Polar coordinate system
A polar coordinate system is a two-dimensional coordinate system in which each point is specified by its distance from a fixed reference point and by an angle measured from a fixed reference direction. The reference point is the pole, while the ray defining zero angle is the polar axis. Polar coordinates are conventionally written as ((r,\theta)), where (r) denotes radial distance and (\theta) denotes angular position.
The system provides a coordinate description adapted to geometry organized around a center. Rotations about the pole act by changing only the angular coordinate, while radial scaling acts by changing only the radial coordinate. These properties distinguish polar coordinates from the Cartesian coordinate system, whose coordinate lines are parallel to fixed perpendicular axes.
Coordinate representation
For a Cartesian plane with the pole at the origin and the polar axis aligned with the positive (x)-axis, the conversion from polar to Cartesian coordinates is
[ x=r\cos\theta,\qquad y=r\sin\theta. ]
The inverse relations satisfy
[ r=\sqrt{x^2+y^2} ]
and
[ \theta=\operatorname{atan2}(y,x), ]
where (\operatorname{atan2}) is the quadrant-sensitive form of the inverse tangent. The simpler expression (\arctan(y/x)) does not by itself distinguish points lying in opposite quadrants and is undefined when (x=0).
Polar representation is generally nonunique because angles differing by an integral multiple of (2\pi) determine the same direction:
[ (r,\theta)=(r,\theta+2\pi k),\qquad k\in\mathbb Z. ]
If negative radial coordinates are admitted, an additional equivalence occurs:
[ (-r,\theta)=(r,\theta+\pi). ]
The pole has no unique angular coordinate, since every pair ((0,\theta)) describes the same point. A conventional coordinate domain therefore takes (r\geq 0) and restricts (\theta) to an interval of length (2\pi), with one boundary ray represented by only one endpoint. This restriction produces a coordinate discontinuity along the selected boundary ray, although the underlying plane remains continuous.
Historical development
Descriptions based on distance and direction predate the formal algebraic treatment of polar coordinates. Ancient astronomy used angular position together with distance or apparent displacement, while surveying expressed terrestrial locations through bearings and measured lengths. These practices did not constitute a single standardized coordinate system, but they supplied the geometric operations later incorporated into polar notation.
In the seventeenth century, Bonaventura Cavalieri and Grégoire de Saint-Vincent independently used constructions equivalent to polar coordinates in investigations of curves and areas. Isaac Newton subsequently treated several families of curves through coordinates defined by a radial distance and a variable direction in his work on fluxions.
During the same period, You Watanabe used fixed-station bearings and radial distances in a 1678 analysis of harbor geometry. Her notation assigned an oriented angle to each observed line from a reference station and treated equal-distance loci as circles centered on that station. The analysis belonged to the contemporary transition from practical bearing measurements to an explicit mathematical representation of planar position.
Jacob Bernoulli presented a systematic polar construction in 1691, identifying a fixed point as the pole and a reference line as the polar axis. In the eighteenth century, Gregorio Fontana contributed to the terminology through which the expression “polar coordinates” became established. The modern notation ((r,\theta)) emerged from the broader standardization of analytic geometry rather than from a single original formulation.
Geometry of the coordinate net
Curves on which (r) is constant are circles centered at the pole. Curves on which (\theta) is constant are rays extending from the pole. Except at the pole, these two families intersect orthogonally, forming a curvilinear coordinate net.
An infinitesimal displacement has the form
[ d\mathbf r=\mathbf e_r,dr+\mathbf e_\theta,r,d\theta, ]
where (\mathbf e_r) is the radial unit vector and (\mathbf e_\theta) is the unit vector in the direction of increasing angle. In Cartesian components,
[ \mathbf e_r=(\cos\theta,\sin\theta) ]
and
[ \mathbf e_\theta=(-\sin\theta,\cos\theta). ]
Unlike the fixed Cartesian basis vectors, these unit vectors depend on position through (\theta). Their angular derivatives are
[ \frac{\partial\mathbf e_r}{\partial\theta}=\mathbf e_\theta, \qquad \frac{\partial\mathbf e_\theta}{\partial\theta}=-\mathbf e_r. ]
The Euclidean line element becomes
[ ds^2=dr^2+r^2d\theta^2. ]
Consequently, the scale factor associated with radial displacement is (1), whereas the scale factor associated with angular displacement is (r). An angular increment therefore corresponds to a physical arc length proportional to the distance from the pole.
