Parabolic coordinates are a two-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
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
in which the
coordinate lines are
confocal In geometry, confocal means having the same foci: confocal conic sections.
* For an optical cavity consisting of two mirrors, confocal means that they share their foci. If they are identical mirrors, their radius of curvature, ''R''mirror, equals ' ...
parabola
In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...
s.
A three-dimensional version of parabolic coordinates is obtained by rotating the two-dimensional
system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...
about the symmetry axis of the parabolas.
Parabolic coordinates have found many applications, e.g., the treatment of the
Stark effect
The Stark effect is the shifting and splitting of spectral lines of atoms and molecules due to the presence of an external electric field. It is the electric-field analogue of the Zeeman effect, where a spectral line is split into several compon ...
and the
potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gravi ...
of the edges.
Two-dimensional parabolic coordinates
Two-dimensional parabolic coordinates
are defined by the equations, in terms of Cartesian coordinates:
:
:
The curves of constant
form confocal parabolae
:
that open upwards (i.e., towards
), whereas the curves of constant
form confocal parabolae
:
that open downwards (i.e., towards
). The foci of all these parabolae are located at the origin.
The Cartesian coordinates
and
can be converted to parabolic coordinates by:
:
:
Two-dimensional scale factors
The scale factors for the parabolic coordinates
are equal
:
Hence, the infinitesimal element of area is
:
and the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
equals
:
Other differential operators such as
and
can be expressed in the coordinates
by substituting
the scale factors into the general formulae
found in
orthogonal coordinates In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...
.
Three-dimensional parabolic coordinates
The two-dimensional parabolic coordinates form the basis for two sets of three-dimensional
orthogonal coordinates In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...
. The
parabolic cylindrical coordinates
In mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in the
perpendicular z-direction. Hence, the coordinate surfaces ...
are produced by projecting in the
-direction.
Rotation about the symmetry axis of the parabolae produces a set of
confocal paraboloids, the coordinate system of tridimensional parabolic coordinates. Expressed in terms of cartesian coordinates:
:
:
:
where the parabolae are now aligned with the
-axis,
about which the rotation was carried out. Hence, the azimuthal angle
is defined
:
The surfaces of constant
form confocal paraboloids
:
that open upwards (i.e., towards
) whereas the surfaces of constant
form confocal paraboloids
:
that open downwards (i.e., towards
). The foci of all these paraboloids are located at the origin.
The
Riemannian metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
associated with this coordinate system is
:
Three-dimensional scale factors
The three dimensional scale factors are:
:
:
:
It is seen that the scale factors
and
are the same as in the two-dimensional case. The infinitesimal volume element is then
:
and the Laplacian is given by
:
Other differential operators such as
and
can be expressed in the coordinates
by substituting
the scale factors into the general formulae
found in
orthogonal coordinates In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...
.
See also
*
Parabolic cylindrical coordinates
In mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in the
perpendicular z-direction. Hence, the coordinate surfaces ...
*
Orthogonal coordinate system In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q'd'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate su ...
*
Curvilinear coordinates
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is locally inve ...
Bibliography
*
*
*
*
* Same as Morse & Feshbach (1953), substituting ''u''
''k'' for ΞΎ
''k''.
*
External links
*
MathWorld description of parabolic coordinates
{{Orthogonal coordinate systems
Orthogonal coordinate systems