polar coordinate systems

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane 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) is called the ''pole'', and the ray 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 degrees 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.

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 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 Hipparchus (190–120 BC) 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. 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.

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 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 professor Julian Lowell Coolidge'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. Blaise Pascal 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'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 degrees or radians (2 rad being equal to 360°). Degrees are traditionally used in navigation, surveying, 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 from the pole horizontally to the right, and the polar angle increases to positive angles for ccw rotations, whereas in navigation ( bearing, heading) 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:
$\begin x &= r \cos \varphi, \\ y &= r \sin \varphi. \end$

The Cartesian coordinates ''x'' and ''y'' can be converted to polar coordinates ''r'' and ''φ'' with ''r'' ≥ 0 and ''φ'' in the interval (−, ] by:
$\begin r &= \sqrt = \operatorname(x,y)\\ \varphi &= \operatorname(y, x), \end$
where hypot is the Pythagorean addition, Pythagorean sum and atan2 is a common variation on the arctangent function defined as
$\operatorname(y, x) = \begin \arctan\left(\frac\right) & \mbox x > 0\\ \arctan\left(\frac\right) + \pi & \mbox x < 0 \mbox y \ge 0\\ \arctan\left(\frac\right) - \pi & \mbox x < 0 \mbox y < 0\\ \frac & \mbox x = 0 \mbox y > 0\\ -\frac & \mbox x = 0 \mbox y < 0\\ \text & \mbox x = 0 \mbox y = 0. \end$

If ''r'' is calculated first as above, then this formula for ''φ'' may be stated more simply using the arccosine function:
$\varphi = \begin \arccos\left(\frac\right) & \mbox y \ge 0 \mbox r \neq 0 \\ -\arccos\left(\frac\right) & \mbox y < 0 \\ \text & \mbox r = 0. \end$

Complex numbers

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
$z = x + iy$
where ''i'' is the imaginary unit, or can alternatively be written in polar form as
$z = r(\cos\varphi + i\sin\varphi)$
and from there, by Euler's formula, as
$z = re^ = r \exp i \varphi.$
where ''e'' is Euler's number, and ''φ'', expressed in radians, is the principal value of the complex number function arg 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:
$z = r \operatorname\mathrm \varphi = r \angle \varphi .$

For the operations of multiplication, division, exponentiation, and root extraction 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: $r_0 e^\, r_1 e^ = r_0 r_1 e^$
; Division: $\frac = \frace^$
; Exponentiation (De Moivre's formula): $\left(re^\right)^n = r^n e^$
; Root Extraction (Principal root): \sqrt = \sqrt e^

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 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 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, lemniscate, 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 $(r_0, \gamma)$ and radius ''a'' is
$r^2 - 2 r r_0 \cos(\varphi - \gamma) + r_0^2 = a^2.$

This can be simplified in various ways, to conform to more specific cases, such as the equation
$r(\varphi)=a$
for a circle with a center at the pole and radius ''a''.

When or the origin lies on the circle, the equation becomes
$r = 2 a\cos(\varphi - \gamma).$

In the general case, the equation can be solved for , giving
$r = r_0 \cos(\varphi - \gamma) + \sqrt$
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
$\varphi = \gamma,$
where $\gamma$ is the angle of elevation of the line; that is, $\varphi = \arctan m$, where $m$ is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line $\varphi = \gamma$ perpendicularly at the point $(r_0, \gamma)$ has the equation
$r(\varphi) = r_0 \sec(\varphi - \gamma).$

Otherwise stated $(r_0, \gamma)$ is the point in which the tangent intersects the imaginary circle of radius $r_0$

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,
$r(\varphi) = a\cos\left(k\varphi + \gamma_0\right)$

for any constant γ₀ (including 0). If ''k'' is an integer, these equations will produce a ''k''-petaled rose if ''k'' is odd, 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 ''a'' directly represents the length or amplitude of the petals of the rose, while ''k'' relates to their spatial frequency. The constant γ₀ can be regarded as a phase angle.

Archimedean spiral

The Archimedean spiral is a spiral discovered by Archimedes which can also be expressed as a simple polar equation. It is represented by the equation
$r(\varphi) = a + b\varphi.$
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 lies along the polar axis) is given by:
$r =$
where ''e'' is the eccentricity and $\ell$ is the semi-latus rectum (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. The special case of the latter results in a circle of the radius $\ell$.

