Matplotlib streamplot with streamlines that don't break or end - numpy

Question

I'd like to make a streamplot with lines that don't stop when they get too close together. I'd rather each streamline be calculated in both directions until it hits the edge of the window. The result is there'd be some areas where they'd all jumble up. But that's what I want.
I there anyway to do this in matplotlib? If not, is there another tool I can use for this that could interface with python/numpy?
import numpy as np
import matplotlib.pyplot as plt
Y,X = np.mgrid[-10:10:.01, -10:10:.01]
U, V = Y**2, X**2
plt.streamplot(X,Y, U,V, density=1)
plt.show(False)

Answer 1

Ok, I've figured out I can get mostly what I want by turning up the density a lot and using custom start points. I'm still interested if there is a better or alternate way to do this.
Here's my solution. Doesn't it look so much better?
import numpy as np
import matplotlib.pyplot as plt
Y,X = np.mgrid[-10:10:.01, -10:10:.01]
y,x = Y[:,0], X[0,:]
U, V = Y**2, X**2
stream_points = np.array(zip(np.arange(-9,9,.5), -np.arange(-9,9,.5)))
plt.streamplot(x,y, U,V, start_points=stream_points, density=35)
plt.show(False)
Edit: By the way, there seems to be some bug in streamplot such that start_points keyword only works if you use 1d arrays for the grid data. See Python Matplotlib Streamplot providing start points

Answer 2

As of Matplotlib version 3.6.0, an optional parameter broken_streamlines has been added for disabling streamline breaks.
Adding it to your snippet produces the following result:
import numpy as np
import matplotlib.pyplot as plt
Y,X = np.mgrid[-10:10:.01, -10:10:.01]
U, V = Y**2, X**2
plt.streamplot(X,Y, U,V, density=1, broken_streamlines=False)
plt.show(False)
Note
This parameter just extends the streamlines which were originally drawn (as in the question). This means that the streamlines in the modified plot above are much more uneven than the result obtained in the other answer, with custom start_points. The density of streamlines on any stream plot does not represent the magnitude of U or V at that point, only their direction. See the documentation for the density parameter of matplotlib.pyplot.streamplot for more details on how streamline start points are chosen by default, when they aren't specified by the optional start_points parameter.
For accurate streamline density, consider using matplotlib.pyplot.contour, but be aware that contour does not show arrows.
Choosing start points automatically
It may not always be easy to choose a set of good starting points automatically. However, if you know the streamfunction corresponding to the flow you wish to plot you can use matplotlib.pyplot.contour to produce a contour plot (which can be hidden from the output), and then extract a suitable starting point from each of the plotted contours.
In the following example, psi_expression is the streamfunction corresponding to the flow. When modifying this example for your own needs, make sure to update both the line defining psi_expression, as well as the one defining U and V. Ensure these both correspond to the same flow.
The density of the streamlines can be altered by changing contour_levels. Here, the contours are uniformly distributed.
import numpy as np
import matplotlib.pyplot as plt
import sympy as sy
x, y = sy.symbols("x y")
psi_expression = x**3 - y**3
psi_function = sy.lambdify((x, y), psi_expression)
Y, X = np.mgrid[-10:10:0.01, -10:10:0.01]
psi_evaluated = psi_function(X, Y)
U, V = Y**2, X**2
contour_levels = np.linspace(np.amin(psi_evaluated), np.amax(psi_evaluated), 30)
# Draw a temporary contour plot.
temp_figure = plt.figure()
contour_plot = plt.contour(X, Y, psi_evaluated, contour_levels)
plt.close(temp_figure)
points_list = []
# Iterate over each contour.
for collection in contour_plot.collections:
# Iterate over each segment in this contour.
for path in collection.get_paths():
middle_point = path.vertices[len(path.vertices) // 2]
points_list.append(middle_point)
# Reshape python list into numpy array of coords.
stream_points = np.reshape(np.array(points_list), (-1, 2))
plt.streamplot(X, Y, U, V, density=1, start_points=stream_points, broken_streamlines=False)
plt.show(False)