**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__. 2^{nd}ed. Springer-Verlag: Berlin, Heidelberg, New York. 1971. ISBN 0-387-02732-7
This blog is
property of Edward Shore, 2017.

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