A polar grid with several angles labeled in degrees

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. The polar coordinate system is especially useful in situations where the relationship between two points is most easily expressed in terms of angles and distance; in the more familiar Cartesian or rectangular coordinate system, such a relationship can only be found through trigonometric formulae. Image File history File links Polar_graph_paper. ... Image File history File links Polar_graph_paper. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... 2-dimensional renderings (ie. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ... An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. ... Distance is a numerical description of how far apart objects are at any given moment in time. ... Fig. ... Wikibooks has a book on the topic of Trigonometry Trigonometry (from Greek trigōnon triangle + metron measure[1]) is a branch of mathematics that deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled triangles). ...

As the coordinate system is two-dimensional, each point is determined by two polar coordinates: the radial coordinate and the angular coordinate. The radial coordinate (usually denoted as r) denotes the point's distance from a central point known as the pole (equivalent to the origin in the Cartesian system). The angular coordinate (also known as the polar angle or the azimuth angle, and usually denoted by θ or t) denotes the positive or anticlockwise (counterclockwise) angle required to reach the point from the 0° ray or polar axis (which is equivalent to the positive x-axis in the Cartesian coordinate plane).^[1] Azimuth is the horizontal component of a direction (compass direction), measured around the horizon, usually from the north toward the east — i. ... This article is about angles in geometry. ... The Clockwise direction A clockwise motion is one that proceeds like the clocks hands: from the top to the right, then down and then to the left, and back to the top. ... In Euclidean geometry, a ray, or half-line, given two distinct points A (the origin) and B on the ray, is set of points C on the line containing points A and B such that A is not strictly between C and B. O----O-----*---> A B C In geometric... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

History

See also: History of trigonometric functions

The concepts of angle and radius were already used by ancient peoples of the 1st millennium BCE. The astronomer 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.^[2] 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. The history of trigonometric functions may span about 4000 years. ... BCE is a TLA that may stand for: Before the Common Era, date notation equivalent to BC (e. ... A recreation of the famous Library of Alexandria Greek astronomy is the astronomy of those who spoke Greek in classical antiquity. ... Hipparchus. ... A chord of a curve is a geometric line segment whose endpoints both lie on the curve. ... Archimedes (Greek: c. ... An Archimedean spiral is a curve which in polar coordinates (r, θ) can be described by the equation with real numbers a and b. ...

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.^[3] 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. Harvard University (incorporated as The President and Fellows of Harvard College) is a private university in Cambridge, Massachusetts, USA and a member of the Ivy League. ... Julian Lowell Coolidge (September 28, 1873 - March 5, 1954) was an American mathematician and a professor and chairman of the Mathematics Department at Harvard University. ... Grégoire de Saint-Vincent (1584 Bruges - 1667 Ghent), a Jesuit, was a mathematician who independently discovered the Mercator series, the expansion of log(1 + x) in ascending powers of x. ... Coins illustrating Cavalieris principle Bonaventura Francesco Cavalieri (in Latin, Cavalerius) (1598–November 30, 1647) was an Italian mathematician best known today for Cavalieris principle, which states that the volumes of two objects are equal if the areas of corresponding cross-sections are in all cases equal. ... An Archimedean spiral is a curve which in polar coordinates (r, θ) can be described by the equation with real numbers a and b. ... Blaise Pascal (pronounced ), (June 19, 1623–August 19, 1662) was a French mathematician, physicist, and religious philosopher. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane...

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.^[4] 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. Method of Fluxions was a book by Isaac Newton. ... Sir Isaac Newton, (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist, regarded by many as the greatest figure in the history of science. ... Acta Eruditorum (Latin: reports, acts of the scholars) was the first scientific journal of the German lands, published from 1682 to 1782. ... Jakob Bernoulli. ... Curvature is the amount by which a geometric object deviates from being flat. ...

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.^[5][6] Alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them.^[3] Gregorio Fontana (December 7, 1735 Villa di Nogaredo - August 24, 1803 Milan) was an Italian mathematician. ... The English language is a West Germanic language that originates in England. ... George Peacock George Peacock (April 9, 1791 – November 8, 1858) was an English mathematician. ... Sylvestre François de Lacroix (April 28, 1765–May 24, 1843) was a French mathematician. ... Alexis Claude Clairault (or Clairaut) ( May 3, 1713 - May 17, 1765) was a French mathematician. ... Leonhard Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ...

