POLAR COORDINATE SYSTEM : Comprehensive Encyclopedia

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 sheet of polar graph paper.

1 Plotting and graphing
- 1.1 Plotting points with polar coordinates
- 1.2 Converting between polar and Cartesian coordinates
2 Polar equations
3 Complex numbers
4 History
5 See also
- 5.1 Other coordinate systems
6 References

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, if you had the coordinates (3,60°), the point would be plotted 3 units from the origin on the 60° ray. 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.

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

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

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

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

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 if replacing θ by −θ in its equation produces a mathematically equivalent equation, symmetric about the vertical (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 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

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

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:

$r(\cos\theta+i\sin\theta) \,$ , abbreviated $r \mbox{ cis } \theta \,$
$r e^{i\theta} \,$

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): $(r \mbox{ cis } \theta)^n = r^n \mbox{ cis } (n\theta) \,$

History

See also History of trigonometric functions

The Greeks used the concepts of angle and radius. The astromer Hipparchus (190-120BC) tabulated a table of chord function giving the length of the chord for each angle, and there are references to him using polar coordinates in establishing stellar positions.^[1] In On Spirals Archimedean describes his famous spiral, a function whose radius depends on the angle. The Greek work did not extend to a full coordinate system.

There are various accounts of who first introduced polar coordinates. The full history of the subject is described in Julian Lowell Coolidge's Origin of Polar Coordinates.^[2]^[3]

Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts at about the same time. St Vincent wrote about them privately in 1625 and published in 1647, whilst Cavalieri published in 1635 with a correct version appearing in 1653. According to research by Harvard mathematics professor and author Julian Lowell Coolidge,^[4] Bonaventura Cavalieri was the first mathematician to utilize the concept of polar coordinates, which he used to solve a problem relating to the area within an Archimedian spiral. Blaise Pascal subsequently used polar coordinates to solve problems relating to parabolas. Isaac Newton was the first to look upon polar coordinates as a method of locating any point in the plane, while Jacob Bernoulli developed the first polar equations. Alexis Clairaut and Leonhard Euler are credited with extending the concept of polar coordinates to three dimensions.

References

^ Friendly, Michael. Milestones in the History of Thematic Cartography, Statistical Graphics, and Data Visualization. Retrieved on [[September 10, 2006]].
^ The MacTutor History of Mathematics archive: Coolidge's Origin of Polar Coordinates
^ Coolidge, Julian (1952). "The Origin of Polar Coordinates". American Mathematical Monthly 59: 78-85.
^ The MacTutor History of Mathematics archive: Coolidge's Origin of Polar Coordinates

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

POLAR COORDINATE SYSTEM

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