Polar coordinate system

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 (r, φ + n × 360°) and (−r, φ + 180° + n × 360°) = (−r, φ + (2n + 1) × 180°), where n is an arbitrary integer.[10] Moreover, the pole itself can be expressed as (0, φ) for any angle φ.[11]

Where a unique representation is needed for any point besides the pole, it is usual to limit r to positive numbers (r > 0) and φ to either the interval [0, 360°) or the interval (−180°, 180°], which in radians are [0, 2π) or (−π, π].[12] 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 (−90°, 90°]. In all cases a unique azimuth for the pole (r = 0) must be chosen, e.g., φ = 0.

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

An illustration of a complex number plotted on the complex plane using Euler's formula

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

$${\displaystyle z=x+iy}$$

where i is the imaginary unit, or can alternatively be written in polar form as

$${\displaystyle z=r(\cos \varphi +i\sin \varphi )}$$

and from there, by Euler's formula,[14] as

$${\displaystyle z=re^{i\varphi }=r\exp i\varphi .}$$

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 cis and angle notations:

$${\displaystyle z=r\operatorname {\mathrm {cis} } \varphi =r\angle \varphi .}$$

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

${\displaystyle r_{0}e^{i\varphi _{0}}\,r_{1}e^{i\varphi _{1}}=r_{0}r_{1}e^{i\left(\varphi _{0}+\varphi _{1}\right)}}$

Division

${\displaystyle {\frac {r_{0}e^{i\varphi _{0}}}{r_{1}e^{i\varphi _{1}}}}={\frac {r_{0}}{r_{1}}}e^{i(\varphi _{0}-\varphi _{1})}}$

Exponentiation (De Moivre's formula)

${\displaystyle \left(re^{i\varphi }\right)^{n}=r^{n}e^{in\varphi }}$

Root Extraction (Principal root)

${\displaystyle {\sqrt[{n}]{re^{i\varphi }}}={\sqrt[{n}]{r}}e^{i\varphi \over n}}$

A circle with equation r(φ) = 1

The general equation for a circle with a center at ${\displaystyle (r_{0},\gamma )}$ and radius a is

$${\displaystyle r^{2}-2rr_{0}\cos(\varphi -\gamma )+r_{0}^{2}=a^{2}.}$$

This can be simplified in various ways, to conform to more specific cases, such as the equation

$${\displaystyle r(\varphi )=a}$$

for a circle with a center at the pole and radius a.[15]

When r₀ = a or the origin lies on the circle, the equation becomes

$${\displaystyle r=2a\cos(\varphi -\gamma ).}$$

In the general case, the equation can be solved for r, giving

$${\displaystyle r=r_{0}\cos(\varphi -\gamma )+{\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\varphi -\gamma )}}}$$

The solution with a minus sign in front of the square root gives the same curve.

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

$${\displaystyle \varphi =\gamma ,}$$

where ${\displaystyle \gamma }$ is the angle of elevation of the line; that is, ${\displaystyle \varphi =\arctan m}$, where ${\displaystyle m}$ is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line ${\displaystyle \varphi =\gamma }$ perpendicularly at the point ${\displaystyle (r_{0},\gamma )}$ has the equation

$${\displaystyle r(\varphi )=r_{0}\sec(\varphi -\gamma ).}$$

Otherwise stated ${\displaystyle (r_{0},\gamma )}$ is the point in which the tangent intersects the imaginary circle of radius ${\displaystyle r_{0}}$

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

A polar rose is a mathematical curve that looks like a petaled flower, and that can be expressed as a simple polar equation,

$${\displaystyle r(\varphi )=a\cos \left(k\varphi +\gamma _{0}\right)}$$

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 2k-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.

One arm of an Archimedean spiral with equation r(φ) = φ / 2π for 0 < φ < 6πpi

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

$${\displaystyle r(\varphi )=a+b\varphi .}$$

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 φ > 0 and one for φ < 0. The two arms are smoothly connected at the pole. If a = 0, 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.

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:

$${\displaystyle r={\ell \over {1-e\cos \varphi }}}$$

where e is the eccentricity and ${\displaystyle \ell }$ 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 the radius ${\displaystyle \ell }$.

The graphs of two polar functions ${\displaystyle r=f(\theta )}$ and ${\displaystyle r=g(\theta )}$ have possible intersections of three types:

1. In the origin, if the equations ${\displaystyle f(\theta )=0}$ and ${\displaystyle g(\theta )=0}$ have at least one solution each.
2. All the points ${\displaystyle [g(\theta _{i}),\theta _{i}]}$ where ${\displaystyle \theta _{i}}$ are solutions to the equation ${\displaystyle f(\theta +2k\pi )=g(\theta )}$ where ${\displaystyle k}$ is an integer.
3. All the points ${\displaystyle [g(\theta _{i}),\theta _{i}]}$ where ${\displaystyle \theta _{i}}$ are solutions to the equation ${\displaystyle f(\theta +(2k+1)\pi )=-g(\theta )}$ where ${\displaystyle k}$ is an integer.