Plotting points with polar coordinates

The points (3,60°) and (4,210°) on a polar coordinate system

Each point in the polar coordinate system can be described with the two polar coordinates, which are usually called 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.^[1] Image File history File links Example for circular coordinate system File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Example for circular coordinate system File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Azimuth is the horizontal component of a direction (compass direction), measured around the horizon, usually from the north toward the east — i. ... In Euclidean geometry, a ray, or half-line, given two distinct points A (the origin) and B on the ray, is set of points C on the line containing points A and B such that A is not strictly between C and B. O----O-----*---> A B C In geometric... Fig. ...

For example, the polar coordinates (3, 60°) would be plotted as a point 3 units from the pole 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 (the ray reflected about the origin, which differs from the original ray by 180°). Point plotting is an elementary skill required in analytic geometry. ...

One important aspect of the polar coordinate system, not present in the Cartesian coordinate system, is that a single point can be expressed with an infinite number of different coordinates. This is because any number of multiple revolutions can be made around the central pole without affecting the actual location of the point plotted. In general, the point (r, θ) can be represented as (r, θ ± n×360°) or (−r, θ ± (2n + 1)180°), where n is any integer.^[7] The integers are commonly denoted by the above symbol. ...

The arbitrary coordinates (0, θ) are conventionally used to represent the pole, as regardless of the θ coordinate, a point with radius 0 will always be on the pole.^[8] To get a unique representation of a point, it is usual to limit r to non-negative numbers r ≥ 0 and θ to the interval [0, 360°) or (−180°, 180°] (or, in radian measure, [0, 2π) or (−π, π]).^[9] A negative number is a number that is less than zero, such as −3. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

Angles in polar notation are generally expressed in either degrees or radians, using the conversion 2π rad = 360°. The choice depends largely on the context. Navigation applications use degree measure, while some physics applications (specifically rotational mechanics) and almost all mathematical literature on calculus use radian measure.^[10] Some common angles, measured in radians. ... When a circles diameter is 1, its circumference is Ï€. The mathematical constant Ï€ is an irrational real number, approximately equal to 3. ... Table of geography, hydrography, and navigation, from the 1728 Cyclopaedia. ... Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ... Calculus (from Latin, pebble or little stone) is a major area in mathematics where infinitesimal data yields global information. ...

Converting between polar and Cartesian coordinates

A diagram illustrating the conversion formulae

The two polar coordinates r and θ can be converted to the Cartesian coordinates x and y by using the trigonometric functions sine and cosine: Image File history File links Polar_to_cartesian. ... Image File history File links Polar_to_cartesian. ... Fig. ... All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other...

while the two Cartesian coordinates x and y can be converted to polar coordinate r by

(by a simple application of the Pythagorean theorem).

To determine the angular coordinate θ, the following two ideas must be considered: In mathematics, the Pythagorean theorem or Pythagoras theorem is a relation in Euclidean geometry among the three sides of a right triangle. ...

• For r = 0, θ can be set to any real value.
• For r ≠ 0, to get a unique representation for θ, it must be limited to an interval of size 2π. Conventional choices for such an interval are [0, 2π) and (−π, π].

To obtain θ in the interval [0, 2π), the following may be used (arctan denotes the inverse of the tangent function): In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other...

To obtain θ in the interval (−π, π], the following may be used:^[11]

One may avoid having to keep track of the numerator and denominator signs by use of the atan2 function, which has separate arguments for the numerator and the denominator. Atan2 is a two-parameter function for computing the arctangent in the C programming language. ...

Polar equations

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 θ. The resulting curve then consists of points of the form (r(θ), θ) and can be regarded as the graph of the polar function r. In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ... Partial plot of a function f. ... relation graph theory In mathematics, the graph of a function f is the collection of all ordered pairs (x,f(x)). In particular, graph means the graphical representation of this collection, in the form of a curve or surface, together with axes, etc. ...

Different forms of symmetry can be deduced from the equation of a polar function r. If r(−θ) = r(θ) the curve will be symmetrical about the horizontal (0°/180°) ray, if r(π−θ) = r(θ) it will be symmetric about the vertical (90°/270°) ray, and if r(θ−α°) = r(θ) it will be rotationally symmetric α° counterclockwise about the pole. Sphere symmetry group o. ... The triskelion appearing on the Isle of Man flag. ... The Clockwise direction A clockwise motion is one that proceeds like the clocks hands: from the top to the right, then down and then to the left, and back to the top. ...

