Points in the polar coordinate system with pole O
and polar axis L
. In green, the point with radial coordinate 3 and angular coordinate 60 degrees, or (3,60°). In blue, the point (4,210°).
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction.
The fixed point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the fixed direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is the angular coordinate, polar angle, or azimuth.
The concepts of angle and radius were already used by ancient peoples of the 1st millennium BCE. The Greek astronomer and astrologer Hipparchus (190–120 BCE) created a table of 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, Archimedes describes the Archimedean spiral, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system.
From the 8th century CE onward, astronomers developed methods for approximating and calculating the direction to Makkah (qibla)—and its distance—from any location on the Earth. From the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately. The calculation is essentially the conversion of the equatorial polar coordinates of Mecca (i.e. its longitude and latitude) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the great circle through the given location and the Earth's poles, and whose polar axis is the line through the location and its antipodal point.
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 Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs.
In Method of Fluxions (written 1671, published 1736), Sir Isaac Newton examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems. 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'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 was the first to actually develop them.
A polar grid with several angles labeled in degrees
The radial coordinate is often denoted by r, and the angular coordinate by φ, θ or t. The angular coordinate is specified as φ by ISO standard 31-11.
Angles in polar notation are generally expressed in either degrees or radians (2π rad being equal to 360°). Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics and mathematical physics.
In many contexts, a positive angular coordinate means that the angle φ is measured counterclockwise from the axis.
In mathematical literature, the polar axis is often drawn horizontal and pointing to the right.
Uniqueness of polar coordinates [
Adding any number of full turns (360°) to the angular coordinate does not change the corresponding direction. Also, a negative radial coordinate is best interpreted as the corresponding positive distance measured in the opposite direction. Therefore, the same point can be expressed with an infinite number of different polar coordinates (r, φ ± n×360°) or (−r, φ ± (2n + 1)180°), where n is any integer. Moreover, the pole itself can be expressed as (0, φ) for any angle φ.
Where a unique representation is needed for any point, it is usual to limit r to non-negative numbers (r ≥ 0) and φ to the interval [0, 360°) or (−180°, 180°] (in radians, [0, 2π) or (−π, π]). One must also choose a unique azimuth for the pole, e.g., φ = 0.
Converting between polar and Cartesian coordinates [
A diagram illustrating the relationship between polar and Cartesian coordinates.
A curve on the Cartesian plane can be mapped into polar coordinates. In this animation,
is mapped onto
. Click on image for details.
The polar coordinates r and φ can be converted to the Cartesian coordinates x and y by using the trigonometric functions sine and cosine:
The Cartesian coordinates x and y can be converted to polar coordinates r and φ with r ≥ 0 and φ in the interval (−π, π] by:
- (as in the Pythagorean theorem or the Euclidean norm), and
where atan2 is a common variation on the arctangent function defined as
The value of φ above is the principal value of the complex number function arg applied to x+iy. An angle in the range [0, 2π) may be obtained by adding 2π to the value in case it is negative.
Polar equation of a curve [
The equation defining an algebraic curve expressed in polar coordinates is known as a polar equation. In many cases, such an equation can simply be specified by defining r as a function of φ. The resulting curve then consists of points of the form (r(φ), φ) and can be regarded as the graph of the polar function r.
Different forms of symmetry can be deduced from the equation of a polar function r. If r(−φ) = r(φ) the curve will be symmetrical about the horizontal (0°/180°) ray, if r(π − φ) = r(φ) it will be symmetric about the vertical (90°/270°) ray, and if r(φ − α) = r(φ) it will be rotationally symmetric α 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, lemniscate, limaçon, and cardioid.
For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.
A circle with equation r(φ) = 1
The general equation for a circle with a center at (r0, ) and radius a is
This can be simplified in various ways, to conform to more specific cases, such as the equation
for a circle with a center at the pole and radius a.
When r0 = a, or when the origin lies on the circle, the equation becomes
In the general case, the equation can be solved for r, giving
the solution with a minus sign in front of the square root gives the same curve.
A polar rose with equation r(φ) = 2 sin 4φ
Radial lines (those running through the pole) are represented by the equation
where ɣ is the angle of elevation of the line; that is, ɣ = arctan m where m is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line φ = ɣ perpendicularly at the point (r0, ɣ) has the equation
Otherwise stated (r0, ɣ) is the point in which the tangent intersects the imaginary circle of radius r0.
Polar rose [
A polar rose is a famous mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,
for any constant ɣ0 (including 0). If k is an integer, these equations will produce a k-petaled rose if k is odd, or a 2k-petaled rose if k is even. If k is rational but not an integer, a rose-like shape may form but with overlapping petals. Note that these equations never define a rose with 2, 6, 10, 14, etc. petals. The variable a represents the length of the petals of the rose.
Archimedean spiral [
One arm of an Archimedean spiral with equation r(φ) = φ / 2π
for 0 < φ < 6π
The Archimedean spiral is a famous spiral that was discovered by Archimedes, which also can be expressed as a simple polar equation. It is represented by the equation
Changing the parameter a will turn the spiral, while b controls the distance between the arms, which for a given spiral is always constant. The Archimedean spiral has two arms, one for φ > 0 and one for φ < 0. The two arms are smoothly connected at the pole. Taking the mirror image of one arm across the 90°/270° line will yield the other arm. This curve is notable as one of the first curves, after the conic sections, to be described in a mathematical treatise, and as being a prime example of a curve that is best defined by a polar equation.