Using x = r cos φ and y = r sin φ, 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

$${\displaystyle {\begin{aligned}r{\frac {du}{dr}}&=r{\frac {\partial u}{\partial x}}\cos \varphi +r{\frac {\partial u}{\partial y}}\sin \varphi =x{\frac {\partial u}{\partial x}}+y{\frac {\partial u}{\partial y}},\\[2pt]{\frac {du}{d\varphi }}&=-{\frac {\partial u}{\partial x}}r\sin \varphi +{\frac {\partial u}{\partial y}}r\cos \varphi =-y{\frac {\partial u}{\partial x}}+x{\frac {\partial u}{\partial y}}.\end{aligned}}}$$

Hence, we have the following formulae:

$${\displaystyle {\begin{aligned}r{\frac {d}{dr}}&=x{\frac {\partial }{\partial x}}+y{\frac {\partial }{\partial y}}\\[2pt]{\frac {d}{d\varphi }}&=-y{\frac {\partial }{\partial x}}+x{\frac {\partial }{\partial y}}.\end{aligned}}}$$

Using the inverse coordinates transformation, an analogous reciprocal relationship can be derived between the derivatives. Given a function u(r,φ), it follows that

$${\displaystyle {\begin{aligned}{\frac {du}{dx}}&={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial x}}+{\frac {\partial u}{\partial \varphi }}{\frac {\partial \varphi }{\partial x}},\\[2pt]{\frac {du}{dy}}&={\frac {\partial u}{\partial r}}{\frac {\partial r}{\partial y}}+{\frac {\partial u}{\partial \varphi }}{\frac {\partial \varphi }{\partial y}},\end{aligned}}}$$

or

$${\displaystyle {\begin{aligned}{\frac {du}{dx}}&={\frac {\partial u}{\partial r}}{\frac {x}{\sqrt {x^{2}+y^{2}}}}-{\frac {\partial u}{\partial \varphi }}{\frac {y}{x^{2}+y^{2}}}\\[2pt]&=\cos \varphi {\frac {\partial u}{\partial r}}-{\frac {1}{r}}\sin \varphi {\frac {\partial u}{\partial \varphi }},\\[2pt]{\frac {du}{dy}}&={\frac {\partial u}{\partial r}}{\frac {y}{\sqrt {x^{2}+y^{2}}}}+{\frac {\partial u}{\partial \varphi }}{\frac {x}{x^{2}+y^{2}}}\\[2pt]&=\sin \varphi {\frac {\partial u}{\partial r}}+{\frac {1}{r}}\cos \varphi {\frac {\partial u}{\partial \varphi }}.\end{aligned}}}$$

Hence, we have the following formulae:

$${\displaystyle {\begin{aligned}{\frac {d}{dx}}&=\cos \varphi {\frac {\partial }{\partial r}}-{\frac {1}{r}}\sin \varphi {\frac {\partial }{\partial \varphi }}\\[2pt]{\frac {d}{dy}}&=\sin \varphi {\frac {\partial }{\partial r}}+{\frac {1}{r}}\cos \varphi {\frac {\partial }{\partial \varphi }}.\end{aligned}}}$$

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.

$${\displaystyle {\begin{aligned}x&=r(\varphi )\cos \varphi \\y&=r(\varphi )\sin \varphi \end{aligned}}}$$

Differentiating both equations with respect to φ yields

$${\displaystyle {\begin{aligned}{\frac {dx}{d\varphi }}&=r'(\varphi )\cos \varphi -r(\varphi )\sin \varphi \\[2pt]{\frac {dy}{d\varphi }}&=r'(\varphi )\sin \varphi +r(\varphi )\cos \varphi .\end{aligned}}}$$

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

$${\displaystyle {\frac {dy}{dx}}={\frac {r'(\varphi )\sin \varphi +r(\varphi )\cos \varphi }{r'(\varphi )\cos \varphi -r(\varphi )\sin \varphi }}.}$$

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

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 0 < b − a < 2π. The length of L is given by the following integral

$${\displaystyle L=\int _{a}^{b}{\sqrt {\left[r(\varphi )\right]^{2}+\left[{\tfrac {dr(\varphi )}{d\varphi }}\right]^{2}}}d\varphi }$$

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

Let R denote the region enclosed by a curve r(φ) and the rays φ = a and φ = b, where 0 < b − a ≤ 2π. Then, the area of R is

$${\displaystyle {\frac {1}{2}}\int _{a}^{b}\left[r(\varphi )\right]^{2}\,d\varphi .}$$

