Bipolar coordinates

Bipolar coordinates are a two-dimensional orthogonal curvilinear coordinate system defined relative to two fixed foci, or poles, separated by a distance $2a, where the coordinate curves consist of circles that either pass through both poles (constant angular coordinate \xi) or have the poles as inverse points (constant radial-like coordinate \eta).^[1] The transformation from Cartesian coordinates (x, y) to bipolar coordinates (\xi, \eta) is given by

x = a \frac{\sinh \eta}{\cosh \eta - \cos \xi}, \quad y = a \frac{\sin \xi}{\cosh \eta - \cos \xi},

with \xi \in [0, 2\pi) and \eta \in (-\infty, \infty), ensuring the system covers the entire plane except the poles themselves.^[1]^[2] The scale factors for both coordinates are equal, h_\xi = h_\eta = \frac{a}{\cosh \eta - \cos \xi}, confirming orthogonality and enabling the separation of variables in Laplace's equation.^[1] In three dimensions, bipolar coordinates extend to a cylindrical form (\xi, \eta, z) by including the axial coordinate z unchanged, with scale factor h_z = 1, resulting in coordinate surfaces that are circular cylinders and planes parallel to the xy-plane.^[1]^[3] This system is particularly suited for problems exhibiting symmetry around two parallel axes, such as the electrostatic potential between two charged cylindrical conductors or the magnetic vector potential due to parallel current-carrying wires.^[1] Bipolar coordinates find applications in various fields of physics and engineering, including steady-state heat conduction in regions with embedded cylindrical pipes, fluid dynamics around pairs of cylinders, and diffusion processes in domains with circular inclusions.^[3] They also appear in electromagnetic theory for analyzing transmission lines and in mathematical modeling of axisymmetric flows, such as locomotion within surfactant-laden droplets.^[1] The separability of the Helmholtz and Laplace equations in this system facilitates analytical solutions to boundary value problems that are intractable in Cartesian coordinates.^[3]

Definition and Geometry

Definition

Bipolar coordinates constitute a two-dimensional orthogonal curvilinear coordinate system in the Euclidean plane, defined relative to two fixed foci separated by a distance $2a where a > 0. The foci are conventionally placed at F_1 = (-a, 0) and F_2 = (a, 0) along the x-axis in Cartesian coordinates.^[4]^[5] The coordinates are denoted by \sigma and \tau, where \sigma \in [0, 2\pi) represents the angle subtended by the segment joining the two foci at a point P in the plane, and \tau \in (-\infty, \infty) is the hyperbolic coordinate given by \tau = \ln(d_1 / d_2), with d_1 and d_2 being the Euclidean distances from P to F_1 and F_2, respectively.^[6]^[7] The transformation from bipolar to Cartesian coordinates (x, y) is expressed as:

x = \frac{a \sinh \tau}{\cosh \tau - \cos \sigma}, \quad y = \frac{a \sin \sigma}{\cosh \tau - \cos \sigma}.

These relations map the entire plane excluding the foci, with the denominator ensuring the coordinates cover non-singular points.^[4]^[8] The level sets of constant \sigma and constant \tau form two families of circles known as Apollonian circles, which are orthogonal to each other and pass through or are centered relative to the foci, providing the geometric basis for the coordinate system's utility in problems with circular symmetry around two points.^[4]^[5]

Coordinate Curves

In bipolar coordinates, the curves of constant σ form a family of circles that pass through the two foci located at (±a, 0) on the x-axis, where a > 0 is the scale parameter. These circles, known as Apollonian circles, have their centers on the y-axis at (0, a cot σ), with σ ranging from 0 to 2π (excluding σ = π, where the curve degenerates). The radius of each such circle is a |csc σ|. The Cartesian equation for a fixed σ is given by

x^2 + (y - a \cot \sigma)^2 = a^2 \csc^2 \sigma.

This family constitutes a confocal pencil of circles, where all members intersect at the foci but do not intersect elsewhere within the family.^[4]^[5] The curves of constant τ form another family of circles that are orthogonal to the constant-σ family. These circles do not pass through the foci; instead, they enclose one focus or the other depending on the sign of τ, which ranges from -∞ to +∞. The centers lie on the x-axis at (a coth τ, 0), and the radius is a |csch τ|. The Cartesian equation for a fixed τ is

(x - a \coth \tau)^2 + y^2 = a^2 \csch^2 \tau.

