**HP Prime: Parabolic Coordinates**

**The Formulas**

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

**Derivation**

The formulas to
find the rectangular coordinates are given.
We can derive the formulas for the parabolic coordinates by the
following:

Assume that μ >
0 and v > 0 (neither are zero). Then:

x = μ * v * cos
ϕ

y = μ * v * sin
ϕ

x / y = (μ * v
* cos ϕ)/( μ * v * sin ϕ)

x / y = 1 / tan
ϕ

y / x = tan ϕ

ϕ = atan (y /
x)

Express μ in
terms of z:

z = 1/2 * (μ^2 –
v^2)

2 * z = μ^2 –
v^2

μ^2 = 2*z + v^2

*Since μ is positive, only the positive square root is considered:*

μ = √(2*z +
v^2)

Can we find an
expression for v? If we can, we have
found our formulas:

x = μ * v * cos
ϕ

*Square both sides:*

x^2 = μ^2 * v^2
* cos^2 ϕ (I)

*Note that for a given variable w, cos(atan w) = 1 / √(w^2 + 1)*

*In this case, w = y/x or:*

cos^2 ϕ

= (cos (atan ϕ))^2

= √(1 / ((y/x)^2
+ 1))^2

= 1 / ((y/x)^2
+ 1)

= x^2 / x^2 * 1
/ ((y/x)^2 + 1)

= x^2 / (x^2 +
y^2)

*Back to (I):*

x^2 = μ^2 * v^2
* cos^2 ϕ (I)

‘x^2 = (2*z +
v^2) * v^2 * x^2 / (x^2 + y^2)

*Assuming x≠0, divide both sides by x^2:*

1 = (2*z + v^2)
* v^2 * 1 / (x^2 + y^2)

1 = (2*z*v^2 +
v^4) * 1/(x^2 + y^2)

0 = 1/(x^2 +
y^2) * v^4 + (2*z)/(x^2 + y^2) * v^2 – 1

*Here we have a quadratic equations in the form of Av^4 + Bv^2 + C = 0 where:*

A = 1/(x^2 +
y^2)

B = (2*z)/(x^2
+ y^2)

C = -1

*The solution is v^2 = (-B + √(B^2 – 4*A*C)/(2*A). Remember that v > 0, so only positive roots will be considered. Then:*

-B/(2*A) = -(2*z)/(x^2 + y^2) * (x^2 + y^2)/2 = -z

*And:*

B^2 – 4*A*C =
(4*z^2 + 4*(x^2 + y^2))/(x^2 + y^2)^2

√( B^2 – 4*A*C)
= 2 * √(x^2 + y^2 + z^2)/(x^2 + y^2)

√( B^2 – 4*A*C)/(2*A)
= √(x^2 + y^2 + z^2)

*Hence:*

v^2 = -z + √(x^2
+ y^2 + z^2)

v = √(-z + √(x^2
+ y^2 + z^2))

**HP Prime Program PBC2REC (Parabolic to Rectangular)**

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)**

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