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Conical coordinates

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Conical coordinates

Conical coordinates are a three-dimensional orthogonal coordinate system consisting of concentric spheres (described by their radius r) and by two families of perpendicular cones, aligned along the z- and x-axes, respectively.

Contents

Basic definitions

The conical coordinates ( r , μ , ν ) are defined by

x = r μ ν b c y = r b ( μ 2 − b 2 ) ( ν 2 − b 2 ) ( b 2 − c 2 ) z = r c ( μ 2 − c 2 ) ( ν 2 − c 2 ) ( c 2 − b 2 )

with the following limitations on the coordinates

ν 2 < c 2 < μ 2 < b 2 .

Surfaces of constant r are spheres of that radius centered on the origin

x 2 + y 2 + z 2 = r 2 ,

whereas surfaces of constant μ and ν are mutually perpendicular cones

x 2 μ 2 + y 2 μ 2 + b 2 + z 2 μ 2 − c 2 = 0

and

x 2 ν 2 + y 2 ν 2 − b 2 + z 2 ν 2 + c 2 = 0.

In this coordinate system, both Laplace's equation and the Helmholtz equation are separable.

Scale factors

The scale factor for the radius r is one (hr = 1), as in spherical coordinates. The scale factors for the two conical coordinates are

h μ = r μ 2 − ν 2 ( b 2 − μ 2 ) ( μ 2 − c 2 )

and

h ν = r μ 2 − ν 2 ( b 2 − ν 2 ) ( c 2 − ν 2 ) .

Alternate Definition

An alternative set of conical coordinates have been derived

ξ = r cos ⁡ ( ϕ sin ⁡ θ ) ψ = r sin ⁡ ( ϕ sin ⁡ θ ) ζ = θ ,

where { r , θ , ϕ } are spherical polar coordinates. The corresponding inverse relations are

r = ξ 2 + ψ 2 ϕ = 1 sin ⁡ ζ arctan ⁡ ( ψ ξ ) θ = ζ .

The infinitesimal Euclidean distance between two points in these coordinates

d s 2 = d ξ 2 + d ψ 2 + ( ξ 2 + ψ 2 ) ( 1 + arctan ⁡ ( ψ ξ ) 2 cot ⁡ ζ 2 ) d ζ 2 + 2 ψ arctan ⁡ ( ψ ξ ) cot ⁡ ζ d ξ d ζ − 2 ξ arctan ⁡ ( ψ ξ ) cot ⁡ ζ d ψ d ζ .

If the path between any two points is constrained to surface of the cone given by ζ = π 4 then the geodesic distance between any two points

{ ξ 1 , ψ 1 , ζ 1 = π 4 } and { ξ 2 , ψ 2 , ζ 2 = π 4 } is

s 12 2 = ( ξ 1 − ξ 2 ) 2 + ( ψ 1 − ψ 2 ) 2 .

References

Conical coordinates Wikipedia
(Text) CC BY-SA


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