The polar coordinate system is a two-dimensional coordinate system in which points are given by an angle and a distance from the pole, called the origin in the Cartesian coordinate system.

The two polar coordinates r (the radial coordinate) and θ (the angular coordinate, polar angle, or azimuth angle, sometimes represented as φ or t) are defined in terms of Cartesian coordinates by

$x\; =\; r\; \backslash cos\; \backslash theta\; \backslash ,$

$y\; =\; r\; \backslash sin\; \backslash theta\; \backslash ,$

where r is the radial distance from the pole, and θ is the counterclockwise angle from the 0° ray (sometimes called the polar axis), which is the section of the Cartesian x-axis from the origin eastward.

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

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

$\backslash theta\; =\; \backslash arctan\; \backslash frac\{y\}\{x\}\backslash qquad\; x\; \backslash ne\; 0\; \backslash ,$

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

For example, if you had the coordinates (3, 60°), the point would be plotted 3 units from the origin on the 60° ray, at the Cartesian point $(\backslash textstyle\backslash frac\{3\}\{2\},\backslash textstyle\backslash frac\{3\backslash sqrt3\}\{2\})$. If you had the coordinates (−3, 240°), the point would be in the same location, because −3 units on the 240° ray is the same as 3 units on its opposite ray, the 60° ray.

Polar equations

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The equation of a line or curve expressed in polar coordinates is known as a polar equation, and is usually written with r as a function of θ. A polar line/curve is symmetric about the 0°/180° ray if replacing θ by −θ in its equation produces in an equivalent equation, symmetric about the 90°/270° ray if replacing θ by π−θ produces an equivalent equation, and symmetric about the pole if replacing r by −r produces an equivalent equation. Any polar line/curve can be rotated α° counterclockwise about the pole by substituting θ−α in the equation for θ.

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

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

$\backslash theta\; =\; \backslash arctan(m)\; \backslash ,$, 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,

$r(\backslash theta)\; =\; \backslash frac\{r\_0\}\{\backslash cos(\backslash theta-\backslash varphi)\}\; \backslash ,$.

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

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

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

A horizontal line has the equation

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

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

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

$r(\backslash theta)=a\; \backslash ,$

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

$r(\backslash theta)=2r\_0\; \backslash cos(\backslash theta-\backslash varphi)\; \backslash ,$

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

$r^2\; -\; 2\; r\; r\_0\; \backslash cos(\backslash theta\; -\; \backslash varphi)\; +\; r\_0^2\; =\; a^2\; \backslash ,$

Limaçon

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

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

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

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

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

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.

Cardioid

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

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

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

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

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

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.

Lemniscate

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

$r^2\; =\; a\; \backslash cos\; 2\backslash theta\; \backslash ,$

for a horizontal lemniscate, or

$r^2\; =\; a\; \backslash sin\; 2\backslash theta\; \backslash ,$

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.

Polar Rose

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

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

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

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.

Archimedean spiral

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

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

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° ray will yield the other arm.

Complex numbers

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

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

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

$a\; =\; r\; \backslash cos\; \backslash theta\; \backslash ,$

$b\; =\; r\; \backslash sin\; \backslash theta\; \backslash ,$

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

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

See also

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• 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
• Hyperbolic coordinates
• Stereographic projection
• Parallel coordinates
• Geocentric coordinates

References

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Coordinate systems

Polární soustava souřadnic | Polært koordinatsystem | Polarkoordinaten | Coordenadas polares | Coordonnées polaires | 極座標系 | 극좌표 | poolcoördinaten | Polárna sústava súradníc | Koordinaatisto | Polära koordinater