Elliptic coordinate system
The elliptic coordinate system is a two-dimensional orthogonal curvilinear coordinate system in the Euclidean plane, characterized by families of confocal ellipses and hyperbolas that serve as its level curves.[1] In this system, the coordinates (\mu, \nu) transform to Cartesian coordinates (x, y) via the equations x = c \cosh \mu \cos \nu and y = c \sinh \mu \sin \nu, where c > 0 is a fixed scale parameter representing half the distance between the foci located at (\pm c, 0).[2] The coordinate \mu (ranging from 0 to \infty) parameterizes the confocal ellipses, starting from the degenerate ellipse (the line segment between the foci) at \mu = 0 and expanding outward, while \nu (ranging from 0 to $2\pi) parameterizes the confocal hyperbolas, which branch into the first and third quadrants or second and fourth quadrants depending on the value.[2] The scale factors for both coordinates are equal, h_\mu = h_\nu = c \sqrt{\sinh^2 \mu + \sin^2 \nu}, ensuring orthogonality and making the system conformal, which simplifies the Laplacian and other differential operators.[2] This coordinate system finds applications in mathematical physics, particularly for boundary value problems exhibiting elliptical symmetry, such as solving Laplace's or Helmholtz's equation inside elliptical domains for electrostatic potentials or acoustic waves.[2] It is also employed in the analysis of integrable dynamical systems, like billiards confined to elliptical tables, where the confocal property preserves integrability under reflections.[3] In three dimensions, the elliptic system extends naturally to elliptic cylindrical coordinates by adjoining a linear z-coordinate, with z unchanged and scale factor h_z = 1, facilitating solutions for cylindrical geometries with elliptical cross-sections, such as in waveguide propagation.[2] Further generalizations include prolate and oblate spheroidal coordinates, which are obtained by rotating the elliptic system about an axis to form surfaces of revolution for axisymmetric problems in quantum mechanics and fluid dynamics.[4][5]Two-Dimensional Fundamentals
Definition
The two-dimensional elliptic coordinate system is an orthogonal curvilinear coordinate system in the plane, parameterized by μ and ν, which transform to Cartesian coordinates (x, y) via the equations x = a \cosh \mu \cos \nu y = a \sinh \mu \sin \nu where a > 0 is the interfocal distance parameter.[6] These equations arise from the geometry of confocal conic sections, with constant-μ curves forming ellipses and constant-ν curves forming hyperbolas.[6] The coordinate μ ranges over [0, ∞), functioning as a radial-like parameter that quantifies progression from the innermost degenerate ellipse (the line segment joining the foci at μ = 0) to larger enclosing ellipses as μ increases.[6] The coordinate ν spans [0, 2π), serving as an angular parameter that traces the elliptic "longitude" around the foci.[6] The foci of this system are fixed at the points (±a, 0) on the x-axis, ensuring the confocal property where every ellipse and hyperbola in the coordinate grid shares these two points.[7] This formulation is motivated by elliptical geometry and facilitates the solution of boundary value problems with elliptical symmetry, such as those governed by Laplace's equation in electrostatics or hydrodynamics.[8]Geometric Interpretation
The elliptic coordinate system in two dimensions provides a geometric framework where points in the plane are parameterized by distances relative to two fixed foci located at (\pm a, 0). Constant \mu curves correspond to confocal ellipses centered at the origin, sharing the same foci, with the semi-major axis along the x-direction given by a \cosh \mu and the semi-minor axis by a \sinh \mu, where \mu \geq 0.[9] These ellipses are highly elongated near \mu = 0, degenerating to the line segment between the foci as \mu \to 0, and become less elongated, approaching circular shapes as \mu increases.[2] Constant \nu curves, for $0 \leq \nu < 2\pi, form confocal hyperbolae also sharing the foci at (\pm a, 0), with branches opening along the y-direction. These hyperbolae have asymptotes oriented at angles \nu and \pi - \nu relative to the positive x-axis, reflecting the angular parameterization that traces the curve's orientation. The confocal nature ensures that every ellipse and hyperbola in the family intersects at right angles, establishing the orthogonality of the coordinate system at all intersection points.[2] This perpendicularity arises from the geometric properties of confocal conics, where the tangent directions satisfy the condition for orthogonal trajectories.[9] The coordinates \mu and \nu directly relate to the distances d_1 and d_2 from a point to the foci, with \mu determined by \cosh \mu = (d_1 + d_2)/(2a), capturing the sum of distances characteristic of ellipses, and \nu by \cos \nu = (d_1 - d_2)/(2a), adjusted via the imaginary unit i in some formulations to align with hyperbolic differences.[9] This distance-based interpretation underscores the system's utility for problems with elliptical or hyperbolic symmetry, such as potential theory around elongated objects. The parametric equations x = a \cosh \mu \cos \nu, y = a \sinh \mu \sin \nu serve as the foundation for visualizing these curves.[2]Scale Factors and Metrics
In μ-ν Coordinates
In the μ-ν formulation of the two-dimensional elliptic coordinate system, the position is parameterized byx = a \cosh \mu \cos \nu, \quad y = a \sinh \mu \sin \nu,
where a > 0 is a scale parameter, \mu \geq 0, and $0 \leq \nu < 2\pi.[10] These parametric equations define confocal ellipses and hyperbolas as level curves of μ and ν, respectively. The scale factors are derived from the partial derivatives of the position vector \mathbf{r} = (x, y). Specifically,
h_\mu = \left| \frac{\partial \mathbf{r}}{\partial \mu} \right| = a \sqrt{\sinh^2 \mu + \sin^2 \nu},
obtained by computing
\left( \frac{\partial x}{\partial \mu} \right)^2 + \left( \frac{\partial y}{\partial \mu} \right)^2 = a^2 (\sinh^2 \mu \cos^2 \nu + \cosh^2 \mu \sin^2 \nu) = a^2 (\sinh^2 \mu + \sin^2 \nu).
