Parabolic Cylindrical Coordinates

[Eddie W. Shore]

Parabolic Cylindrical Coordinates

Link: http://edspi31415.blogspot.com/2017/03/h...rical.html

The relationship and conversion factors between parabolic cylindrical coordinates (μ, v, ϕ) and rectangular coordinates (x, y, z) are as follows:

x = 1/2 * (μ^2 – v^2)
y = μ * v
z = z

μ = √(x + √(x^2 + y^2))
v = y / μ
z = z
(note the sequence)

where μ ≥ 0

HP Prime Program PCC2REC (Parabolic Cylindrical to Rectangular)

Code:

EXPORT PCC2REC(μ,v,z)
BEGIN
// 2017-02-27 EWS
// Parabolic Cylindrical
// to Rectangular
// μ≥0
LOCAL x:=1/2*(μ^2-v^2);
LOCAL y:=μ*v;
RETURN {x,y,z};
END;

HP Prime Program REC2PBC (Rectangular to Parabolic Cylindrical)

Code:

EXPORT REC2PCC(x,y,z)
BEGIN
// 2017-02-27 EWS
// Rectangular to
// Parabolic Cylindrical
// μ≥0
LOCAL μ:=√(x+√(x^2+y^2));
LOCAL v:=y/μ;
RETURN {μ,v,z};
END;

Example

μ = 2, v = 3, z = 1
Result: x = -2.5, y = 6, z = 1

Source:
P. Moon and D.E. Spencer. Field Theory Handbook: Including Coordinate Systems Differential Equations and Their Solutions. 2nd ed. Springer-Verlag: Berlin, Heidelberg, New York. 1971. ISBN 0-387-02732-7