In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the polar coordinate system is a
two-dimensional coordinate system
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
in which each
point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
on a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
is determined by a
distance from a reference point and an
angle from a reference direction. The reference point (analogous to the origin of a
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
) is called the ''pole'', and the
ray
Ray may refer to:
Fish
* Ray (fish), any cartilaginous fish of the superorder Batoidea
* Ray (fish fin anatomy), a bony or horny spine on a fin
Science and mathematics
* Ray (geometry), half of a line proceeding from an initial point
* Ray (g ...
from the pole in the reference direction is the ''polar axis''. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''
azimuth''.
Angles in polar notation are generally expressed in either
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
s or
radians (2
rad being equal to 360°).
Grégoire de Saint-Vincent and
Bonaventura Cavalieri independently introduced the concepts in the mid-17th century, though the actual term "polar coordinates" has been attributed to
Gregorio Fontana in the 18th century. The initial motivation for the introduction of the polar system was the study of
circular and
orbital motion
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a p ...
.
Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as
spiral
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
Helices
Two major definitions of "spiral" in the American Heritage Dictionary are:[cylindrical
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infini ...]
and
spherical coordinate systems.
History
The concepts of angle and radius were already used by ancient peoples of the first millennium
BC. The
Greek astronomer and
astrologer
Astrology is a range of divinatory practices, recognized as pseudoscientific since the 18th century, that claim to discern information about human affairs and terrestrial events by studying the apparent positions of celestial objects. Dif ...
Hipparchus (190–120 BC) created a table of
chord
Chord may refer to:
* Chord (music), an aggregate of musical pitches sounded simultaneously
** Guitar chord a chord played on a guitar, which has a particular tuning
* Chord (geometry), a line segment joining two points on a curve
* 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
''On Spirals'' ( el, Περὶ ἑλίκων) is a treatise by Archimedes, written around 225 BC. Notably, Archimedes employed the Archimedean spiral in this book to square the circle and trisect an angle.
Contents
Preface
Archimedes begins ''O ...
'',
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
describes the
Archimedean spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...
, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system.
From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to
Mecca (
qibla)—and its distance—from any location on the Earth. From the 9th century onward they were using
spherical trigonometry and
map projection methods to determine these quantities accurately. The calculation is essentially the conversion of the
equatorial polar coordinates of Mecca (i.e. its
longitude and
latitude) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the
great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
through the given location and the Earth's poles and whose polar axis is the line through the location and its
antipodal point.
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
Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher le ...
professor
Julian Lowell Coolidge
Julian Lowell Coolidge (September 28, 1873 – March 5, 1954) was an American mathematician, historian and a professor and chairman of the Harvard University Mathematics Department.
Biography
Born in Brookline, Massachusetts, he graduated from Ha ...
's ''Origin of Polar Coordinates.''
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
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...
.
Blaise Pascal
Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic Church, Catholic writer.
He was a child prodigy who was educated by his father, a tax collector in Rouen. Pa ...
subsequently used polar coordinates to calculate the length of
parabolic arcs.
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. 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.
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
George Peacock FRS (9 April 1791 – 8 November 1858) was an English mathematician and Anglican cleric. He founded what has been called the British algebra of logic.
Early life
Peacock was born on 9 April 1791 at Thornton Hall, Denton, nea ...
's 1816 translation of
Lacroix's ''Differential and Integral Calculus''.
Alexis Clairaut was the first to think of polar coordinates in three dimensions, and
Leonhard Euler was the first to actually develop them.
Conventions
The radial coordinate is often denoted by ''r'' or
''ρ'', and the angular coordinate by
''φ'',
''θ'', or ''t''. The angular coordinate is specified as ''φ'' by
ISO standard
31-11. However, in mathematical literature the angle is often denoted by θ instead.
Angles in polar notation are generally expressed in either
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
s or
radians (2
rad being equal to 360°). Degrees are traditionally used in
navigation,
surveying
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...
, and many applied disciplines, while radians are more common in mathematics and mathematical
physics.
The angle ''φ'' is defined to start at 0° from a ''reference direction'', and to increase for rotations in either
clockwise (cw) or counterclockwise (ccw) orientation. For example, in mathematics, the reference direction is usually drawn as a
ray
Ray may refer to:
Fish
* Ray (fish), any cartilaginous fish of the superorder Batoidea
* Ray (fish fin anatomy), a bony or horny spine on a fin
Science and mathematics
* Ray (geometry), half of a line proceeding from an initial point
* Ray (g ...
from the pole horizontally to the right, and the polar angle increases to positive angles for ccw rotations, whereas in navigation (
bearing,
heading
Heading can refer to:
* Heading (metalworking), a process which incorporates the extruding and upsetting processes
* Headline, text at the top of a newspaper article
* Heading (navigation), the direction a person or vehicle is facing, usually si ...
) the 0°-heading is drawn vertically upwards and the angle increases for cw rotations. The polar angles decrease towards negative values for rotations in the respectively opposite orientations.
