Polar coordinate system

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which points are given by an angle and a distance from the pole, known as the origin in the more common Cartesian coordinate system. The polar coordinate system is used in many fields, including mathematics, physics, engineering, navigation, and robotics, as it is straightforward to determine the relationship between two points in terms of angles and distance, something that can only be done in the Cartesian coordinate system by using trigonometric and other formulae. Because of the circular nature of the polar coordinate system, some curves can be expressed only by a polar equation, and for many others, it is much simpler to describe the curve with an equation in polar rather than Cartesian form.

A polar grid with several angles labeled.

Plotting and graphing

Plotting points with polar coordinates

The points (3,60°) and (4,210°)

As with all two-dimensional coordinate systems, there are two polar coordinates: r (the radial coordinate) and θ (the angular coordinate, polar angle, or azimuth angle, sometimes represented as φ or t). The r coordinate represents the radial distance from the pole, and the θ coordinate represents the anticlockwise (counterclockwise) angle from the 0° ray (sometimes called the polar axis), known as the positive x-axis on the Cartesian coordinate plane.

For example, the polar coordinates (3,60°) would be plotted as a point 3 units from the origin on the 60° ray. The coordinates (−3,240°) would also be plotted at this point because a negative radial distance is measured as a positive distance on the opposite ray (240° - 180° = 60°).

Converting between polar and Cartesian coordinates

The two polar coordinates r and θ can be converted to Cartesian coordinates by

$x=r\cos \theta \,$

$y=r\sin \theta \,$

From those two formulas, conversion formulas in terms of x and y are derived, including

$r={\sqrt {x^{2}+y^{2}}}\,$

$\theta =\arctan {\frac {y}{x}}\qquad x\neq 0\,$

If x = 0, then if y is positive θ = 90° (π/2 radians) and if y is negative θ = 270° (3π/2 radians).

Polar equations

The equation of a curve expressed in polar coordinates is known as a polar equation, and is usually written with r as a function of θ. A polar curve is symmetric about the horizontal (0°/180°) ray, because if θ is replaced by −θ in its equation, then it produces a mathematically equivalent equation. A polar curve is also symmetric about the vertical (90°/270°) ray by replacing θ with π−θ, and about the pole by replacing r with −r. Any polar curve can be rotated α° counterclockwise about the pole by substituting θ−α in the equation for θ.

A circle with equation r(θ) = 1.

Circle

The are several ways to write the polar equation of a circle, which conform to circles at different locations and of different sizes.

For a circle with a center at the pole and radius a the equation is

$r(\theta )=a\,$

For a circle with a center at (r₀, φ) and radius r₀ the equation is

$r(\theta )=2r_{0}\cos(\theta -\varphi )\,$

For any circle with a center at (r₀, φ) and radius a the equation is

$r^{2}-2rr_{0}\cos(\theta -\varphi )+r_{0}^{2}=a^{2}\,$

Line

A line can be expressed as a polar equation if it runs through the pole or if it is perpendicular to another line which does.

If a line does run through the pole, its equation can be represented by the equation

$\theta =\varphi \,$, where φ is the angle of elevation of the line, or

$\theta =\arctan(m)\,$, where m is the slope of the line in the Cartesian coordinate system.

If a line does not run through the pole, but runs through the point (r₀, φ), and is perpendicular to the line θ = φ its equation is,

$r(\theta )={r_{0}}\sec(\theta -\varphi )\,$.

From this it is derived that a vertical line has the equation

$r(\theta )=a\sec(\theta )\,$,

where a is the distance of the line from the 90°/270° line. If it is east of the line, a is positive, and if it is west, a is negative.

A horizontal line has the equation

$r(\theta )=a\csc(\theta )\,$,

where a is the distance of the line from the 0°/180° line. If it is north of the line, a is positive, and if it is south, a is negative.