Curves within this family do not intersect each other.^[4]^[5] Regarding asymptotic behavior, as σ approaches 0 or 2π, the constant-σ circles grow large with centers moving far along the y-axis, causing the arcs between the foci to approach the line segment on the x-axis connecting the two foci. For constant-τ curves, as τ approaches ±∞, the circles shrink toward the foci, with centers approaching (±a, 0) and radii tending to zero.^[5]

Mathematical Properties

Orthogonality

In curvilinear coordinate systems, orthogonality requires that the basis vectors along the coordinate directions are perpendicular, meaning their dot product vanishes: \mathbf{e}_\sigma \cdot \mathbf{e}_\tau = 0. This condition is equivalent to \frac{\partial \mathbf{r}}{\partial \sigma} \cdot \frac{\partial \mathbf{r}}{\partial \tau} = 0, where \mathbf{r}(\sigma, \tau) is the position vector in Cartesian coordinates.$$] For bipolar coordinates, the position vector is [ \mathbf{r}(\sigma, \tau) = \left( a \frac{\sinh \tau}{\cosh \tau - \cos \sigma}, , a \frac{\sin \sigma}{\cosh \tau - \cos \sigma} \right),

with foci at $(\pm a, 0)$ and parameters $\sigma \in [0, 2\pi)$ and $\tau \in (-\infty, \infty)$.$$\] To verify the condition explicitly, denote $D = \cosh \tau - \cos \sigma$. The partial derivatives are \[ \frac{\partial x}{\partial \sigma} = -\frac{a \sinh \tau \sin \sigma}{D^2}, \quad \frac{\partial y}{\partial \sigma} = \frac{a (\cosh \tau \cos \sigma - 1)}{D^2},

\frac{\partial x}{\partial \tau} = \frac{a (\cosh^2 \tau - \cosh \tau \cos \sigma - \sinh^2 \tau)}{D^2} = \frac{a (1 - \cosh \tau \cos \sigma)}{D^2}, \quad \frac{\partial y}{\partial \tau} = -\frac{a \sin \sigma \sinh \tau}{D^2}.

The dot product is then

\frac{\partial x}{\partial \sigma} \frac{\partial x}{\partial \tau} + \frac{\partial y}{\partial \sigma} \frac{\partial y}{\partial \tau} = -\frac{a^2 \sinh \tau \sin \sigma}{D^4} \left[ (\cosh \tau D - \sinh^2 \tau) + (\cos \sigma D - \sin^2 \sigma) \right].

The expression in brackets simplifies to \cosh \tau D + \cos \sigma D - \sinh^2 \tau - \sin^2 \sigma = (\cosh^2 \tau - \sinh^2 \tau) - (\cos^2 \sigma + \sin^2 \sigma) = 1 - 1 = 0, yielding zero overall and confirming orthogonality.$$] The scale factors, which are the magnitudes of these basis vectors, are both h_\sigma = h_\tau = a / D. An alternative confirmation arises from the geometric interpretation: the curves of constant \sigma and constant \tau form two conjugate families of Apollonian circles, which intersect orthogonally by definition, as every circle in one family is perpendicular to every circle in the other.[$$ This orthogonal system covers the entire xy-plane except the line segment joining the foci (where \tau = 0 and \sigma is undefined), ensuring the coordinates are well-defined almost everywhere.$$]

