**Kernel:**Python 3

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# 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:

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:

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:

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:

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):

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:

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:

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:

## 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?

We could create a scatter plot of the points to get an idea of the surface we're sampling from:

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):

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 MÃ¶bius strip using this, as we'll see next.

### Example: Visualizing a MÃ¶bius strip

A MÃ¶bius 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 MÃ¶bius 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:

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 MÃ¶bius strip, we must have the strip makes half a twist during a full loop, or $\Delta\phi = \Delta\theta/2$.

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:

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 MÃ¶bius strip. This can be accomplished as follows:

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