A limacon with equation r(θ) = 3/4 + 3/2 cosθ

Limaçon

A limaçon, also known as a limaçon of Pascal, is a heart-shaped mathematical curve. It is given by the equations

$r(\theta )=a+b\cos \theta \,$

for a limaçon centered on the 0°/180° line, or

$r(\theta )=a+b\sin \theta \,$

for a limaçon centered on the 90°/270° line.

There are three types of limaçons, depending on the relationship between a and b. If a>b, then it is a dimpled limaçon, if a<b, it is a limaçon with an inner loop, and if a=b, it is a cardioid. A limaçon can be produced as a locus by tracing the path of a chosen point of a circle which rolls without slipping around another circle which is fixed.

A cardioid with equation r(θ) = 1 − sinθ.

Cardioid

A cardioid is a special limaçon where a and b are equal. It is it given by the equations

$r(\theta )=a+a\cos \theta \,$

for a cardioid centered on the 0°/180° line, or

$r(\theta )=a+a\sin \theta \,$

for a cardioid centered on the 90°/270° line.

Cardioids got their name from the greek kardioeides, literally heart shape, because of their resemblance to a heart. A cardioid can be produced as a locus by tracing the path of a chosen point of a circle which rolls without slipping around another circle which is fixed but which has the same radius as the rolling circle.

A lemniscate with equation r² = cos2θ.

Lemniscate

A lemniscate is a mathematical curve which looks like a figure eight. It is it given by the equations

$r^{2}=a\cos 2\theta \,$

for a horizontal lemniscate, or

$r^{2}=a\sin 2\theta \,$

for a vertical one.

The lemniscate was first described in 1694 by Jakob Bernoulli as a modification of an ellipse, which is the locus of points for which the sum of the distances to each of two fixed focal points is a constant. A lemniscate, by contrast, is the locus of points for which the product of these distances is constant.

A polar rose with equation r(θ) = 2sin4θ.

Polar Rose

A polar rose is a mathematical curve which looks like a petalled flower. It is given by the equations

$r(\theta )=a\cos k\theta \,$ OR

$r(\theta )=a\sin k\theta \,$

If k is an integer, these equations will produce a k-petalled rose if k is odd, or a 2k-petalled rose if k is even. If k is not an integer, a disc is formed, as the number of petals is also not an integer. Note that with these equations it is impossible to make a rose with 2 more than a multiple of 4 (2, 6, 10, etc.) petals. The variable a represents the length of the petals of the rose.

One arm of an Archimedean spiral with equation r(θ) = θ for 0 < θ < 6π.

Archimedean spiral

The Archimedean spiral is a spiral that was discovered by Archimedes. It is represented by the equation:

$r(\theta )=a+b\theta \,$.

Changing the parameter a will turn the spiral, while b controls the distance between the arms.

Note that the Archimedean spiral has two arms, one for θ > 0 and one for θ < 0. The two arms are smoothly connected at the origin. Taking the mirror image of one arm across the 90°/270° line will yield the other arm.

Complex numbers

Complex numbers, written in rectangular form as a + bi, can also be expressed in polar form in two different ways:

1. $r(\cos \theta +i\sin \theta )\,$, abbreviated $r{\mbox{ cis }}\theta \,$
2. $re^{i\theta }\,$

of which both are equivalent as per Euler's formula. To convert between rectangular and polar complex numbers, the following conversion formulas are used:

$a=r\cos \theta \,$

$b=r\sin \theta \,$

and therefore $r={\sqrt {a^{2}+b^{2}}}\,$

For the operations of multiplication, division, and exponentiation, and finding roots of complex numbers, it is much easier to use polar complex numbers than rectangular complex numbers. In abbreviated form:

• Multiplication: $(r{\mbox{ cis }}\theta )*(R{\mbox{ cis }}\varphi )=rR{\mbox{ cis }}(\theta +\varphi )\,$
• Division: ${\frac {r{\mbox{ cis }}\theta }{R{\mbox{ cis }}\varphi }}={\frac {r}{R}}{\mbox{ cis }}(\theta -\varphi )\,$
• Exponentiation (De Moivre's formula): $(r{\mbox{ cis }}\theta )^{n}=r^{n}{\mbox{ cis }}(n\theta )\,$