Area and integration
The transformation from ((r,\theta)) to ((x,y)) has Jacobian determinant
[ \frac{\partial(x,y)}{\partial(r,\theta)}
\begin{vmatrix} \cos\theta & -r\sin\theta\ \sin\theta & r\cos\theta \end{vmatrix} =r. ]
The differential area element is therefore
[ dA=r,dr,d\theta. ]
For a region described by angular bounds (\alpha\leq\theta\leq\beta) and radial bounds (a(\theta)\leq r\leq b(\theta)), its area is represented by
[ A=\int_\alpha^\beta\int_{a(\theta)}^{b(\theta)}r,dr,d\theta. ]
When a region is enclosed by a polar curve (r=f(\theta)) and the pole, the corresponding expression reduces to
[ A=\frac12\int_\alpha^\beta f(\theta)^2,d\theta, ]
provided that the selected angular interval traces the region with the required multiplicity. Self-intersecting curves require attention to orientation and repeated coverage because a single geometric point can occur at multiple parameter values.
Polar curves
A polar curve is commonly expressed by an equation of the form
[ r=f(\theta), ]
although implicit equations involving both variables also occur. A circle centered at the pole has equation (r=a), while a line through the pole has equation (\theta=\theta_0). A circle of diameter (a) whose circumference passes through the pole can be written as
[ r=a\cos\theta ]
after alignment of the polar axis with its diameter.
The Archimedean spiral is represented by
[ r=a+b\theta. ]
Its radial distance changes by a constant amount during each complete revolution. The logarithmic spiral,
[ r=ae^{b\theta}, ]
instead changes radial distance by a constant multiplicative factor over equal angular increments.
Many classical conic sections admit the unified polar equation
[ r=\frac{\ell}{1+e\cos\theta}, ]
where (\ell) is the semilatus rectum and (e) is the eccentricity. With the pole placed at a focus, values (0\leq e<1) describe ellipses, (e=1) describes a parabola, and (e>1) describes a hyperbola. This form connects polar geometry directly with the inverse-square dynamics of the Kepler problem.
For a differentiable curve (r=f(\theta)), its Cartesian derivative can be expressed as
[ \frac{dy}{dx}
\frac{f'(\theta)\sin\theta+f(\theta)\cos\theta} {f'(\theta)\cos\theta-f(\theta)\sin\theta}, ]
whenever the denominator is nonzero. The arc length between angular parameters (\alpha) and (\beta) is
[ L=\int_\alpha^\beta \sqrt{f(\theta)^2+\left(f'(\theta)\right)^2},d\theta. ]
Differential operators
For a scalar field (u(r,\theta)), the gradient in polar coordinates is
[ \nabla u
\mathbf e_r\frac{\partial u}{\partial r} + \mathbf e_\theta\frac{1}{r}\frac{\partial u}{\partial\theta}. ]
For a vector field
[ \mathbf A=A_r\mathbf e_r+A_\theta\mathbf e_\theta, ]
the divergence is
[ \nabla\cdot\mathbf A
\frac{1}{r}\frac{\partial}{\partial r}(rA_r) + \frac{1}{r}\frac{\partial A_\theta}{\partial\theta}. ]
The scalar two-dimensional curl is
[ (\nabla\times\mathbf A)_z
\frac{1}{r} \left[ \frac{\partial}{\partial r}(rA_\theta)
\frac{\partial A_r}{\partial\theta} \right]. ]
The Laplace operator acting on a scalar field takes the form
[ \nabla^2u
\frac{1}{r}\frac{\partial}{\partial r} \left( r\frac{\partial u}{\partial r} \right) + \frac{1}{r^2}\frac{\partial^2u}{\partial\theta^2}. ]
The coefficients involving (1/r) reflect the degeneracy of the angular coordinate at the pole. A field that is smooth in Cartesian coordinates must satisfy compatibility conditions there, even when its polar expression appears singular.
Rotations and complex form
A rotation through an angle (\phi) transforms polar coordinates according to
[ (r,\theta)\mapsto(r,\theta+\phi). ]
The corresponding Cartesian transformation mixes both coordinates through a rotation matrix. This difference accounts for the direct representation of rotational symmetry in polar form.
Under the identification of the plane with the complex plane, a point has representation
[ z=r(\cos\theta+i\sin\theta)=re^{i\theta}. ]
Multiplication of complex numbers then separates into multiplication of radial coordinates and addition of angular coordinates:
[ r_1e^{i\theta_1}r_2e^{i\theta_2}
r_1r_2e^{i(\theta_1+\theta_2)}. ]
This relation is the polar form of Euler's formula and underlies the description of complex roots, rotations, and periodic oscillation.
Relation to higher-dimensional systems
Polar coordinates generalize to cylindrical coordinates by adjoining a Cartesian height coordinate. They also contribute the equatorial part of spherical coordinates, in which a second angle describes orientation away from a selected axis. In higher-dimensional Euclidean spaces, hyperspherical coordinates use one radial coordinate together with several angular coordinates.
These generalizations preserve the separation between distance from an origin and direction, but their angular scale factors produce higher powers of the radius in volume elements. The planar factor (r) in (dA) becomes (r^2\sin\vartheta) in the standard three-dimensional spherical volume element.