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

A planimeter, which mechanically computes polar integrals

This result can be found as follows. First, the interval [a, b] is divided into n subintervals, where n is some positive integer. Thus Δφ, the angle measure of each subinterval, is equal to b − a (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 Δφ and arc length r(φ_i)Δφ. The area of each constructed sector is therefore equal to

$${\displaystyle \left[r(\varphi _{i})\right]^{2}\pi \cdot {\frac {\Delta \varphi }{2\pi }}={\frac {1}{2}}\left[r(\varphi _{i})\right]^{2}\Delta \varphi .}$$

Hence, the total area of all of the sectors is

$${\displaystyle \sum _{i=1}^{n}{\tfrac {1}{2}}r(\varphi _{i})^{2}\,\Delta \varphi .}$$

As the number of subintervals n is increased, the approximation of the area improves. Taking n → ∞, 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.

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:

$${\displaystyle J=\det {\frac {\partial (x,y)}{\partial (r,\varphi )}}={\begin{vmatrix}{\frac {\partial x}{\partial r}}&{\frac {\partial x}{\partial \varphi }}\\[2pt]{\frac {\partial y}{\partial r}}&{\frac {\partial y}{\partial \varphi }}\end{vmatrix}}={\begin{vmatrix}\cos \varphi &-r\sin \varphi \\\sin \varphi &r\cos \varphi \end{vmatrix}}=r\cos ^{2}\varphi +r\sin ^{2}\varphi =r.}$$

Hence, an area element in polar coordinates can be written as

$${\displaystyle dA=dx\,dy\ =J\,dr\,d\varphi =r\,dr\,d\varphi .}$$

Now, a function, that is given in polar coordinates, can be integrated as follows:

$${\displaystyle \iint _{R}f(x,y)\,dA=\int _{a}^{b}\int _{0}^{r(\varphi )}f(r,\varphi )\,r\,dr\,d\varphi .}$$

Here, R is the same region as above, namely, the region enclosed by a curve r(φ) and the rays φ = a and φ = b. The formula for the area of R is retrieved by taking f identically equal to 1.

A graph of ${\displaystyle f(x)=e^{-x^{2}}}$ and the area between the function and the ${\displaystyle x}$-axis, which is equal to ${\displaystyle {\sqrt {\pi }}}$.

A more surprising application of this result yields the Gaussian integral:

$${\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\,dx={\sqrt {\pi }}.}$$

Vector calculus can also be applied to polar coordinates. For a planar motion, let ${\displaystyle \mathbf {r} }$ be the position vector (r cos(φ), r sin(φ)), with r and φ depending on time t.

We define the unit vectors

$${\displaystyle {\hat {\mathbf {r} }}=(\cos(\varphi ),\sin(\varphi ))}$$

in the direction of ${\displaystyle \mathbf {r} }$ and

$${\displaystyle {\hat {\boldsymbol {\varphi }}}=(-\sin(\varphi ),\cos(\varphi ))={\hat {\mathbf {k} }}\times {\hat {\mathbf {r} }}\ ,}$$

in the plane of the motion perpendicular to the radial direction, where ${\displaystyle {\hat {\mathbf {k} }}}$ is a unit vector normal to the plane of the motion.

Then

$${\displaystyle {\begin{aligned}\mathbf {r} &=(x,\ y)=r(\cos \varphi ,\ \sin \varphi )=r{\hat {\mathbf {r} }}\ ,\\{\dot {\mathbf {r} }}&=\left({\dot {x}},\ {\dot {y}}\right)={\dot {r}}(\cos \varphi ,\ \sin \varphi )+r{\dot {\varphi }}(-\sin \varphi ,\ \cos \varphi )={\dot {r}}{\hat {\mathbf {r} }}+r{\dot {\varphi }}{\hat {\boldsymbol {\varphi }}}\ ,\\{\ddot {\mathbf {r} }}&=\left({\ddot {x}},\ {\ddot {y}}\right)\\&={\ddot {r}}(\cos \varphi ,\ \sin \varphi )+2{\dot {r}}{\dot {\varphi }}(-\sin \varphi ,\ \cos \varphi )+r{\ddot {\varphi }}(-\sin \varphi ,\ \cos \varphi )-r{\dot {\varphi }}^{2}(\cos \varphi ,\ \sin \varphi )\\&=\left({\ddot {r}}-r{\dot {\varphi }}^{2}\right){\hat {\mathbf {r} }}+\left(r{\ddot {\varphi }}+2{\dot {r}}{\dot {\varphi }}\right){\hat {\boldsymbol {\varphi }}}\\&=\left({\ddot {r}}-r{\dot {\varphi }}^{2}\right){\hat {\mathbf {r} }}+{\frac {1}{r}}\;{\frac {d}{dt}}\left(r^{2}{\dot {\varphi }}\right){\hat {\boldsymbol {\varphi }}}.\end{aligned}}}$$

Position vector r, always points radially from the origin.