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. Rose with k = 7 petals Some samples of k=n/d. ... An Archimedean spiral is a curve which in polar coordinates (r, θ) can be described by the equation with real numbers a and b. ... A lemniscate of Bernoulli In mathematics, the Lemniscate of Bernoulli is a figure-eight shaped algebraic curve described by a Cartesian equation of the form: Graphing this equation produces a curve similar to the symbol. ... In mathematics, limaçons (pronounced with a soft c), also known as limaçons of Pascal, are heart-shaped mathematical curves. ... In geometry, the cardioid is an epicycloid which has one and only one cusp. ...

For the circle, line, and polar rose below, it is understood that there are no restrictions on the domain and range of the curve.

Circle

A circle with equation r(θ) = 1

The general equation for a circle with a center at (r₀, φ) and radius a is Image File history File links Download high resolution version (2043x2043, 126 KB) Summary A circle with equation r=1 Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ... Image File history File links Download high resolution version (2043x2043, 126 KB) Summary A circle with equation r=1 Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. ...

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.^[12]

Line

Radial lines (those running through the pole) are represented by the equation

,

where φ is the angle of elevation of the line; that is, φ = arctan m where m is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line θ = φ perpendicularly at the point (r₀, φ) has the equation Look up Slope in Wiktionary, the free dictionary. ... Fig. ...

Polar rose

A polar rose with equation r(θ) = 2 sin 4θ

A polar rose is a famous mathematical curve that looks like a petalled flower, and that can be expressed as a simple polar equation, Image File history File links Download high resolution version (2043x2043, 160 KB) Summary A polar rose with equation r=2sin(4θ) Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version... Image File history File links Download high resolution version (2043x2043, 160 KB) Summary A polar rose with equation r=2sin(4θ) Licensing I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version... In mathematics, a rose is a sinusoid plotted in polar coordinates. ...

for any constant φ₀ (including 0). 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 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.
In mathematics, the parity of an object refers to whether it is even or odd. ... In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ...

Archimedean spiral

One arm of an Archimedean spiral with equation r(θ) = θ for 0 < θ < 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 The picture of an Archimedean spiral was produced using the octave plotting program with the input # Output to png file: set terminal png small color; set output archimedian_spiral. ... The picture of an Archimedean spiral was produced using the octave plotting program with the input # Output to png file: set terminal png small color; set output archimedian_spiral. ... An Archimedean spiral is a curve which in polar coordinates (r, &#952;) can be described by the equation with real numbers a and b. ... Archimedes (Greek: c. ...

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 θ > 0 and one for θ < 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.
Wikibooks has more on the topic of Conic section Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ...

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: Image File history File links Elps-slr. ... Image File history File links Elps-slr. ... The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae. ...

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 . (This page refers to eccentricity in mathematics. ... In a conic section, the latus rectum is the chord parallel to the directrix through the focus. ... In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... For other uses, see Ellipse (disambiguation). ...

Complex numbers

An illustration of a complex number z plotted on the complex plane

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 Image File history File links Imaginarynumber2. ... Image File history File links Imaginarynumber2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

where i is the imaginary unit, or can alternatively be written in polar form (via the conversion formulae given above) as In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ...

and from there as

where e is Euler's number, which are equivalent as shown by Euler's formula.^[13] (Note that this formula, like all those involving exponentials of angles, assumes that the angle θ is expressed in radians.) To convert between the rectangular and polar forms of a complex number, the conversion formulae given above can be used. e is the unique number such that the value of the derivative (slope of a tangent line) of f (x)=ex (blue curve) at the point x=0 is exactly 1. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

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: In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. ...

• Multiplication:

• Division:

• Exponentiation (De Moivre's formula):

de Moivres formula, named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and any integer n it holds that The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. ...

Calculus

Calculus can be applied to equations expressed in polar coordinates.^[14][15] Calculus (from Latin, pebble or little stone) is a major area in mathematics where infinitesimal data yields global information. ...

The angular coordinate θ is expressed in radians throughout this section, which is the conventional choice when doing calculus.

Differential calculus

We have the following formulas:

To find the Cartesian slope of the tangent line to a polar curve r(θ) at any given point, the curve is first expressed as a system of parametric equations.

Differentiating both equations with respect to θ yields For a non-technical overview of the subject, see Calculus. ...

Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point (r, r(θ)):

Integral calculus

The integration region R is bounded by the curve r(θ) and the rays θ = a and θ = b.

Let R denote the region enclosed by a curve r(θ) and the rays θ = a and θ = b, where 0 < b − a < 2π. Then, the area of R is Image File history File links Polar_coordinates_integration_region. ... Image File history File links Polar_coordinates_integration_region. ...

