Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system ( λ , μ , ν ) that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces, the ellipsoidal coordinate system is not produced by rotating or projecting any two-dimensional orthogonal coordinate system.
The Cartesian coordinates ( x , y , z ) can be produced from the ellipsoidal coordinates ( λ , μ , ν ) by the equations
x 2 = ( a 2 + λ ) ( a 2 + μ ) ( a 2 + ν ) ( a 2 − b 2 ) ( a 2 − c 2 ) y 2 = ( b 2 + λ ) ( b 2 + μ ) ( b 2 + ν ) ( b 2 − a 2 ) ( b 2 − c 2 ) z 2 = ( c 2 + λ ) ( c 2 + μ ) ( c 2 + ν ) ( c 2 − b 2 ) ( c 2 − a 2 ) where the following limits apply to the coordinates
− λ < c 2 < − μ < b 2 < − ν < a 2 . Consequently, surfaces of constant λ are ellipsoids
x 2 a 2 + λ + y 2 b 2 + λ + z 2 c 2 + λ = 1 , whereas surfaces of constant μ are hyperboloids of one sheet
x 2 a 2 + μ + y 2 b 2 + μ + z 2 c 2 + μ = 1 , because the last term in the lhs is negative, and surfaces of constant ν are hyperboloids of two sheets
x 2 a 2 + ν + y 2 b 2 + ν + z 2 c 2 + ν = 1 because the last two terms in the lhs are negative.
Scale factors and differential operators
For brevity in the equations below, we introduce a function
S ( σ ) = d e f ( a 2 + σ ) ( b 2 + σ ) ( c 2 + σ ) where σ can represent any of the three variables ( λ , μ , ν ) . Using this function, the scale factors can be written
h λ = 1 2 ( λ − μ ) ( λ − ν ) S ( λ ) h μ = 1 2 ( μ − λ ) ( μ − ν ) S ( μ ) h ν = 1 2 ( ν − λ ) ( ν − μ ) S ( ν ) Hence, the infinitesimal volume element equals
d V = ( λ − μ ) ( λ − ν ) ( μ − ν ) 8 − S ( λ ) S ( μ ) S ( ν ) d λ d μ d ν and the Laplacian is defined by
∇ 2 Φ = 4 S ( λ ) ( λ − μ ) ( λ − ν ) ∂ ∂ λ [ S ( λ ) ∂ Φ ∂ λ ] + Other differential operators such as ∇ ⋅ F and ∇ × F can be expressed in the coordinates ( λ , μ , ν ) by substituting the scale factors into the general formulae found in orthogonal coordinates.