Velocity vector v, always tangent to the path of motion.

Acceleration vector a, not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.

Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 2d space, but a plane in any higher dimension.

The term ${\displaystyle r{\dot {\varphi }}^{2}}$ is sometimes referred to as the centripetal acceleration, and the term ${\displaystyle 2{\dot {r}}{\dot {\varphi }}}$ as the Coriolis acceleration. For example, see Shankar.[18]

Note: these terms, that appear when acceleration is expressed in polar coordinates, are a mathematical consequence of differentiation; they appear whenever polar coordinates are used. In planar particle dynamics these accelerations appear when setting up Newton's second law of motion in a rotating frame of reference. Here these extra terms are often called fictitious forces; fictitious because they are simply a result of a change in coordinate frame. That does not mean they do not exist, rather they exist only in the rotating frame.

Inertial frame of reference S and instantaneous non-inertial co-rotating frame of reference S′. The co-rotating frame rotates at angular rate Ω equal to the rate of rotation of the particle about the origin of S′ at the particular moment t. Particle is located at vector position r(t) and unit vectors are shown in the radial direction to the particle from the origin, and also in the direction of increasing angle ϕ normal to the radial direction. These unit vectors need not be related to the tangent and normal to the path. Also, the radial distance r need not be related to the radius of curvature of the path.

For a particle in planar motion, one approach to attaching physical significance to these terms is based on the concept of an instantaneous co-rotating frame of reference.[19] To define a co-rotating frame, first an origin is selected from which the distance r(t) to the particle is defined. An axis of rotation is set up that is perpendicular to the plane of motion of the particle, and passing through this origin. Then, at the selected moment t, the rate of rotation of the co-rotating frame Ω is made to match the rate of rotation of the particle about this axis, dφ/dt. Next, the terms in the acceleration in the inertial frame are related to those in the co-rotating frame. Let the location of the particle in the inertial frame be (r(t), φ(t)), and in the co-rotating frame be (r′(t), φ′(t)). Because the co-rotating frame rotates at the same rate as the particle, dφ′/dt = 0. The fictitious centrifugal force in the co-rotating frame is mrΩ², radially outward. The velocity of the particle in the co-rotating frame also is radially outward, because dφ′/dt = 0. The fictitious Coriolis force therefore has a value −2m(dr/dt)Ω, pointed in the direction of increasing φ only. Thus, using these forces in Newton's second law we find:

$${\displaystyle {\boldsymbol {F}}+{\boldsymbol {F}}_{\text{cf}}+{\boldsymbol {F}}_{\text{Cor}}=m{\ddot {\boldsymbol {r}}}\ ,}$$

where over dots represent time differentiations, and F is the net real force (as opposed to the fictitious forces). In terms of components, this vector equation becomes:

$${\displaystyle {\begin{aligned}F_{r}+mr\Omega ^{2}&=m{\ddot {r}}\\F_{\varphi }-2m{\dot {r}}\Omega &=mr{\ddot {\varphi }}\ ,\end{aligned}}}$$

which can be compared to the equations for the inertial frame:

$${\displaystyle {\begin{aligned}F_{r}&=m{\ddot {r}}-mr{\dot {\varphi }}^{2}\\F_{\varphi }&=mr{\ddot {\varphi }}+2m{\dot {r}}{\dot {\varphi }}\ .\end{aligned}}}$$

This comparison, plus the recognition that by the definition of the co-rotating frame at time t it has a rate of rotation Ω = dφ/dt, shows that we can interpret the terms in the acceleration (multiplied by the mass of the particle) as found in the inertial frame as the negative of the centrifugal and Coriolis forces that would be seen in the instantaneous, non-inertial co-rotating frame.

For general motion of a particle (as opposed to simple circular motion), the centrifugal and Coriolis forces in a particle's frame of reference commonly are referred to the instantaneous osculating circle of its motion, not to a fixed center of polar coordinates. For more detail, see centripetal force.

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.[20] Thus, an aircraft traveling 5 nautical miles due east will be traveling 5 units at heading 90 (read zero-niner-zero by air traffic control).[21]

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.

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 + 0.5sin(ϕ) at its target design frequency.[22] The pattern shifts toward omnidirectionality at lower frequencies.

• Adams, Robert; Christopher Essex (2013). Calculus: a complete course (Eighth ed.). Pearson Canada Inc. ISBN 978-0-321-78107-9.
• Anton, Howard; Irl Bivens; Stephen Davis (2002). Calculus (Seventh ed.). Anton Textbooks, Inc. ISBN 0-471-38157-8.
• Finney, Ross; George Thomas; Franklin Demana; Bert Waits (June 1994). Calculus: Graphical, Numerical, Algebraic (Single Variable Version ed.). Addison-Wesley Publishing Co. ISBN 0-201-55478-X.