Symmetric streamplot with matplotlib

Question

I'm trying to plot the streamlines of a magnetic field around a sphere using matplotlib, and it does work quite nicely. However, the resulting image is not symmetric, but it should be (I think).

This is the code used to generate the image. Excuse the length, but I thought it would be better than just posting a non-working snippet. Also, it's not very pythonic; that's because I converted it from Matlab, which was easier than I expected.

from __future__ import division
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Circle

def cart2spherical(x, y, z):
    r = np.sqrt(x**2 + y**2 + z**2)
    phi = np.arctan2(y, x)
    theta = np.arccos(z/r)
    if r == 0:
        theta = 0
    return (r, theta, phi)

def S(theta, phi):
    S = np.array([[np.sin(theta)*np.cos(phi), np.cos(theta)*np.cos(phi), -np.sin(phi)],
                  [np.sin(theta)*np.sin(phi), np.cos(theta)*np.sin(phi),  np.cos(phi)],
                  [np.cos(theta),             -np.sin(theta),             0]])
    return S

def computeB(r, theta, phi, a=1, muR=100, B0=1):
    delta = (muR - 1)/(muR + 2)
    if r > a:
        Bspherical = B0*np.array([np.cos(theta) * (1 + 2*delta*a**3 / r**3),
                                  np.sin(theta) * (delta*a**3 / r**3 - 1),
                                  0])
        B = np.dot(S(theta, phi), Bspherical)
    else:
        B = 3*B0*(muR / (muR + 2)) * np.array([0, 0, 1])
    return B

Z, X = np.mgrid[-2.5:2.5:1000j, -2.5:2.5:1000j]
Bx = np.zeros(np.shape(X))
Bz = np.zeros(np.shape(X))
Babs = np.zeros(np.shape(X))
for i in range(len(X)):
    for j in range(len(Z)):
        r, theta, phi = cart2spherical(X[0, i], 0, Z[j, 0])
        B = computeB(r, theta, phi)
        Bx[i, j], Bz[i, j] = B[0], B[2]
        Babs[i, j] = np.sqrt(B[0]**2 + B[1]**2 + B[2]**2)

fig=plt.figure()
ax=fig.add_subplot(111)

plt.streamplot(X, Z, Bx, Bz, color='k', linewidth=0.8*Babs, density=1.3,
               minlength=0.9, arrowstyle='-')
ax.add_patch(Circle((0, 0), radius=1, facecolor='none', linewidth=2))
plt.axis('equal')
plt.axis('off')
fig.savefig('streamlines.pdf', transparent=True, bbox_inches='tight', pad_inches=0)

Answer 1

Use a mask to separate the two regions of interest:

mask = np.sqrt(X**2+Z**2)<1

BX_OUT = Bx.copy()
BZ_OUT = Bz.copy()
BX_OUT[mask] = None
BZ_OUT[mask] = None
plt.streamplot(X, Z, BX_OUT, BZ_OUT, color='k', 
               arrowstyle='-', density=2)

BX_IN = Bx.copy()
BZ_IN = Bz.copy()
BX_IN[~mask] = None
BZ_IN[~mask] = None
plt.streamplot(X, Z, BX_IN, BZ_IN, color='r', 
               arrowstyle='-', density=2)

The resulting plot isn't exactly symmetric, but by giving the algorithm a hint, it's far closer than what you had before. Play with the density of the grid via meshgrid and the density parameter to achieve the effect you are looking for.

Answer 2

First of all, for curiosity, why would you want to plot symmetric data? Why plotting half of isn't fine?

Said that, this is a possible hack. You can use mask arrays as Hooked suggested to plot half of it:

mask = X>0
BX_OUT = Bx.copy()
BZ_OUT = Bz.copy()
BX_OUT[mask] = None
BZ_OUT[mask] = None
res = plt.streamplot(X, Z, BX_OUT, BZ_OUT, color='k', 
           arrowstyle='-',linewidth=1,density=2)

then you save in res the result from streamplot, extract the lines and plot them with the opposite X coordinate.

lines = res.lines.get_paths()
for l in lines:
    plot(-l.vertices.T[0],l.vertices.T[1],'k')

I used this hack to extract streamlines and arrows from a 2D plot, then apply a 3D transformation and plot it with mplot3d. A picture is in one of my questions here.