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