23.2 Programs and Computational Aspects
269
Elliptic Coordinates The corresponding transformation to cartesian coordinates is
given by Eqs. (23.29). The inverse transformation is based on
r
2
1 = x
2
+ y
2
+
z +
R
2
2
(23.36a)
r
2
2 = x
2
+ y
2
+
z −
R
2
2
(23.36b)
μ =
r 1 + r 2
2
and
ν =
r 1 − r 2
2
.
(23.36c)
For H
+
2 r 1 and r 2 can be interpreted as the distance between the two nuclei to the
electron and R as the distance between the two nuclei.
The Class CoordTrafo
The class CoordTrafo supports the transformation between all coordinate systems listed in Table 23.1. The syntax is obj = CoordTrafo(was, x1, x2,
x3, scal, scal2). The input variable “was” is built by the transformation
acronyms listed in Table 23.1. For example, the transformation from spherical
coordinates to cartesian coordinates is given by was = ’sphe2cart’, the transformation from an oblate spheroidal coordinate system to cylindrical coordinates is
given by was = ’obla2cyli’. In most cases the corresponding acronym of
the coordinate system is given by the first four letters. “xi”, i ∈ [1, 2, 3] are the
coordinates, either of the same size or scalar values. The limits for each coordinate
system are listed in the equations above, but not proven by the class CoordTrafo.
In some cases transformation and inverse transformation are not bijective, e.g., in
case of vanishing coordinate values. “scal” and “scal2” are optional scalar values
(scaling factors of the corresponding coordinate system) with default values 1. In
Table 23.1 Coordinate systems supported by the SPECFUNPHYS class CoordTrafo
Coordinate system
Acronym
Transformation example
Cartesian
cart
Cartesian to spherical: cart2sphe
Spherical
sphe
Spherical to cartesian: sphe2cart
Cylindrical
cyli
Spherical to cylindrical: sphe2cyli
Prolate spheriodal
prol
prolate spheroidal to cartesian: prol2cart
Parabolic
para
Parabolic to spherical: para2sphe
Elliptic cylindrical
elcy
Cartesian to elliptic cylindrical: cart2elcy
Oblate spheroidal
obla
Oblate spheroidal to cartesian: obla2cart
Parabolic cylindrical
pacy
Parabolic cylind. to cylindrical: pacy2cyli
Paraboloidal
paal
Paraboloidal to cartesian: paal2cart
Semiparabolic
semi
Semiparabolic to cartesian: semi2cart
Elliptic
elli
Elliptic to parabolic: elli2para
269
Elliptic Coordinates The corresponding transformation to cartesian coordinates is
given by Eqs. (23.29). The inverse transformation is based on
r
2
1 = x
2
+ y
2
+
z +
R
2
2
(23.36a)
r
2
2 = x
2
+ y
2
+
z −
R
2
2
(23.36b)
μ =
r 1 + r 2
2
and
ν =
r 1 − r 2
2
.
(23.36c)
For H
+
2 r 1 and r 2 can be interpreted as the distance between the two nuclei to the
electron and R as the distance between the two nuclei.
The Class CoordTrafo
The class CoordTrafo supports the transformation between all coordinate systems listed in Table 23.1. The syntax is obj = CoordTrafo(was, x1, x2,
x3, scal, scal2). The input variable “was” is built by the transformation
acronyms listed in Table 23.1. For example, the transformation from spherical
coordinates to cartesian coordinates is given by was = ’sphe2cart’, the transformation from an oblate spheroidal coordinate system to cylindrical coordinates is
given by was = ’obla2cyli’. In most cases the corresponding acronym of
the coordinate system is given by the first four letters. “xi”, i ∈ [1, 2, 3] are the
coordinates, either of the same size or scalar values. The limits for each coordinate
system are listed in the equations above, but not proven by the class CoordTrafo.
In some cases transformation and inverse transformation are not bijective, e.g., in
case of vanishing coordinate values. “scal” and “scal2” are optional scalar values
(scaling factors of the corresponding coordinate system) with default values 1. In
Table 23.1 Coordinate systems supported by the SPECFUNPHYS class CoordTrafo
Coordinate system
Acronym
Transformation example
Cartesian
cart
Cartesian to spherical: cart2sphe
Spherical
sphe
Spherical to cartesian: sphe2cart
Cylindrical
cyli
Spherical to cylindrical: sphe2cyli
Prolate spheriodal
prol
prolate spheroidal to cartesian: prol2cart
Parabolic
para
Parabolic to spherical: para2sphe
Elliptic cylindrical
elcy
Cartesian to elliptic cylindrical: cart2elcy
Oblate spheroidal
obla
Oblate spheroidal to cartesian: obla2cart
Parabolic cylindrical
pacy
Parabolic cylind. to cylindrical: pacy2cyli
Paraboloidal
paal
Paraboloidal to cartesian: paal2cart
Semiparabolic
semi
Semiparabolic to cartesian: semi2cart
Elliptic
elli
Elliptic to parabolic: elli2para