Python:Plotting Surfaces

There are many problems in engineering that require examining a 2-D domain. For example, if you want to determine the distance from a specific point on a flat surface to any other flat surface, you need to think about both the x and y coordinate. There are various other functions that need x and y coordinates.

Introductory Links

To better understand how plotting works in Python, start with reading the following pages from the Tutorials page:

• Usage Guide
• Pyplot tutorial
• Image tutorial
• The mplot3d Toolkit
• 3D plotting examples gallery

Also, there are several excellent tutorials out there! For example:

• Three-Dimensional Plotting in Matplotlib from the Python Data Science Handbook by Jake VanderPlas.

Individual Patches

One way to create a surface is to generate lists of the x, y, and z coordinates for each location of a patch. Python can make a surface from the points specified by the matrices and will then connect those points by linking the values next to each other in the matrix. For example, if x, y, and z are 2x2 matrices, the surface will generate group of four lines connecting the four points and then fill in the space among the four lines:

import numpy as np
from mpl_toolkits.mplot3d import axes3d
import matplotlib.pyplot as plt

fig = plt.figure(num=1)
fig.clf()
ax = fig.add_subplot(111, projection='3d')

x = np.array([[1, 3], [2, 4]])
y = np.array([[5, 6], [7, 8]])
z = np.array([[9, 12], [10, 11]])

ax.plot_surface(x, y, z)
ax.set(xlabel='x', ylabel='y', zlabel='z')

fig.savefig('PatchExOrig_py.png')

Note that the four "corners" above are not all co-planar; Python will thus create the patch using two triangles - to show this more clearly, you can tell Python to change the view by specifying the elevation (angle up from the xy plane) and azimuth (angle around the xy plane):

ax.view_init(elev=30, azim=45)
plt.draw()

which yields the following image:

You can add more patches to the surface by increasing the size of the matrices. For example, adding another column will add two more intersections to the surface:

fig = plt.figure(num=1)
fig.clf()
ax = fig.add_subplot(111, projection='3d')

x = np.array([[1, 3, 5], [2, 4, 6]])
y = np.array([[5, 6, 5], [7, 8, 9]])
z = np.array([[9, 12, 12], [10, 11, 12]])

ax.plot_surface(x, y, z)
ax.set(xlabel='x', ylabel='y', zlabel='z')
ax.view_init(elev=30, azim=220)

Note the rotation to better see the two different patches.

The meshgrid Command

Much of the time, rather than specifying individual patches, you will have functions of two parameters to plot. Numpy's meshgrid command is specifically used to create matrices that will represent two parameters. For example, note the output to the following Python commands:

In [1]: (x, y) = np.meshgrid(np.arange(-2, 2.1, 1), np.arange(-1, 1.1, .25))
In [2]: x
Out[2]: 
array([[-2., -1.,  0.,  1.,  2.],
       [-2., -1.,  0.,  1.,  2.],
       [-2., -1.,  0.,  1.,  2.],
       [-2., -1.,  0.,  1.,  2.],
       [-2., -1.,  0.,  1.,  2.],
       [-2., -1.,  0.,  1.,  2.],
       [-2., -1.,  0.,  1.,  2.],
       [-2., -1.,  0.,  1.,  2.],
       [-2., -1.,  0.,  1.,  2.]])
In [3]: y
Out[3]: 
array([[-1.  , -1.  , -1.  , -1.  , -1.  ],
       [-0.75, -0.75, -0.75, -0.75, -0.75],
       [-0.5 , -0.5 , -0.5 , -0.5 , -0.5 ],
       [-0.25, -0.25, -0.25, -0.25, -0.25],
       [ 0.  ,  0.  ,  0.  ,  0.  ,  0.  ],
       [ 0.25,  0.25,  0.25,  0.25,  0.25],
       [ 0.5 ,  0.5 ,  0.5 ,  0.5 ,  0.5 ],
       [ 0.75,  0.75,  0.75,  0.75,  0.75],
       [ 1.  ,  1.  ,  1.  ,  1.  ,  1.  ]])

The first argument gives the values that the first output variable should include, and the second argument gives the values that the second output variable should include. Note that the first output variable x basically gives an x coordinate and the second output variable y gives a y coordinate. This is useful if you want to plot a function in 2-D. Note that the stopping values for the arange commands are just past where we wanted to end.

Examples Using 2 Independent Variables

For example, to plot z=x+y over the ranges of x and y specified above - the code would be:

fig = plt.figure(num=1)
fig.clf()
ax = fig.add_subplot(111, projection='3d')

(x, y) = np.meshgrid(np.arange(-2, 2.1, 1), np.arange(-1, 1.1, .25))
z = x + y

ax.plot_surface(x, y, z)
ax.set(xlabel='x', ylabel='y', zlabel='z', title='z = x + y')

and the graph is:

If you want to make it more colorful, you can import colormaps and then use one; here is the complete code:

import numpy as np
from mpl_toolkits.mplot3d import axes3d
import matplotlib.pyplot as plt
from matplotlib import cm

fig = plt.figure(num=1)
fig.clf()
ax = fig.add_subplot(111, projection='3d')

(x, y) = np.meshgrid(np.arange(-2, 2.1, 1), np.arange(-1, 1.1, .25))
z = x + y

ax.plot_surface(x, y, z, cmap=cm.copper)
ax.set(xlabel='x', ylabel='y', zlabel='z', title='z = x + y')

To see all the colormaps, after importing the cm group just type

help(cm)

to see the names or go to Colormap Reference to see the colors.

