# Three-Dimensional Plotting in Matplotlib

Matplotlib was initially designed with only two-dimensional plotting in mind. Around the time of the 1.0 release, some three-dimensional plotting utilities were built on top of Matplotlib’s two-dimensional display, and the result is a convenient (if somewhat limited) set of tools for three-dimensional data visualization. three-dimensional plots are enabled by importing the `mplot3d`

toolkit, included with the main Matplotlib installation:In [1]:

frommpl_toolkitsimportmplot3d

Once this submodule is imported, a three-dimensional axes can be created by passing the keyword `projection='3d'`

to any of the normal axes creation routines:In [2]:

%matplotlibinlineimportnumpyasnpimportmatplotlib.pyplotasplt

In [3]:

fig = plt.figure() ax = plt.axes(projection='3d')

With this three-dimensional axes enabled, we can now plot a variety of three-dimensional plot types. Three-dimensional plotting is one of the functionalities that benefits immensely from viewing figures interactively rather than statically in the notebook; recall that to use interactive figures, you can use `%matplotlib notebook`

rather than `%matplotlib inline`

when running this code.

## Three-dimensional Points and Lines

The most basic three-dimensional plot is a line or collection of scatter plot created from sets of (x, y, z) triples. In analogy with the more common two-dimensional plots discussed earlier, these can be created using the `ax.plot3D`

and `ax.scatter3D`

functions. The call signature for these is nearly identical to that of their two-dimensional counterparts, so you can refer to Simple Line Plots and Simple Scatter Plots for more information on controlling the output. Here we’ll plot a trigonometric spiral, along with some points drawn randomly near the line:In [4]:

ax = plt.axes(projection='3d')# Data for a three-dimensional linezline = np.linspace(0, 15, 1000) xline = np.sin(zline) yline = np.cos(zline) ax.plot3D(xline, yline, zline, 'gray')# Data for three-dimensional scattered pointszdata = 15 * np.random.random(100) xdata = np.sin(zdata) + 0.1 * np.random.randn(100) ydata = np.cos(zdata) + 0.1 * np.random.randn(100) ax.scatter3D(xdata, ydata, zdata, c=zdata, cmap='Greens');

Notice that by default, the scatter points have their transparency adjusted to give a sense of depth on the page. While the three-dimensional effect is sometimes difficult to see within a static image, an interactive view can lead to some nice intuition about the layout of the points.

## Three-dimensional Contour Plots

Analogous to the contour plots we explored in Density and Contour Plots, `mplot3d`

contains tools to create three-dimensional relief plots using the same inputs. Like two-dimensional `ax.contour`

plots, `ax.contour3D`

requires all the input data to be in the form of two-dimensional regular grids, with the Z data evaluated at each point. Here we’ll show a three-dimensional contour diagram of a three-dimensional sinusoidal function:In [5]:

deff(x, y):returnnp.sin(np.sqrt(x ** 2 + y ** 2)) x = np.linspace(-6, 6, 30) y = np.linspace(-6, 6, 30) X, Y = np.meshgrid(x, y) Z = f(X, Y)

In [6]:

fig = plt.figure() ax = plt.axes(projection='3d') ax.contour3D(X, Y, Z, 50, cmap='binary') ax.set_xlabel('x') ax.set_ylabel('y') ax.set_zlabel('z');

Sometimes the default viewing angle is not optimal, in which case we can use the `view_init`

method to set the elevation and azimuthal angles. In the following example, we’ll use an elevation of 60 degrees (that is, 60 degrees above the x-y plane) and an azimuth of 35 degrees (that is, rotated 35 degrees counter-clockwise about the z-axis):In [7]:

ax.view_init(60, 35) fig

Out[7]:

Again, note that this type of rotation can be accomplished interactively by clicking and dragging when using one of Matplotlib’s interactive backends.