Conic sections [
Ellipse, showing semi-latus rectum
A conic section with one focus on the pole and the other somewhere on the 0° ray (so that the conic's major axis lies along the polar axis) is given by:
where e is the eccentricity and is the semi-latus rectum (the perpendicular distance at a focus from the major axis to the curve). If e > 1, this equation defines a hyperbola; if e = 1, it defines a parabola; and if e < 1, it defines an ellipse. The special case e = 0 of the latter results in a circle of radius .
Complex numbers [
An illustration of a complex number z
plotted on the complex plane
An illustration of a complex number plotted on the complex plane using Euler's formula
Every complex number can be represented as a point in the complex plane, and can therefore be expressed by specifying either the point's Cartesian coordinates (called rectangular or Cartesian form) or the point's polar coordinates (called polar form). The complex number z can be represented in rectangular form as
where i is the imaginary unit, or can alternatively be written in polar form (via the conversion formulae given above) as
and from there as
where e is Euler's number, which are equivalent as shown by Euler's formula. (Note that this formula, like all those involving exponentials of angles, assumes that the angle φ is expressed in radians.) To convert between the rectangular and polar forms of a complex number, the conversion formulae given above can be used.
For the operations of multiplication, division, and exponentiation of complex numbers, it is generally much simpler to work with complex numbers expressed in polar form rather than rectangular form. From the laws of exponentiation:
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 x = r cos φ and y = r sin φ , one can derive a relationship between derivatives in Cartesian and polar coordinates. For a given function, u(x,y), it follows that
Hence, we have the following formulae:
Using the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a function u(r,φ), it follows that
Hence, we have the following formulae:
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.
Differentiating both equations with respect to φ yields
Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point (r(φ), φ):
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 0 < b − a < 2π. The length of L is given by the following integral
Integral calculus (area) [
The integration region R
is bounded by the curve r
) and the rays φ
Let R denote the region enclosed by a curve r(φ) and the rays φ = a and φ = b, where 0 < b − a ≤ 2π. Then, the area of R is
The region R
is approximated by n
sectors (here, n
, which mechanically computes polar integrals
This result can be found as follows. First, the interval [a, b] is divided into n subintervals, where n is an arbitrary positive integer. Thus Δφ, the length of each subinterval, is equal to b − a (the total length of the interval), divided by n, the number of subintervals. For each subinterval i = 1, 2, …, n, let φi be the midpoint of the subinterval, and construct a sector with the center at the pole, radius r(φi), central angle Δφ and arc length r(φi)Δφ. The area of each constructed sector is therefore equal to
Hence, the total area of all of the sectors is
As the number of subintervals n is increased, the approximation of the area continues to improve. In the limit as n → ∞, the sum becomes the Riemann sum for the above integral.
A mechanical device that computes area integrals is the planimeter, which measures the area of plane figures by tracing them out: this replicates integration in polar coordinates by adding a joint so that the 2-element linkage effects Green's theorem, converting the quadratic polar integral to a linear integral.
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 determinant of the coordinate conversion formula has to be considered:
Hence, an area element in polar coordinates can be written as
Now, a function that is given in polar coordinates can be integrated as follows:
Here, R is the same region as above, namely, the region enclosed by a curve r(φ) and the rays φ = a and φ = b.
The formula for the area of R mentioned above is retrieved by taking f identically equal to 1. A more surprising application of this result yields the Gaussian integral
Vector calculus [
Vector calculus can also be applied to polar coordinates. For a planar motion, let be the position vector (rcos(φ), rsin(φ)), with r and φ depending on time t.
We define the unit vectors
in the direction of r and
in the plane of the motion perpendicular to the radial direction, where is a unit vector normal to the plane of the motion.
Centrifugal and Coriolis terms [
The term is sometimes referred to as the centrifugal term, and the term as the Coriolis term. For example, see Shankar. Although these equations bear some resemblance in form to the centrifugal and Coriolis effects found in rotating reference frames, nonetheless these are not the same things. For example, the physical centrifugal and Coriolis forces appear only in non-inertial frames of reference. In contrast, these terms that appear when acceleration is expressed in polar coordinates are a mathematical consequence of differentiation; these terms appear wherever polar coordinates are used. In particular, these terms appear even when polar coordinates are used in inertial frames of reference, where the physical centrifugal and Coriolis forces never appear.
Inertial frame of reference S
and instantaneous non-inertial co-rotating frame of reference S′
. The co-rotating frame rotates at angular rate Ω equal to the rate of rotation of the particle about the origin of S′
at the particular moment t
. Particle is located at vector position r
) and unit vectors are shown in the radial direction to the particle from the origin, and also in the direction of increasing angle θ
normal to the radial direction. These unit vectors need not be related to the tangent and normal to the path. Also, the radial distance r
need not be related to the radius of curvature of thee path.