A python module for converting between cartesian and polar/spherical coordinates in any number of dimensions.
There are several conventions to choose from:
| dimension | name | parameters |
|---|---|---|
| 2-d | polar | r, phi |
| 3-d | spherical | r, theta, phi |
| n-d | n-spherical | r, phi1, phi2 .. phi[n-1] |
r == 0 -> by convention use phi == 0
r != 0 -> 0 <= phi < 2\*PI
x = r * cos(phi)
y = r * sin(phi)
r = sqrt(x^2 + y^2)
phi = atan2(y,x)
r==0 -> by convention: phi == theta == 0
r!=0 -> 0<=theta<2\*PI, 0<=phi<=PI
x = r * sin(phi) * cos(theta)
y = r * sin(phi) * sin(theta)
z = r * cos(phi)
r = sqrt(x^2+y^2+z^2)
phi = arccos(z/r)
theta = atan2(y,x)
x1 = r * cos(phi1)
x2 = r * sin(phi1) * cos(phi2)
x3 = r * sin(phi1) * sin(phi2) * cos(phi3)
...
x[i] = r * prod(sin(phi[j]), j=1..i-1) * cos(phi[i])
...
x[n-1] = r * prod(sin(phi[j]), j=1..n-2) * cos(phi[n-1])
x[n] = r * prod(sin(phi[j]), j=1..n-1)
note: when we add a dummy variable phi[n+1] == 0, the formula's become even more uniform.
(helper function)
r[i] = sqrt(sum(x[j]^2, j=i..n))
r = r[1]
phi1 = arccos(x1/r[1])
phi2 = arccos(x2/r[2])
...
phi[i] = arccos(x[i]/r[i])
...
phi[n-1] = arccos(x[n-1]/r[n-1]) for x[n]>=0
= 2*PI - arccos(x[n-1]/r[n-1]) for x[n] < 0