The region R is approximated by n sectors (here, n = 5).

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 Δθ, 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 θ_i be the midpoint of the subinterval, and construct a sector with the center at the pole, radius r(θ_i), central angle Δθ and arc length . The area of each constructed sector is therefore equal to . Hence, the total area of all of the sectors is Image File history File links Polar_coordinates_integration_Riemann_sum. ... Image File history File links Polar_coordinates_integration_Riemann_sum. ... A circular sector or circle sector also known as a pie piece is the portion of a circle enclosed by two radii and an arc. ...

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. In mathematics, a Riemann sum is a method for approximating the values of integrals. ...

Generalization

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: Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... In calculus, the substitution rule is a tool for finding antiderivatives and integrals. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

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(θ) and the rays θ = a and θ = 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 The integral of any Gaussian function (named after Carl Friedrich Gauss) is quickly reducible to the Gaussian integral This integral cannot be computed by elementary means since the function has no simple antiderivative. ...

Vector calculus

Vector calculus can also be applied to polar coordinates. Let be the position vector , with r and θ depending on time t, be a unit vector in the direction and be a unit vector at right angles to . The first and second derivatives of position are Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ...

Three dimensions

The polar coordinate system is extended into three dimensions with two different coordinate systems, the cylindrical and spherical coordinate systems, both of which include two-dimensional or planar polar coordinates as a subset. In essence, the cylindrical coordinate system extends polar coordinates by adding an additional distance coordinate, while the spherical system instead adds an additional angular coordinate.

Cylindrical coordinates

2 points plotted with cylindrical coordinates

Main article: Cylindrical coordinate system

The cylindrical coordinate system is a coordinate system that essentially extends the two-dimensional polar coordinate system by adding a third coordinate measuring the height of a point above the plane, similar to the way in which the Cartesian coordinate system is extended into three dimensions. The third coordinate is usually denoted h, making the three cylindrical coordinates (r, θ, h). Image File history File links Example for a cylindrical coordinate system File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Image File history File links Example for a cylindrical coordinate system File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... 2 points plotted with cylindrical coordinates The cylindrical coordinate system is a three-dimensional coordinate system which essentially extends circular polar coordinates by adding a third coordinate (usually denoted ) which measures the height of a point above the plane. ... Fig. ...

The three cylindrical coordinates can be converted to Cartesian coordinates by

Spherical coordinates

A point plotted using spherical coordinates

Main article: Spherical coordinate system

Polar coordinates can also be extended into three dimensions using the coordinates (ρ, φ, θ), where ρ is the distance from the origin, φ is the angle from the z-axis (called the colatitude or zenith and measured from 0 to 180°) and θ is the angle from the x-axis (as in the polar coordinates). This coordinate system, called the spherical coordinate system, is similar to the latitude and longitude system used for Earth, with the origin in the centre of Earth, the latitude δ being the complement of φ, determined by δ = 90° − φ, and the longitude l being measured by l = θ − 180°.^[16] Image File history File links Download high resolution version (1000x1000, 16 KB)Spherical Coordinates Drawn by the Jacob Rus in Adobe Illustrator using fonts from AMS LaTeX, to replace earlier drawing by Cryogen File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to... Image File history File links Download high resolution version (1000x1000, 16 KB)Spherical Coordinates Drawn by the Jacob Rus in Adobe Illustrator using fonts from AMS LaTeX, to replace earlier drawing by Cryogen File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to... A point plotted using the spherical coordinate system In mathematics, the spherical coordinate system is a coordinate system for representing geometric figures in three dimensions using three coordinates: the radial distance of a point from a fixed origin, the zenith angle from the positive z-axis, and the azimuth angle... Latitude, usually denoted symbolically by the Greek letter phi, , gives the location of a place on Earth north or south of the equator. ... Longitude, sometimes denoted by the Greek letter λ (lambda),[1][2] describes the location of a place on Earth east or west of a north-south line called the Prime Meridian. ...

The three spherical coordinates are converted to Cartesian coordinates by

Applications

Polar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane. They are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point. For instance, the examples above show how elementary polar equations suffice to define curves — such as the Archimedean spiral — whose equation in the Cartesian coordinate system would be much more intricate. Moreover, many physical systems — such as those concerned with bodies moving around a central point or with phenomena originating from a central point — are simpler and more intuitive to model using polar coordinates. The initial motivation for the introduction of the polar system was the study of circular and orbital motion. In physics, circular motion is rotation along a circle: a circular path or a circular orbit. ... In physics, orbital motion is the either a motion of a planet in a planetary orbit, or a motion of an electron around the nucleus of an atom, or any other motion of parts of a bound system. ...