Intersection of two polar curves

The graphs of two polar functions $r = f(\theta)$ and $r = g(\theta)$ have possible intersections of three types:
# In the origin, if the equations $f(\theta) = 0$ and $g(\theta) = 0$ have at least one solution each.
# All the points (\theta_i),\theta_i/math> where $\theta_i$ are solutions to the equation $f(\theta+2k\pi)=g(\theta)$ where $k$ is an integer.
# All the points (\theta_i),\theta_i/math> where $\theta_i$ are solutions to the equation $f(\theta+(2k+1)\pi)=-g(\theta)$ where $k$ is an integer.

Calculus

Calculus can be applied to equations expressed in polar coordinates.

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

Differential calculus

Using and , one can derive a relationship between derivatives in Cartesian and polar coordinates. For a given function, ''u''(''x'',''y''), it follows that (by computing its total derivatives)
or
\begin
r \frac
&= r \frac \cos\varphi +
r \frac \sin\varphi
= x \frac + y \frac, \\ pt \frac
&= - \frac r \sin\varphi +
\frac r \cos\varphi
= -y \frac + x \frac.
\end

Hence, we have the following formulae:
\begin
r \frac &= x \frac + y \frac \\ pt \frac &= -y \frac + x \frac.
\end

Using the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a function ''u''(''r'',''φ''), it follows that
\begin
\frac
&= \frac\frac +
\frac\frac, \\ pt \frac
&= \frac\frac +
\frac\frac,
\end
or
\begin
\frac
&= \frac\frac -
\frac\frac \\ pt &= \cos \varphi \frac -
\frac \sin\varphi \frac, \\ pt \frac
&= \frac\frac +
\frac\frac \\ pt &= \sin\varphi \frac +
\frac \cos\varphi \frac.
\end

Hence, we have the following formulae:
\begin
\frac
&= \cos \varphi \frac -
\frac \sin\varphi \frac \\ pt \frac
&= \sin \varphi \frac +
\frac \cos\varphi \frac.
\end

To find the Cartesian slope of the tangent line to a polar curve ''r''(''φ'') at any given point, the curve is first expressed as a system of parametric equations.
$\begin x &= r(\varphi)\cos\varphi \\ y &= r(\varphi)\sin\varphi \end$

Differentiating both equations with respect to ''φ'' yields
\begin
\frac &= r'(\varphi)\cos\varphi - r(\varphi)\sin\varphi \\ pt \frac &= r'(\varphi)\sin\varphi + r(\varphi)\cos\varphi.
\end

Dividing the second equation by the first yields the Cartesian slope of the tangent line to the curve at the point :
$\frac = \frac.$

For other useful formulas including divergence, gradient, and Laplacian in polar coordinates, see curvilinear coordinates.

Integral calculus (arc length)

The arc length (length of a line segment) defined by a polar function is found by the integration over the curve ''r''(''φ''). Let ''L'' denote this length along the curve starting from points ''A'' through to point ''B'', where these points correspond to ''φ'' = ''a'' and ''φ'' = ''b'' such that . The length of ''L'' is given by the following integral
$L = \int_a^b \sqrt d\varphi$

Integral calculus (area)

Let ''R'' denote the region enclosed by a curve ''r''(''φ'') and the rays ''φ'' = ''a'' and ''φ'' = ''b'', where . Then, the area of ''R'' is
\frac12\int_a^b \left (\varphi)\right2\, d\varphi.

This result can be found as follows. First, the interval is divided into ''n'' subintervals, where ''n'' is some positive integer. Thus Δ''φ'', the angle measure of each subinterval, is equal to (the total angle measure of the interval), divided by ''n'', the number of subintervals. For each subinterval ''i'' = 1, 2, ..., ''n'', let ''φ''_''i'' be the midpoint of the subinterval, and construct a sector with the center at the pole, radius ''r''(''φ''_''i''), central angle Δ''φ'' and arc length ''r''(''φ''_''i'')Δ''φ''. The area of each constructed sector is therefore equal to
\left (\varphi_i)\right2 \pi \cdot \frac = \frac\left (\varphi_i)\right2 \Delta \varphi.
Hence, the total area of all of the sectors is
$\sum_^n \tfrac12r(\varphi_i)^2\,\Delta\varphi.$

As the number of subintervals ''n'' is increased, the approximation of the area improves. Taking , the sum becomes the Riemann sum for the above integral.

A mechanical device that computes area integrals is the planimeter, which measures the area of plane figures by tracing them out: this replicates integration in polar coordinates by adding a joint so that the 2-element linkage effects Green's theorem, converting the quadratic polar integral to a linear integral.

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