绘制二维向量（二维）：如何在 Python 中的球坐标中 plot 向量场

Question

How to plot a vector field in Spherical Coordinates in Python like this equation:

Is it possible to plot a graph (2D or 3D) of vector field in spherical coordinates using matplotlib?

Need some help.

Answer 1

I have ignored the normalisation by r^3 in H .

import numpy as np
import matplotlib.pyplot as plt

pts = 100
r, theta = np.linspace(-1, 1, pts), np.linspace(0, 2*np.pi, pts)

r_comp = 2*np.cos(theta)
theta_comp = np.sin(theta)


x_comp = r_comp * np.sin(theta_comp)
y_comp = r_comp * np.cos(theta_comp)

plt.figure()
plt.quiver(x_comp, y_comp, r_comp, theta_comp)

Answer 2

Since you posted this on Physics.SE , I assume your equation is using the physics convention for spherical coordinates , with theta=0 at the north pole.

You can easily plot 3D vector fields using SageMath , which is a huge mathematics / algebra system built on top of Python. If you can write Python, it's easy to start working with SageMath. You don't even need to install it: you can run Sage scripts in your Web browser, on the SageMathCell server.

Sage has sophisticated support for vector fields , with a variety of coordinate systems built-in. It can do Cartesian plots working in spherical coordinates , but it's allegedly rather slow, and I haven't experimented with it.

The code below creates an interactive 3D plot of your vector field, doing the coordinate transformations explicitly. Sage lets us define symbolic variables, which can be used to create equations and do algebra and calculus. Sage syntax is almost identical to plain Python, although as a mathematical convenience it permits the use of ^ as well as ** for exponentiation (Sage uses ^^ for exclusive-or).

The Sage plot_vector_field3d function works with Cartesian coordinates. It expects its function argument to be a list, tuple, or vector of functions that each take x, y, z args. So we need to transform those args to polar form, call the H function with the polar args, then transform the polar value returned by H to Cartesian. By working with symbolic variables and functions, we can transform the H function itself, as this code illustrates.

# Create some symbolic variables
x, y, z = var('x,y,z')
rad, th, phi = var('rad,th,ph')

# Transform polar coords to Cartesian
def sphere_xyz(R, theta, phi):
    r = R * sin(theta)
    return r * cos(phi), r * sin(phi), R * cos(theta)

# Transform Cartesian coords to polar
def xyz_sphere(X, Y, Z):
    r2 = X^2 + Y^2
    r = sqrt(r2)
    R = sqrt(r2 + Z^2)
    return R, atan2(r, Z), atan2(Y, X)

# The equation in polar coords
H_polar(rad, th, ph) = (2 * cos(th) / rad^3, sin(th) / rad^3, 0)

# Transform equation to Cartesian
R, theta, phi = xyz_sphere(x, y, z)
H_xyz(x, y, z) = sphere_xyz(*H_polar(rad=R, th=theta, ph=phi))
print(H_xyz)

lo, hi = 3, 6
P = plot_vector_field3d(H_xyz,
  (x, lo, hi), (y, lo, hi), (z, lo, hi),
  colors='rainbow', plot_points=[6, 6, 6])

perspective = False
projection = "perspective" if perspective else "orthographic"
P.show(frame=True, projection=projection)

The script prints this for the transformed version of the H equation.

(x, y, z) |--> (2*z*sin(sqrt(x^2 + y^2)/(x^2 + y^2 + z^2)^2)/(x^2 + y^2 + z^2)^2, 0, 2*z*cos(sqrt(x^2 + y^2)/(x^2 + y^2 + z^2)^2)/(x^2 + y^2 + z^2)^2)

Here's a screenshot of the plot, using an orthographic projection.

It's a lot easier to see the structure in the interactive 3D view .

You can drag the view with the mouse & zoom with the mouse scroll wheel, the shift & ctrl keys can also be used. On touchscreens, use one finger to rotate, two fingers to pan & zoom.