HP Prime: Parabolic Cylindrical Coordinates
(This is
post # 700! Yay! Thank you for sticking
with me on this wonderful journey!)
The Formulas
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
Derivation
The formulas to
find the rectangular coordinates are given.
We can derive the formulas for the parabolic cylindrical coordinates by
the following:
Obviously the z
coordinate remains the same in both systems.
Assume μ ≠ 0. Then:
y = μ * v
v = y / μ
With
substitution and simplification:
x = 1/2 * (μ^2 –
v^2)
2*x = μ^2 – v^2
2*x = μ^2 – (y^2/μ^2)
2*x*μ^2 = μ^4 –
y^2
0 = μ^4 – (2*x)*μ^2
– y^2
We have a
quadratic equation in terms of μ^2.
Since μ≥0, only the positive root is considered:
μ^2 = ( 2*x + √(4*x^2
+ 4*y^2) ) /2
μ^2 = ( 2*x +
2*√(x^2 + y^2) )/2
μ^2 = x + √(x^2
+ y^2)
μ = √( x + √(x^2
+ y^2) )
HP Prime Program PCC2REC (Parabolic
Cylindrical to Rectangular)
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)
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
This blog is
property of Edward Shore, 2017.