Polar coordinate system

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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 \cos \theta \,]

[y = r \sin \theta \,]

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 = \sqrt \,]

[\theta = \arctan \frac\qquad x \ne 0 \,]

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 [(\textstyle\frac,\textstyle\frac)]. 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.

Contents

1 Polar equations

1.1 Line
1.2 Circle
1.3 Limaçon
1.4 Cardioid
1.5 Lemniscate
1.6 Polar Rose
1.7 Archimedean spiral

2 Complex numbers
3 See also

3.1 Other coordinate systems

4 References

Polar equations

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

[\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) = \frac \,].

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° 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(\theta) = a\csc(\theta) \,],

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(\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 - 2 r r_0 \cos(\theta - \varphi) + r_0^2 = a^2 \,]

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° ray, or

[r(\theta) = a + b \sin \theta \,]

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(\theta) = a + a \cos \theta \,]

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

[r(\theta) = a + a \sin \theta \,]

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 \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.

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.

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° ray 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:

[r(\cos\theta+i\sin\theta) \,], abbreviated [r \mbox \theta \,]
[r e^ \,]

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 \,]

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 \theta) * (R \mbox \varphi) = rR \mbox (\theta+\varphi) \,]
Division: [\frac \theta} \varphi} = \frac \mbox (\theta-\varphi) \,]
Exponentiation (De Moivre's formula): [(r \mbox \theta)^n = r^n \mbox (n\theta) \,]

References

, [Polar Coordinates] at MathWorld.

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