Vector calculus

Calculus can be applied to equations expressed in polar coordinates. Let $\mathbf {r}$ be the postion vector $(r\cos(\theta ),r\sin(\theta ))\,$, with r and $\theta$ depending on time t, ${\hat {\mathbf {r} }}$ be a unit vector in the direction $\mathbf {r}$ and ${\hat {\boldsymbol {\theta }}}$ be a unit vector at right angles to $\mathbf {r}$. The first and second derivatives of position are

${\frac {d\mathbf {r} }{dt}}={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\theta }}{\hat {\boldsymbol {\theta }}},$.

${\frac {d^{2}\mathbf {r} }{dt^{2}}}=({\ddot {r}}-r{\dot {\theta }}^{2}){\hat {\mathbf {r} }}+(r{\ddot {\theta }}+2{\dot {r}}{\dot {\theta }}){\hat {\boldsymbol {\theta }}}.$

Let $\mathbf {A}$ be the area swept out by a line joining the focus to a point on the curve. In the limit $d\mathbf {A}$ is half the area of the parallelogram formed by $\mathbf {r}$ and $d\mathbf {r}$,

$dA={\begin{matrix}{\frac {1}{2}}\end{matrix}}|\mathbf {r} \times d\mathbf {r} |$,

and the total area will be the integral of $d\mathbf {A}$ with respect to time.

Kepler's laws of planetary motion

Ellipse, showing semi-latus rectum

Polar coordinates are a natural setting for expressing Kepler's laws of planetary motion. Kepler's first law states that the orbit of a planet around a star forms a ellipse with one focus at the center of mass of the system. The polar equation of a ellipse with one focus at the origin is

$r={l \over (1+e\cos \theta )}$

where l is the semi-latus rectum and e is the eccentricity.

Kepler's second law

Kepler's second law, the law of equal areas states that A line joining a planet and its star sweeps out equal areas during equal intervals of time, that is $d\mathbf {A} \over dt$ is constant. These equations can be derived from Newton's laws of motion, a full derivation of these using polar coordinates is discussed in Kepler's laws of planetary motion.

Like the ellipse, the other conic sections are expressed by the same equation as the ellipse

$r={l \over (1+e\cos \theta )}.$

If e < 1 this equation defines an ellipse, if e = 1 it gives a parabola and if e > 1 it gives a hyperbola.

History

See also History of trigonometric functions

It is known that the Greeks used the concepts of angle and radius. The astronomer Hipparchus (190-120 BC) tabulated 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 his famous spiral, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system.

There are various accounts of who first introduced 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 at about the same time. Saint-Vincent wrote about them privately in 1625 and published in 1647, while Cavalieri published in 1635 with a corrected version appearing in 1653. Cavalieri first utilized polar coordinates to solve a problem relating to the area within an Archimedian 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 was the first to look upon polar coordinates as a method of locating any point in the plane. Newton examined the transformations between polar coordinates and nine other coordinate systems. In 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 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 and Leonhard Euler are credited with extending the concept of polar coordinates to three dimensions.

See also

• List of canonical coordinate transformations
• Point plotting
• Point (geometry)
• Line (mathematics)
• Plane (mathematics)

Other coordinate systems

• Coordinates (mathematics)
• Coordinate systems
• Cylindrical coordinate system
• Curvilinear coordinates
• Orthogonal coordinates
• Elliptic coordinates
• Hyperbolic coordinates
• Stereographic projection
• Parallel coordinates
• Geocentric coordinates

References

• Brown, Richard G. (1997). Andrew M. Gleason (ed.). Advanced Mathematics: Precalculus with Discrete Mathematics and Data Analysis. Evanston, Illinois: McDougal Littell Inc. ISBN 0-395-77114-5.
• Weisstein, Eric W. "Polar Coordinates". MathWorld.
• "Polar coordinates". Retrieved 2006-05-25.
• Ward, Robert L. "Analytic Geometry: Polar Coordinates". Retrieved 2006-05-25.