A system of curvilinear coordinates for which several different notations are commonly used. In this work (u, v, ϕ) is used, whereas Arfken uses (ξ, η, φ) and Moon and Spencer use (η, θ, ψ). The toroidal coordinates are defined by x | = | (a sinh v cos ϕ)/(cosh v - cos u) y | = | (a sinh v sin ϕ)/(cosh v - cos u) z | = | (a sin u)/(cosh v - cos u), where sinh z is the hyperbolic sine and cosh z is the hyperbolic cosine. The coordinates satisfy u element [0, 2π), v element [0, ∞), and ϕ element [0, 2π).

bispherical coordinates | flat-ring cyclide coordinates | Laplace's equation--toroidal coordinates