Does anyone have sample code for plotting ellipsoids? There is one for sphere on matplotlib site, but nothing for ellipsoids. I am trying to plot
x**2 + 2*y**2 + 2*z**2 = c
where c is a constant (like 10) that defines an ellipsoid. I tried the meshgrid(x,y) route, reworked the equation so z is on one side, but the sqrt is a problem. The matplotlib sphere example works with angles, u,v, but I am not sure how to work that for ellipsoid.
Here is how you can do it via spherical coordinates:
# from mpl_toolkits.mplot3d import Axes3D # Not needed with Matplotlib 3.6.3
import matplotlib.pyplot as plt
import numpy as np
fig = plt.figure(figsize=plt.figaspect(1)) # Square figure
ax = fig.add_subplot(111, projection='3d')
coefs = (1, 2, 2) # Coefficients in a0/c x**2 + a1/c y**2 + a2/c z**2 = 1
# Radii corresponding to the coefficients:
rx, ry, rz = 1/np.sqrt(coefs)
# Set of all spherical angles:
u = np.linspace(0, 2 * np.pi, 100)
v = np.linspace(0, np.pi, 100)
# Cartesian coordinates that correspond to the spherical angles:
# (this is the equation of an ellipsoid):
x = rx * np.outer(np.cos(u), np.sin(v))
y = ry * np.outer(np.sin(u), np.sin(v))
z = rz * np.outer(np.ones_like(u), np.cos(v))
# Plot:
ax.plot_surface(x, y, z, rstride=4, cstride=4, color='b')
# Adjustment of the axes, so that they all have the same span:
max_radius = max(rx, ry, rz)
for axis in 'xyz':
getattr(ax, 'set_{}lim'.format(axis))((-max_radius, max_radius))
plt.show()
The resulting plot is similar to
The program above actually produces a nicer looking "square" graphics.
This solution is strongly inspired from the example in Matplotlib's gallery.
Building on EOL's answer. Sometimes you have an ellipsoid in matrix format:
A and c Where A is the ellipsoid matrix and c is a vector representing the centre of the ellipsoid.
import numpy as np
import numpy.linalg as linalg
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
# your ellispsoid and center in matrix form
A = np.array([[1,0,0],[0,2,0],[0,0,2]])
center = [0,0,0]
# find the rotation matrix and radii of the axes
U, s, rotation = linalg.svd(A)
radii = 1.0/np.sqrt(s)
# now carry on with EOL's answer
u = np.linspace(0.0, 2.0 * np.pi, 100)
v = np.linspace(0.0, np.pi, 100)
x = radii[0] * np.outer(np.cos(u), np.sin(v))
y = radii[1] * np.outer(np.sin(u), np.sin(v))
z = radii[2] * np.outer(np.ones_like(u), np.cos(v))
for i in range(len(x)):
for j in range(len(x)):
[x[i,j],y[i,j],z[i,j]] = np.dot([x[i,j],y[i,j],z[i,j]], rotation) + center
# plot
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_wireframe(x, y, z, rstride=4, cstride=4, color='b', alpha=0.2)
plt.show()
plt.close(fig)
del fig
So, not too much new here, but helpful if you've got an ellipsoid in matrix form which is rotated and perhaps not centered at 0,0,0 and want to plot it.
If you have an ellipsoid specified by an arbitrary covariance matrix cov and offset bias, you perform a simpler version of @minillinim's answer by vectorizing the operations.
Starting with
u = np.linspace(0, 2 * np.pi, 100)
v = np.linspace(0, np.pi, 100)
Make a unit sphere
x = np.outer(np.cos(u), np.sin(v))
y = np.outer(np.sin(u), np.sin(v))
z = np.outer(np.ones_like(u), np.cos(v))
sphere = np.stack((x, y, z), axis=-1)[..., None]
Compute the standard deviation matrix, which has the same rotation as the covariance, but scales by the square root of the eigenvalues:
e, v = np.linalg.eig(cov)
s = v @ np.diag(np.sqrt(e)) @ v.T
Now transform the sphere:
ellipsoid = (s @ sphere).squeeze(-1) + bias
You can plot the result pretty much as before:
ax.plot_surface(*ellipsoid.transpose(2, 0, 1), rstride=4, cstride=4, color='b', alpha=0.75)
For reference, u, v have shape (100,), which makes x, y, z into (100, 100) arrays. sphere is (100, 100, 3, 1), which makes it a 100x100 array of 3x1 vectors as far as the broadcasting @ operator is concerned. s @ sphere has the same size, so squeezing out the last unit axis makes it suitable for broadcasting for addition with bias. Finally, ellipsoid.transpose(2, 0, 1) has shape (3, 100, 100), which can be star-expanded as three separate arrays of x-, y- and z-values into the call to plot_surface.