Similarly,
h_\nu = \left| \frac{\partial \mathbf{r}}{\partial \nu} \right| = a \sqrt{\sinh^2 \mu + \sin^2 \nu},
since the system is orthogonal and the expressions symmetrize under differentiation with respect to ν. An equivalent form is h_\mu = h_\nu = a \sqrt{\cosh^2 \mu - \cos^2 \nu}, reflecting the identity \cosh^2 \mu - \sinh^2 \mu = 1 and \cos^2 \nu + \sin^2 \nu = 1.[10] The line element in these coordinates is
ds^2 = h_\mu^2 d\mu^2 + h_\nu^2 d\nu^2 = a^2 (\sinh^2 \mu + \sin^2 \nu) (d\mu^2 + d\nu^2). [10] The infinitesimal area element follows as the product of the scale factors:
dA = h_\mu h_\nu \, d\mu \, d\nu = a^2 (\sinh^2 \mu + \sin^2 \nu) \, d\mu \, d\nu. [10] For the Laplacian operator applied to a scalar function Φ, the general orthogonal form simplifies due to h_\mu = h_\nu = h:
\nabla^2 \Phi = \frac{1}{h^2} \left( \frac{\partial^2 \Phi}{\partial \mu^2} + \frac{\partial^2 \Phi}{\partial \nu^2} \right) = \frac{1}{a^2 (\sinh^2 \mu + \sin^2 \nu)} \left( \frac{\partial^2 \Phi}{\partial \mu^2} + \frac{\partial^2 \Phi}{\partial \nu^2} \right). [10] This form facilitates separation of variables in elliptic coordinates for solving partial differential equations like Laplace's equation.
In σ-τ Coordinates
The σ-τ coordinates provide an alternative parametrization of the elliptic coordinate system, related to the standard μ-ν form by the transformation σ = cosh μ ≥ 1 and τ = cos ν ∈ [-1, 1].[11] This mapping preserves the confocal property of the coordinate curves, with constant-σ surfaces corresponding to ellipses and constant-τ surfaces to hyperbolas, but offers bounded ranges for both parameters, facilitating analysis in regions enclosed by an ellipse.[11] The Cartesian coordinates in terms of σ and τ are given by the parametric equationsx = a \sigma \tau,
y = a \sqrt{(\sigma^2 - 1)(1 - \tau^2)},
where a is half the focal distance. These equations directly follow from substituting the transformation into the standard elliptic parametrization. The relation to distances from the foci at (±a, 0) is d₁ + d₂ = 2a σ and |d₁ - d₂| = 2a |τ|, highlighting how σ scales the sum of distances and τ modulates the difference in a bounded manner.[12] The scale factors for σ and τ are
h_\sigma = a \sqrt{\frac{\sigma^2 - \tau^2}{\sigma^2 - 1}},
h_\tau = a \sqrt{\frac{\sigma^2 - \tau^2}{1 - \tau^2}}.
These ensure orthogonality and are derived from the metric tensor in the transformed coordinates. The infinitesimal area element is then
dA = h_\sigma h_\tau \, d\sigma \, d\tau = a^2 \frac{\sigma^2 - \tau^2}{\sqrt{(\sigma^2 - 1)(1 - \tau^2)}} \, d\sigma \, d\tau.
This form simplifies integrals over elliptic regions by aligning with the bounded parameter space.[13] The σ-τ formulation is particularly advantageous for boundary value problems in bounded domains, such as solving Laplace's equation inside an ellipse, where the finite ranges of σ and τ enable efficient spectral expansions or finite-difference schemes without dealing with unbounded hyperbolic or trigonometric domains.[12] For example, in potential theory, the separability of the Laplacian persists, but the bounded τ interval aids in representing solutions via Fourier-like series over [-1, 1].[11]