Position and navigation

Polar coordinates are used often in navigation, as the destination or direction of travel can be given as an angle and distance from the object being considered. For instance, aircraft use a slightly modified version of the polar coordinates for navigation. In this system, the one generally used for any sort of navigation, the 0° ray is generally called heading 360, and the angles continue in a clockwise direction, rather than counterclockwise, as in the mathematical system. Heading 360 corresponds to magnetic north, while headings 90, 180, and 270 correspond to magnetic east, south, and west, respectively.^[17] Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read niner-zero by air traffic control).^[18] Table of geography, hydrography, and navigation, from the 1728 Cyclopaedia. ... Look up aircraft in Wiktionary, the free dictionary. ... The Clockwise direction A clockwise motion is one that proceeds like the clocks hands: from the top to the right, then down and then to the left, and back to the top. ... This is about the geographic meaning of North Pole. ... Air Traffic Control Towers (ATCTs) at Amsterdams Schiphol Airport Air traffic control (ATC) is a service provided by ground-based controllers who direct aircraft on the ground and in the air. ...

Modeling

Systems displaying radial symmetry provide natural settings for the polar coordinate system, with the central point acting as the pole. A prime example of this usage is the groundwater flow equation when applied to radially symmetric wells. Systems with a radial force are also good candidates for the use of the polar coordinate system. These systems include gravitational fields, which obey the inverse-square law, as well as systems with point sources, such as radio antennas. In biology, radial symmetry is a property of some multicellular organisms. ... The groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through a porous medium (e. ... A central force acting on an object is one whose magnitude depends only on the scalar distance r of the object from the origin and whose direction is along the position vector from the origin to the object. ... “Gravity” redirects here. ... This diagram shows how the law works. ... Look up point source in Wiktionary, the free dictionary. ... A Yagi-Uda beam antenna Short Wave Curtain Antenna (Moosbrunn, Austria) A building rooftop supporting numerous dish and sectored mobile telecommunications antennas (Doncaster, Victoria, Australia) An antenna or aerial is a transducer designed to transmit or receive radio waves which are a class of electromagnetic waves. ...

Radially asymmetric systems may also be modeled with polar coordinates. For example, a microphone's pickup pattern illustrates its proportional response to an incoming sound from a given direction, and these patterns can be represented as polar curves. The curve for a standard cardioid microphone, the most common unidirectional microphone, can be represented as r = 0.5 + 0.5 sin θ.^[19] A microphone, sometimes referred to as a mike or mic (both IPA pronunciation: ), is an acoustic to electric transducer or sensor that converts sound into an electrical signal. ... A microphone, sometimes referred to as a mike or mic (both IPA pronunciation: ), is an acoustic to electric transducer or sensor that converts sound into an electrical signal. ...

See also

• List of canonical coordinate transformations

This is a list of canonical coordinate transformations. ...

References

For the Manfred Mann album, see 2006 (album). ... September 10 is the 253rd day of the Gregorian calendar (254th in leap years). ... Julian Lowell Coolidge (September 28, 1873 - March 5, 1954) was an American mathematician and a professor and chairman of the Mathematics Department at Harvard University. ... For the Manfred Mann album, see 2006 (album). ... September 10 is the 253rd day of the Gregorian calendar (254th in leap years). ... For the Manfred Mann album, see 2006 (album). ... April 13 is the 103rd day of the year (104th in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... September 22 is the 265th day of the year in the Gregorian calendar (266th in leap years). ... For the Manfred Mann album, see 2006 (album). ... May 25 is the 145th day of the year (146th in leap years) in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... November 25 is the 329th (in leap years the 330th) day of the year in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... November 25 is the 329th (in leap years the 330th) day of the year in the Gregorian calendar. ... For the Manfred Mann album, see 2006 (album). ... // 1400 - Owain Glyndŵr declared Prince of Wales by his followers. ... For the Manfred Mann album, see 2006 (album). ... November 26 is the 330th day (331st in leap years) of the year in the Gregorian calendar. ... 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the CE era. ... January 15 is the 15th day of the year in the Gregorian calendar. ...

External links

• FooPlot (online function plotter in polar coordinates)
• Online conversion tool between polar and Cartesian coordinates