Scale Factors

In orthogonal curvilinear coordinate systems, the scale factors h_\sigma and h_\tau quantify the stretching of infinitesimal displacements along the coordinate directions \sigma and \tau, respectively. They are given by h_\sigma = \left| \frac{\partial \mathbf{r}}{\partial \sigma} \right| and h_\tau = \left| \frac{\partial \mathbf{r}}{\partial \tau} \right|, where \mathbf{r}(\sigma, \tau) is the position vector in the plane.^[4] These factors determine the line element ds^2 = h_\sigma^2 d\sigma^2 + h_\tau^2 d\tau^2, reflecting the absence of a cross term due to orthogonality.^[4] For bipolar coordinates, the position vector is \mathbf{r} = \left( a \frac{\sinh \tau}{\cosh \tau - \cos \sigma}, a \frac{\sin \sigma}{\cosh \tau - \cos \sigma} \right), where a > 0 is a scaling constant.^[5] To derive the scale factors, compute the partial derivatives \frac{\partial \mathbf{r}}{\partial \sigma} and \frac{\partial \mathbf{r}}{\partial \tau}, then take their magnitudes. The components involve trigonometric and hyperbolic identities that simplify upon squaring and adding the Cartesian contributions. The resulting expressions are identical: h_\sigma = \frac{a}{\cosh \tau - \cos \sigma} and h_\tau = \frac{a}{\cosh \tau - \cos \sigma}.^[5]^[4] This equality yields the line element ds^2 = \left[ \frac{a}{\cosh \tau - \cos \sigma} \right]^2 (d\sigma^2 + d\tau^2), indicating a conformal metric where distances are scaled isotropically in the (\sigma, \tau)-plane.^[4] The area element follows as dA = h_\sigma h_\tau \, d\sigma \, d\tau = \left[ \frac{a}{\cosh \tau - \cos \sigma} \right]^2 d\sigma \, d\tau.^[5] The equal scale factors simplify computations in integrals over regions bounded by coordinate curves and facilitate separation of variables in partial differential equations, as the metric form resembles that of Cartesian coordinates up to a scalar factor.^[4]

Inverse Transformations

The inverse transformation from Cartesian coordinates (x, y) to bipolar coordinates (\tau, \sigma) in the plane, with foci at (\pm a, 0), expresses the bipolar parameters in terms of distances and angles relative to the foci. Let d_1 = \sqrt{(x + a)^2 + y^2} be the distance from the point (x, y) to the focus at (-a, 0), and d_2 = \sqrt{(x - a)^2 + y^2} the distance to the focus at (a, 0). The coordinate \tau is then given by [ \tau = \ln \left( \frac{d_1}{d_2} \right),

which ranges from $-\infty$ to $\infty$.[](https://arxiv.org/pdf/1503.09183) Equivalently, $\tau$ can be computed using the inverse hyperbolic tangent:

\tau = \artanh \left( \frac{2 a x}{x^2 + y^2 + a^2} \right),

or directly as

\tau = \frac{1}{2} \ln \left( \frac{(x + a)^2 + y^2}{(x - a)^2 + y^2} \right).

These forms are derived from the properties of the coordinate system and ensure consistency with the forward transformation.[](https://www.researchgate.net/publication/273123774_Bipolar_Coordinates_and_the_Two-Cylinder_Capacitor) The angular coordinate $\sigma$, ranging from $-\pi$ to $\pi$, is the difference in angles subtended by the foci at the point $(x, y)$:

\sigma = \theta_1 - \theta_2,

where $\theta_1 = \atan2(y, x + a)$ is the angle from the positive x-axis to the vector from $(-a, 0)$ to $(x, y)$, and $\theta_2 = \atan2(y, x - a)$ is the angle to the vector from $(a, 0)$ to $(x, y)$. This can also be expressed using the argument of the complex ratio:

\sigma = \arg \left( \frac{z + a}{z - a} \right),

with $z = x + i y$. The $\atan2$ function ensures the correct quadrant for each angle, and the result for $\sigma$ is typically adjusted to lie within $[-\pi, \pi]$.[](https://galileoandeinstein.phys.virginia.edu/Elec_Mag/2022_Lectures/EM_15_2D_Electrostatics_Complex_Var_2.html) Special cases arise along the x-axis. For $x > a$ and $y = 0$, both $\theta_1 = 0$ and $\theta_2 = 0$, so $\sigma = 0$. For $x < -a$ and $y = 0$, $\theta_1 = \pi$ and $\theta_2 = \pi$, yielding $\sigma = 0$. However, on the segment between the foci ($-a < x < a$, $y = 0$), $\theta_1 = 0$ and $\theta_2 = \pi$, giving $\sigma = -\pi$ (or equivalently $\pi$ due to periodicity), marking a branch cut where $\sigma$ is discontinuous.[](https://www.reading.ac.uk/maths-and-stats/-/media/project/uor-main/schools-departments/maths/documents/martin-conway-rev.pdf) Numerical implementation requires care for stability, particularly near the branch cut along $[-a, a]$ on the x-axis, where $d_1 \approx d_2$ and $\tau \approx 0$, while small $y$ can cause jumps in $\sigma$ due to the $\atan2$ difference. The $\artanh$ form for $\tau$ avoids potential division by near-zero in logarithmic ratios, and computations should handle the principal branch of the argument to maintain continuity away from the cut.[](https://www.researchgate.net/publication/273123774_Bipolar_Coordinates_and_the_Two-Cylinder_Capacitor) ## Applications and Extensions ### Applications Bipolar coordinates facilitate the separation of variables in Laplace's equation, enabling analytical solutions for boundary value problems in regions bounded by Apollonian circles. In these coordinates $(\sigma, \tau)$, the equation takes the form