To find the distance r from a particular point, say (1,-0.5), you just need to change the function. Since the distance between two points \((x, y)\) and \((x_0, y_0)\) is given by \( r=\sqrt{(x-x_0)^2+(y-y_0)^2} \) the code could be:

fig = plt.figure(num=1)
fig.clf()
ax = fig.add_subplot(111, projection='3d')

(x, y) = np.meshgrid(np.arange(-2, 2.1, 1), np.arange(-1, 1.1, .25))
z = np.sqrt((x-(1))**2 + (y-(-0.5))**2)

ax.plot_surface(x, y, z, cmap=cm.Purples)
ax.set(xlabel='x', ylabel='y', zlabel='z', 
       title='Distance from (1, -0.5)')

and the plot is

Examples Using Refined Grids

You can also use a finer grid to make a better-looking plot:

fig = plt.figure(num=1)
fig.clf()
ax = fig.add_subplot(111, projection='3d')

(x, y) = np.meshgrid(np.linspace(-1.2, 1.2, 20), 
                     np.linspace(-1.2, 1.2, 20))
z = np.sqrt((x-(1))**2 + (y-(-0.5))**2)

ax.plot_surface(x, y, z, cmap=cm.magma)
ax.set(xlabel='x', ylabel='y', zlabel='z', 
       title='Distance from (1, -0.5)')

and the plot is:

Using Other Coordinate Systems

The plotting commands such as plot_surface and plot_wireframe generate surfaces based on matrices of x, y, and z coordinates, respectively, but you can also use other coordinate systems to calculate where the points go. As an example, the function

\( z = e^{-\sqrt{x^2+y^2}}~\cos(4x)~\cos(4y) \)

could be plotted on a rectilinear grid using:

fig = plt.figure(num=1)
fig.clf()
ax = fig.add_subplot(111, projection='3d')

(x, y) = np.meshgrid(np.linspace(-1.8, 1.8, 41), 
                     np.linspace(-1.8, 1.8, 41))
z = np.exp(-np.sqrt(x**2+y**2))*np.cos(4*x)*np.cos(4*y)

ax.plot_surface(x, y, z, cmap=cm.hot)
ax.set(xlabel='x', ylabel='y', zlabel='z', 
       title='Rectilinear Grid')

giving

It could also be plotted on a circular domain using polar coordinates. To do that, r and \(\theta\) coordinates could be generated using meshgrid and the appropriate x, y, and z values could be obtained by noting that \(x=r\cos(\theta)\) and \(y=r\sin(\theta)\). z can then be calculated from any combination of x, y, r, and \(\theta\):

fig = plt.figure(num=1)
fig.clf()
ax = fig.add_subplot(111, projection='3d')

(r, theta) = np.meshgrid(np.linspace(0, 1.8, 41), 
                         np.linspace(0, 2*np.pi, 41))
x = r*np.cos(theta)
y = r*np.sin(theta)
z = np.exp(-r)*np.cos(4*x)*np.cos(4*y)

ax.plot_surface(x, y, z, cmap=cm.hot)
ax.set(xlabel='x', ylabel='y', zlabel='z', 
       title='Circular Grid')

produces:

Color Bars

You may want to add a color bar to indicate the values assigned to particular colors. This may be done using the colorbar() command but to use it you need to have access to a variable that refers to your surface plot - note how the variable chipplot is used below:

fig = plt.figure(num=1)
fig.clf()
ax = fig.add_subplot(111, projection='3d')

(r, theta) = np.meshgrid(np.linspace(0, 1.8, 41), 
                         np.linspace(0, 2*np.pi, 41))
x = r*np.cos(theta)
y = r*np.sin(theta)
z = np.exp(-r)*np.cos(4*x)*np.cos(4*y)

chipplot = ax.plot_surface(x, y, z, cmap=cm.hot)
ax.set(xlabel='x', ylabel='y', zlabel='z', 
       title='Circular Grid')
fig.colorbar(chipplot)

produces

There Is No Try

fig = plt.figure(num=1)
fig.clf()
ax = fig.add_subplot(111, projection='3d')

(theta, phi) = np.meshgrid(np.linspace(0, 2 * np.pi, 41),
                           np.linspace(0, 2 * np.pi, 41))

x = (3 + np.cos(phi)) * np.cos(theta)
y = (3 + np.cos(phi)) * np.sin(theta)
z = np.sin(phi)


def fun(t, f): return (np.cos(f + 2 * t) + 1) / 2


dplot = ax.plot_surface(x, y, z, facecolors=cm.jet(fun(theta, phi)))
ax.set(xlabel='x', 
       ylabel='y', 
       zlabel='z',
       xlim = [-4, 4],
       ylim = [-4, 4],
       zlim = [-4, 4],
       xticks = [-4, -2, 2, 4],
       yticks = [-4, -2, 2, 4],
       zticks = [-1, 0, 1],
       title='Donut!')
fig.savefig('Donut_py.png')

Questions

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