δφ. The area of each constructed sector is therefore equal toHence, the total area of all of the sectors is
As the number of subintervalsn is increased, the approximation of the area improves. Takingn → ∞, the sum becomes theRiemann sum for the above integral.
A mechanical device that computes area integrals is theplanimeter, 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-elementlinkage effectsGreen's theorem, converting the quadratic polar integral to a linear integral.
Hence, an area element in polar coordinates can be written as
Now, a function, that is given in polar coordinates, can be integrated as follows:
Here,R is the same region as above, namely, the region enclosed by a curver(φ) and the raysφ =a andφ =b. The formula for the area ofR is retrieved by takingf identically equal to 1.
A graph of and the area between the function and the-axis, which is equal to.
A more surprising application of this result yields theGaussian integral:
Vector calculus can also be applied to polar coordinates. For a planar motion, let be the position vector(r cos(φ),r sin(φ)), withr andφ depending on timet.
We define anorthonormal basis with three unit vectors:radial, transverse, and normal directions.Theradial direction is defined by normalizing:Radial and velocity directions span theplane of the motion, whose normal direction is denoted:Thetransverse direction is perpendicular to both radial and normal directions:
Then
This equation can be obtain by taking derivative of the function and derivatives of the unit basis vectors.
For a curve in 2D where the parameter is the previous equations simplify to:
Position vectorr, always points radially from the origin.
Velocity vectorv, always tangent to the path of motion.
Acceleration vectora, not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.
Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 2d space, but a plane in any higher dimension.
The term is sometimes referred to as thecentripetal acceleration, and the term as theCoriolis acceleration. For example, see Shankar.[19]
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'ssecond law of motion in a rotating frame of reference. Here these extra terms are often calledfictitious 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.
Inertial frame of referenceS and instantaneous non-inertial co-rotating frame of referenceS′. The co-rotating frame rotates at angular rate Ω equal to the rate of rotation of the particle about the origin ofS′ at the particular momentt. Particle is located at vector positionr(t) and unit vectors are shown in the radial direction to the particle from the origin, and also in the direction of increasing angleϕ normal to the radial direction. These unit vectors need not be related to the tangent and normal to the path. Also, the radial distancer need not be related to the radius of curvature of the path.
For a particle in planar motion, one approach to attaching physical significance to these terms is based on the concept of an instantaneousco-rotating frame of reference.[20] To define a co-rotating frame, first an origin is selected from which the distancer(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 momentt, 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 ismrΩ2, radially outward. The velocity of the particle in the co-rotating frame also is radially outward, becausedφ′/dt = 0. Thefictitious Coriolis force therefore has a value −2m(dr/dt)Ω, pointed in the direction of increasingφ only. Thus, using these forces in Newton's second law we find:where over dots represent derivatives with respect to time, andF is the net real force (as opposed to the fictitious forces). In terms of components, this vector equation becomes:which can be compared to the equations for the inertial frame:
This comparison, plus the recognition that by the definition of the co-rotating frame at timet 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 instantaneousosculating circle of its motion, not to a fixed center of polar coordinates. For more detail, seecentripetal force.
In the modern terminology ofdifferential geometry, polar coordinates providecoordinate charts for thedifferentiable manifoldR2 \ {(0,0)}, the plane minus the origin. In these coordinates, the Euclideanmetric tensor is given byThis can be seen via the change of variables formula for the metric tensor, or by computing thedifferential formsdx,dy via theexterior derivative of the 0-formsx =r cos(θ),y =r sin(θ) and substituting them in the Euclidean metric tensords2 =dx2 +dy2.
An elementary proof of the formula
Let, and be two points in the plane given by their cartesian and polar coordinates. Then
Since, and, we get that
Now we use the trigonometric identity to proceed:
If the radial and angular quantities are near to each other, and therefore near to a common quantity and, we have that. Moreover, the cosine of can be approximated with the Taylor series of the cosine up to linear terms:
so that, and. Therefore, around an infinitesimally small domain of any point,
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 ofcircular andorbital motion.
Polar coordinates are used often innavigation 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 tomagnetic north, while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively.[21] Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (readzero-niner-zero byair traffic control).[22]
Systems displayingradial symmetry provide natural settings for the polar coordinate system, with the central point acting as the pole. A prime example of this usage is thegroundwater flow equation when applied to radially symmetric wells. Systems with aradial force are also good candidates for the use of the polar coordinate system. These systems includegravitational fields, which obey theinverse-square law, as well as systems withpoint sources, such asradio antennas.
Radially asymmetric systems may also be modeled with polar coordinates. For example, amicrophone'spickup 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 asr = 0.5 + 0.5sin(ϕ) at its target design frequency.[23] The pattern shifts toward omnidirectionality at lower frequencies.
^King (2005,p. 169). The calculations were as accurate as could be achieved under the limitations imposed by their assumption that the Earth was a perfect sphere.
^Boyer, C. B. (1949). "Newton as an Originator of Polar Coordinates".American Mathematical Monthly.56 (2). Mathematical Association of America:73–78.doi:10.2307/2306162.JSTOR2306162.
^Lee, Theodore; David Cohen; David Sklar (2005).Precalculus: With Unit-Circle Trigonometry (Fourth ed.). Thomson Brooks/Cole.ISBN0-534-40230-5.
^Stewart, Ian; David Tall (1983).Complex Analysis (the Hitchhiker's Guide to the Plane). Cambridge University Press.ISBN0-521-28763-4.
^Torrence, Bruce Follett;Eve Torrence (1999).The Student's Introduction to Mathematica. Cambridge University Press.ISBN0-521-59461-8.
^Smith, Julius O. (2003)."Euler's Identity".Mathematics of the Discrete Fourier Transform (DFT). W3K Publishing.ISBN0-9745607-0-7. Archived fromthe original on 2006-09-15. Retrieved2006-09-22.
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