\nabla^2 \phi = \frac{(\cosh \tau - \cos \sigma)^2}{a^2} \left( \frac{\partial^2 \phi}{\partial \sigma^2} + \frac{\partial^2 \phi}{\partial \tau^2} \right) = 0,

where $a$ is the scale parameter, leading to solutions expressed as Fourier series in $\sigma$ and hyperbolic functions in $\tau$.[](https://www.researchgate.net/publication/273123774_Bipolar_Coordinates_and_the_Two-Cylinder_Capacitor) In [electrostatics](/page/Electrostatics), bipolar coordinates are particularly useful for computing the potential between two infinite parallel cylindrical conductors, which correspond to constant-$\tau$ boundaries. The potential $\phi$ satisfies the separated Laplace equation with boundary conditions $\phi = V_1$ on one cylinder and $\phi = V_2$ on the other, yielding an explicit expression involving logarithmic terms derived from the separable solutions. The electric field component $E_\sigma$ is given by $E_\sigma = -\frac{1}{h_\sigma} \frac{\partial \phi}{\partial \sigma}$, where $h_\sigma = \frac{a}{\cosh \tau - \cos \sigma}$ is the scale factor, allowing direct calculation of surface charges and [capacitance](/page/Capacitance) per unit length.[](https://www.researchgate.net/publication/273123774_Bipolar_Coordinates_and_the_Two-Cylinder_Capacitor)[](https://asmedigitalcollection.asme.org/fluidsengineering/article/131/5/054501/412201/Inviscid-Flow-Past-Two-Cylinders) For inviscid [fluid dynamics](/page/Fluid_dynamics), bipolar coordinates model [potential flow](/page/Potential_flow) around two circular obstacles or in channels bounded by circular arcs, simplifying the governing Laplace equation for the [velocity potential](/page/Velocity_potential). Analytical solutions for uniform flow past two [cylinders](/page/Cylinder) of arbitrary radii and separation distances are obtained by superposing [dipole](/page/Dipole) and uniform flow terms in these coordinates, applicable to both perpendicular and parallel flow directions relative to the [cylinder axis](/page/Axis).[](https://asmedigitalcollection.asme.org/fluidsengineering/article/131/5/054501/412201/Inviscid-Flow-Past-Two-Cylinders)[](https://www.sciencedirect.com/science/article/abs/pii/S0021999115002235) In heat conduction, bipolar coordinates solve steady-state problems in domains bounded by [Apollonian circles](/page/Apollonian_circles), such as the temperature distribution between two parallel isothermal pipes embedded in a slab. The Laplace equation separates directly, with solutions of the form $T(\sigma, \tau) = A \tau + B$ for constant boundary temperatures, providing the linear temperature profile across the region.[](https://sites.utexas.edu/maple/files/2019/08/37_Elkamel-Bellamine-Subramanian-PDE_Cordinate_Transform-2008.pdf) In modern applications, bipolar coordinates, particularly two-center variants, are used in quantum chemistry for evaluating two-center two-particle integrals via bipolar expansions, aiding in the calculation of molecular properties. Azimuthally expanded formulations also appear in quantum many-body theory for analyzing scattering processes in strongly coupled electron systems.[](https://pubs.aip.org/aip/jcp/article-pdf/45/1/13/18844878/13_1_online.pdf)[](https://pubs.aip.org/aip/jcp/article/158/14/141102/2877795/Quantum-version-of-the-integral-equation-theory) ### Three-Dimensional Extension The primary three-dimensional extension of bipolar coordinates is the bipolar cylindrical (or bicylindrical) coordinate system, obtained by including the Cartesian [z](/page/Z)-coordinate unchanged, resulting in coordinates $(\xi, \eta, z)$ with $\xi \in [0, 2\pi)$, $\eta \in (-\infty, \infty)$, and $z \in (-\infty, \infty)$. The transformation from Cartesian coordinates is

x = a \frac{\sinh \eta}{\cosh \eta - \cos \xi}, \quad y = a \frac{\sin \xi}{\cosh \eta - \cos \xi}, \quad z = z,

