
Conical coordinates, sometimes called sphero-conal or sphero-conical coordinates, are a three-dimensional
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
coordinate system consisting of
concentric spheres (described by their radius ) and by two families of perpendicular elliptic cones, aligned along the - and -axes, respectively. The intersection between one of the cones and the sphere forms a
spherical conic
In mathematics, a spherical conic or sphero-conic is a curve on the sphere, the intersection of the sphere with a concentric elliptic cone. It is the spherical analog of a conic section ( ellipse, parabola, or hyperbola) in the plane, and as in ...
.
Basic definitions
The conical coordinates
are defined by
:
:
:
with the following limitations on the coordinates
:
Surfaces of constant are spheres of that radius centered on the origin
:
whereas surfaces of constant
and
are mutually perpendicular cones
:
and
:
In this coordinate system, both
Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \na ...
and the
Helmholtz equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation
\nabla^2 f = -k^2 f,
where is the Laplace operator (or "Laplacian"), is the eigenva ...
are separable.
Scale factors
The scale factor for the radius is one (), as in
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' mea ...
. The scale factors for the two conical coordinates are
:
and
:
References
Bibliography
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External links
MathWorld description of conical coordinates
{{Orthogonal coordinate systems
Three-dimensional coordinate systems
Orthogonal coordinate systems