Uniqueness of polar coordinates
Adding any number of full
turns (360°) to the angular coordinate does not change the corresponding direction. Similarly, any polar coordinate is identical to the coordinate with the negative radial component and the opposite direction (adding 180° to the polar angle). Therefore, the same point (''r'', ''φ'') can be expressed with an infinite number of different polar coordinates and , where ''n'' is an arbitrary
integer. Moreover, the pole itself can be expressed as (0, ''φ'') for any angle ''φ''.
Where a unique representation is needed for any point besides the pole, it is usual to limit ''r'' to positive numbers () and ''φ'' to either the
interval or the interval , which in radians are or . Another convention, in reference to the usual codomain of the
arctan function, is to allow for arbitrary nonzero real values of the radial component and restrict the polar angle to . In all cases a unique azimuth for the pole (''r'' = 0) must be chosen, e.g., ''φ'' = 0.
Converting between polar and Cartesian coordinates
The polar coordinates ''r'' and ''φ'' can be converted to the Cartesian coordinates ''x'' and ''y'' by using the
trigonometric functions sine and cosine:
The Cartesian coordinates ''x'' and ''y'' can be converted to polar coordinates ''r'' and ''φ'' with ''r'' ≥ 0 and ''φ'' in the interval (−, ] by:
where hypot is the
Pythagorean addition, Pythagorean sum and
atan2
In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi < \theta \leq \pi) between the positive is a common variation on the
arctangent
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...
function defined as
If ''r'' is calculated first as above, then this formula for ''φ'' may be stated more simply using the
arccosine function:
Complex numbers
Every
complex number can be represented as a point in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, 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
where ''i'' is the
imaginary unit, or can alternatively be written in polar form as
and from there, by
Euler's formula, as
where ''e'' is
Euler's number, and ''φ'', expressed in radians, is the
principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positive ...
of the complex number function
arg
Arg or ARG may refer to:
Places
*''Arg'' () means "citadel" in Persian, and may refer to:
**Arg, Iran, a village in Fars Province, Iran
**Arg (Kabul), presidential palace in Kabul, Afghanistan
**Arg, South Khorasan, a village in South Khorasan P ...
applied to ''x'' + ''iy''. To convert between the rectangular and polar forms of a complex number, the conversion formulae given above can be used. Equivalent are the and
angle notations:
For the operations of
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
,
division,
exponentiation, and
root extraction
In mathematics, a radicand, also known as an nth root, of a number ''x'' is a number ''r'' which, when raised to the power ''n'', yields ''x'':
:r^n = x,
where ''n'' is a positive integer, sometimes called the ''degree'' of the root. A roo ...
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:
; Multiplication:
; Division:
; Exponentiation (
De Moivre's formula):
; Root Extraction (Principal root):
Polar equation of a curve
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 ''φ''. The resulting curve then consists of points of the form (''r''(''φ''), ''φ'') and can be regarded as the
graph of the polar function ''r''. Note that, in contrast to Cartesian coordinates, the independent variable ''φ'' is the ''second'' entry in the ordered pair.
Different forms of
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
can be deduced from the equation of a polar function ''r'':
* If the curve will be symmetrical about the horizontal (0°/180°) ray;
* If it will be symmetric about the vertical (90°/270°) ray:
* If it will be
rotationally symmetric
Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
by α clockwise and counterclockwise about the pole.
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
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...
,
lemniscate
In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves. The word comes from the Latin "''lēmniscātus''" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons",. or which alternativel ...
,
limaçon, and
cardioid.
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
The general equation for a circle with a center at
and radius ''a'' is
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''.
When or the origin lies on the circle, the equation becomes
In the general case, the equation can be solved for , giving
The solution with a minus sign in front of the square root gives the same curve.
Line
''Radial'' lines (those running through the pole) are represented by the equation
where
is the angle of elevation of the line; that is,
, where
is the
slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line
perpendicularly at the point
has the equation
Otherwise stated
is the point in which the tangent intersects the imaginary circle of radius
Polar rose
A
polar rose is a mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,
for any constant γ
0 (including 0). If ''k'' is an integer, these equations will produce a ''k''-petaled rose if ''k'' is
odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
, or a 2''k''-petaled 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
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
''a'' directly represents the length or amplitude of the petals of the rose, while ''k'' relates to their spatial frequency. The constant γ
0 can be regarded as a phase angle.
Archimedean spiral
The
Archimedean spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...
is a spiral discovered by
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
which can also be expressed as a simple polar equation. It is represented by the equation
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 and one for . The two arms are smoothly connected at the pole. If , 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 a prime example of a curve best defined by a polar equation.
Conic sections
A
conic section with one focus on the pole and the other somewhere on the 0° ray (so that the conic's
major axis
In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lo ...
lies along the polar axis) is given by:
where ''e'' is the
eccentricity and
is the
semi-latus rectum
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
(the perpendicular distance at a focus from the major axis to the curve). If , this equation defines a
hyperbola; if , it defines a
parabola; and if , it defines an
ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
. The special case of the latter results in a circle of the radius
.
Intersection of two polar curves
The graphs of two polar functions
and
have possible intersections of three types:
# In the origin, if the equations
and
have at least one solution each.
# All the points