with scale factors $h_\xi = h_\eta = \frac{a}{\cosh \eta - \cos \xi}$ and $h_z = 1$. This system produces coordinate surfaces that are circular cylinders (constant $\xi$ or $\eta$) and planes parallel to the xy-plane (constant z), suitable for problems with [translational symmetry](/page/Translational_symmetry) along z but variation in the cross-section, such as the potential around finite-length cylindrical conductors or flows with axial dependence. [Laplace's equation](/page/Laplace's_equation) is separable in these coordinates for axisymmetric cases.[](https://mathworld.wolfram.com/BipolarCylindricalCoordinates.html)[](https://user.xmission.com/~rimrock/Documents/Bipolar%20Coordinates%20and%20the%20Two-Cylinder%20Capacitor.pdf) Another three-dimensional extension is bispherical coordinates, obtained by considering rotational symmetry around the axis joining the two foci located at $(0, 0, \pm a)$. In this system, the coordinates are typically denoted as $(\eta, \mu, \phi)$, where $\eta \in [0, \pi]$ plays the role of a colatitude-like parameter, $\mu \in (-\infty, \infty)$ is a hyperbolic parameter analogous to the 2D bipolar coordinate, and $\phi$ is the azimuthal angle. The transformation to Cartesian coordinates is given by

r = \frac{a \sin \eta}{\cosh \mu - \cos \eta}, \quad z = \frac{a \sinh \mu}{\cosh \mu - \cos \eta},

x = r \cos \phi, \quad y = r \sin \phi,

where $r = \sqrt{x^2 + y^2}$ is the cylindrical radius.[](https://mathworld.wolfram.com/BisphericalCoordinates.html) Constant-$\mu$ surfaces form spheres centered along the z-axis, with radius $a / |\sinh \mu|$ and center at $z = a \coth \mu$, enabling the modeling of axisymmetric problems involving pairs of spheres, such as electrostatic potentials between two conducting spheres.[](https://pubs.aip.org/books/monograph/45/chapter/20669079/Bispherical-coordinates) The scale factors for bispherical coordinates, essential for computing gradients and Laplacians, are

h_\eta = \frac{a}{\cosh \mu - \cos \eta}, \quad h_\mu = \frac{a}{\cosh \mu - \cos \eta}, \quad h_\phi = \frac{a \sin \eta}{\cosh \mu - \cos \eta}.

These ensure orthogonality, as the coordinate surfaces intersect at right angles. [Laplace's equation](/page/Laplace's_equation) is separable in bispherical coordinates, facilitating analytical solutions for [potential theory](/page/Potential_theory) in regions bounded by spherical surfaces, such as heat conduction or fluid flow between eccentric spheres. However, the [Helmholtz equation](/page/Helmholtz_equation) is not separable, limiting direct applications to wave problems without additional approximations.[](https://mathworld.wolfram.com/BisphericalCoordinates.html)[](https://pubs.aip.org/aip/pof/article/31/2/021208/949211/Transport-phenomena-in-bispherical-coordinates) An alternative three-dimensional extension is [toroidal](/page/Toroidal) coordinates, derived by rotating the two-dimensional bipolar system about the axis joining the foci, yielding coordinates $(\sigma, \tau, \phi)$ with $\sigma \in [0, 2\pi)$, $\tau \in [0, \infty)$, and $\phi \in [0, 2\pi)$. The transformation is

r = \frac{a \sinh \tau}{\cosh \tau - \cos \sigma}, \quad z = \frac{a \sin \sigma}{\cosh \tau - \cos \sigma},

x = r \cos \phi, \quad y = r \sin \phi.

Constant-$\tau$ surfaces are tori of revolution, while constant-$\sigma$ surfaces are spheres, making this system suitable for geometries involving rings or [toroidal](/page/Toroidal) structures, such as [magnetic fields](/page/The_Magnetic_Fields) in tokamaks or acoustic [radiation](/page/Radiation) from toroidal sources. The scale factors are

h_\sigma = \frac{a}{\cosh \tau - \cos \sigma}, \quad h_\tau = \frac{a}{\cosh \tau - \cos \sigma}, \quad h_\phi = \frac{a \sinh \tau}{\cosh \tau - \cos \sigma}.

Laplace's equation is separable here as well, supporting applications in plasma physics and electromagnetics for toroidal symmetries.[19][20]