How to interpolate a vector field with Python?

Question

I have a 2D vector field (in fact it's a 3D, but if we know how to do it with 2D, I think it will be easy to generalize to 3D) like this:

import matplotlib.pyplot as plt, numpy as np
x = [0, 0, 1, 1, 2, 2, 0, 1, 2]
y = [1, 2, 1, 2, 1, 2, 1.5, 1.5, 1.5]
u = [0.5, -1, 0, 0, 0.25, 1, 0, 0, 0.75]
v = [1, 1, 1, 1, 1, 1, 1, 1, 1]
plt.quiver(x, y, u, v)
plt.show()

How to interpolate smoothly this vector field?

I know how to use np.polyfit but I don't see how to do an interpolation for a vector field.

Example: I would like to interpolate [0,2]x[1,2] with hundreds of arrows.

Answer 1

Using NumPy's meshgrid and SciPy's interpolate.griddata methods, this might be a fast and feasible solution:

import matplotlib.pyplot as plt
import numpy as np
from scipy import interpolate

x = [0, 0, 1, 1, 2, 2, 0, 1, 2]
y = [1, 2, 1, 2, 1, 2, 1.5, 1.5, 1.5]
u = [0.5, -1, 0, 0, 0.25, 1, 0, 0, 0.75]
v = [1, 1, 1, 1, 1, 1, 1, 1, 1]

plt.figure(1)
plt.quiver(x, y, u, v)

xx = np.linspace(0, 2, 10)
yy = np.linspace(1, 2, 10)
xx, yy = np.meshgrid(xx, yy)

points = np.transpose(np.vstack((x, y)))
u_interp = interpolate.griddata(points, u, (xx, yy), method='cubic')
v_interp = interpolate.griddata(points, v, (xx, yy), method='cubic')

plt.figure(2)
plt.quiver(xx, yy, u_interp, v_interp)
plt.show()

Output of the interpolated plot:

Playing around with the number of points to be created within the np.linspace calls, gives you more or less arrows.

Hope that helps!