Parabolic Coordinates

[Eddie W. Shore]

Parabolic Coordinates

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

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

x = μ * v * cos ϕ
y = μ * v * sin ϕ
z = 1/2 * (μ^2 – v^2)

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

where μ ≥ 0 and v ≥ 0

HP Prime Program PBC2REC (Parabolic to Rectangular)

Code:

EXPORT PBC2REC(u,v,φ)
BEGIN
// Parabolic to Rectangular
// u≥0, v≥0, 0≤φ<2π
// EWS 2017-02-28
LOCAL x:=u*v*COS(φ);
LOCAL y:=u*v*SIN(φ);
LOCAL z:=1/2*(u^2-v^2);
RETURN {x,y,z};
END;

HP Prime Program REC2PBC (Rectangular to Parabolic)

Code:

EXPORT REC2PBC(x,y,z)
BEGIN
// Rectangular to Parabolic
// u≥0, v≥0, 0≤φ<2π
// EWS 2017-02-28
LOCAL φ:=ATAN(y/x);
LOCAL v:=√(−z+√(x^2+y^2+z^2));
LOCAL u:=√(2*z+v^2);
RETURN {u,v,φ};
END;

Examples (angles are in radians)

μ = 1, v = 3, ϕ = 0.4
Result: x ≈ 2.76318, y ≈ 1.16826, z = -4

x = 1.69042, y = 7.9006. z = -2.76432
Result: μ ≈ 2.40311, v ≈ 3.36208, ϕ ≈ 1.36000

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