## Wireframes and Surface Plots

Two other types of three-dimensional plots that work on gridded data are wireframes and surface plots. These take a grid of values and project it onto the specified three-dimensional surface, and can make the resulting three-dimensional forms quite easy to visualize. Here’s an example of using a wireframe:In [8]:

fig = plt.figure() ax = plt.axes(projection='3d') ax.plot_wireframe(X, Y, Z, color='black') ax.set_title('wireframe');

A surface plot is like a wireframe plot, but each face of the wireframe is a filled polygon. Adding a colormap to the filled polygons can aid perception of the topology of the surface being visualized:In [9]:

ax = plt.axes(projection='3d') ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap='viridis', edgecolor='none') ax.set_title('surface');

Note that though the grid of values for a surface plot needs to be two-dimensional, it need not be rectilinear. Here is an example of creating a partial polar grid, which when used with the `surface3D`

plot can give us a slice into the function we’re visualizing:In [10]:

r = np.linspace(0, 6, 20) theta = np.linspace(-0.9 * np.pi, 0.8 * np.pi, 40) r, theta = np.meshgrid(r, theta) X = r * np.sin(theta) Y = r * np.cos(theta) Z = f(X, Y) ax = plt.axes(projection='3d') ax.plot_surface(X, Y, Z, rstride=1, cstride=1, cmap='viridis', edgecolor='none');

## Surface Triangulations

For some applications, the evenly sampled grids required by the above routines is overly restrictive and inconvenient. In these situations, the triangulation-based plots can be very useful. What if rather than an even draw from a Cartesian or a polar grid, we instead have a set of random draws?In [11]:

theta = 2 * np.pi * np.random.random(1000) r = 6 * np.random.random(1000) x = np.ravel(r * np.sin(theta)) y = np.ravel(r * np.cos(theta)) z = f(x, y)

We could create a scatter plot of the points to get an idea of the surface we’re sampling from:In [12]:

ax = plt.axes(projection='3d') ax.scatter(x, y, z, c=z, cmap='viridis', linewidth=0.5);

This leaves a lot to be desired. The function that will help us in this case is `ax.plot_trisurf`

, which creates a surface by first finding a set of triangles formed between adjacent points (remember that x, y, and z here are one-dimensional arrays):In [13]:

ax = plt.axes(projection='3d') ax.plot_trisurf(x, y, z, cmap='viridis', edgecolor='none');

The result is certainly not as clean as when it is plotted with a grid, but the flexibility of such a triangulation allows for some really interesting three-dimensional plots. For example, it is actually possible to plot a three-dimensional Möbius strip using this, as we’ll see next.

### Example: Visualizing a Möbius strip

A Möbius strip is similar to a strip of paper glued into a loop with a half-twist. Topologically, it’s quite interesting because despite appearances it has only a single side! Here we will visualize such an object using Matplotlib’s three-dimensional tools. The key to creating the Möbius strip is to think about it’s parametrization: it’s a two-dimensional strip, so we need two intrinsic dimensions. Let’s call them $\theta$, which ranges from $0$ to $2\pi$ around the loop, and $w$ which ranges from -1 to 1 across the width of the strip:In [14]:

theta = np.linspace(0, 2 * np.pi, 30) w = np.linspace(-0.25, 0.25, 8) w, theta = np.meshgrid(w, theta)

Now from this parametrization, we must determine the *(x, y, z)* positions of the embedded strip.

Thinking about it, we might realize that there are two rotations happening: one is the position of the loop about its center (what we’ve called $\theta$), while the other is the twisting of the strip about its axis (we’ll call this $\phi$). For a Möbius strip, we must have the strip makes half a twist during a full loop, or $\Delta\phi = \Delta\theta/2$.In [15]:

phi = 0.5 * theta

Now we use our recollection of trigonometry to derive the three-dimensional embedding. We’ll define $r$, the distance of each point from the center, and use this to find the embedded $(x, y, z)$ coordinates:In [16]:

# radius in x-y planer = 1 + w * np.cos(phi) x = np.ravel(r * np.cos(theta)) y = np.ravel(r * np.sin(theta)) z = np.ravel(w * np.sin(phi))

Finally, to plot the object, we must make sure the triangulation is correct. The best way to do this is to define the triangulation *within the underlying parametrization*, and then let Matplotlib project this triangulation into the three-dimensional space of the Möbius strip. This can be accomplished as follows:In [17]:

# triangulate in the underlying parametrizationfrommatplotlib.triimportTriangulation tri = Triangulation(np.ravel(w), np.ravel(theta)) ax = plt.axes(projection='3d') ax.plot_trisurf(x, y, z, triangles=tri.triangles, cmap='viridis', linewidths=0.2); ax.set_xlim(-1, 1); ax.set_ylim(-1, 1); ax.set_zlim(-1, 1);

Combining all of these techniques, it is possible to create and display a wide variety of three-dimensional objects and patterns in Matplotlib.