how to draw a plane in 3d matplotlib
Matplotlib was initially designed with just two-dimensional plotting in listen. Effectually the fourth dimension of the 1.0 release, some three-dimensional plotting utilities were built on top of Matplotlib's two-dimensional display, and the outcome is a convenient (if somewhat express) ready of tools for three-dimensional data visualization. three-dimensional plots are enabled by importing the mplot3d toolkit, included with the primary Matplotlib installation:
In [1]:
from mpl_toolkits import mplot3d
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]:
% matplotlib inline import numpy equally np import matplotlib.pyplot as plt
In [3]:
fig = plt . figure () ax = plt . axes ( projection = '3d' )
With this 3-dimensional axes enabled, we can now plot a variety of 3-dimensional plot types. Three-dimensional plotting is ane of the functionalities that benefits immensely from viewing figures interactively rather than statically in the notebook; call back that to use interactive figures, y'all can use %matplotlib notebook rather than %matplotlib inline when running this code.
3-dimensional Points and Lines¶
The most basic three-dimensional plot is a line or collection of scatter plot created from sets of (ten, y, z) triples. In illustration with the more common 2-dimensional plots discussed earlier, these tin 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 tin refer to Simple Line Plots and Simple Scatter Plots for more information on controlling the output. Hither we'll plot a trigonometric spiral, forth with some points drawn randomly near the line:
In [iv]:
ax = plt . axes ( projection = '3d' ) # Data for a 3-dimensional line zline = np . linspace ( 0 , 15 , g ) xline = np . sin ( zline ) yline = np . cos ( zline ) ax . plot3D ( xline , yline , zline , 'gray' ) # Data for three-dimensional scattered points zdata = fifteen * np . random . random ( 100 ) xdata = np . sin ( zdata ) + 0.one * 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 besprinkle points have their transparency adjusted to give a sense of depth on the page. While the three-dimensional effect is sometimes difficult to come across within a static image, an interactive view tin can pb to some dainty 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 3-dimensional relief plots using the same inputs. Like two-dimensional ax.contour plots, ax.contour3D requires all the input data to exist in the form of 2-dimensional regular grids, with the Z data evaluated at each indicate. Here we'll prove a 3-dimensional contour diagram of a three-dimensional sinusoidal function:
In [five]:
def f ( x , y ): render np . sin ( np . sqrt ( 10 ** 2 + y ** 2 )) x = np . linspace ( - 6 , half-dozen , 30 ) y = np . linspace ( - half-dozen , half-dozen , 30 ) Ten , Y = np . meshgrid ( 10 , y ) Z = f ( Ten , Y )
In [half-dozen]:
fig = plt . effigy () ax = plt . axes ( projection = '3d' ) ax . contour3D ( Ten , Y , Z , 50 , cmap = 'binary' ) ax . set_xlabel ( 'x' ) ax . set_ylabel ( 'y' ) ax . set_zlabel ( 'z' );
Sometimes the default viewing bending is not optimal, in which case we tin use the view_init method to set the elevation and azimuthal angles. In the following instance, we'll employ an peak of 60 degrees (that is, threescore degrees above the x-y plane) and an azimuth of 35 degrees (that is, rotated 35 degrees counter-clockwise nearly the z-axis):
Out[vii]:
Again, note that this type of rotation can exist accomplished interactively by clicking and dragging when using one of Matplotlib'southward interactive backends.
Wireframes and Surface Plots¶
Ii other types of three-dimensional plots that work on gridded data are wireframes and surface plots. These take a grid of values and project information technology onto the specified three-dimensional surface, and can make the resulting three-dimensional forms quite easy to visualize. Hither's an example of using a wireframe:
In [8]:
fig = plt . figure () ax = plt . axes ( project = '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. Calculation a colormap to the filled polygons can help perception of the topology of the surface existence visualized:
In [9]:
ax = plt . axes ( projection = '3d' ) ax . plot_surface ( X , Y , Z , rstride = ane , 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 exist rectilinear. Here is an case of creating a fractional polar grid, which when used with the surface3D plot tin requite us a slice into the function nosotros're visualizing:
In [10]:
r = np . linspace ( 0 , six , 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 exist very useful. What if rather than an fifty-fifty draw from a Cartesian or a polar grid, we instead have a set of random draws?
In [11]:
theta = ii * np . pi * np . random . random ( 1000 ) r = vi * np . random . random ( m ) 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 become an thought of the surface nosotros're sampling from:
In [12]:
ax = plt . axes ( project = '3d' ) ax . scatter ( x , y , z , c = z , cmap = 'viridis' , linewidth = 0.five );
This leaves a lot to exist desired. The function that will help us in this example is ax.plot_trisurf, which creates a surface past kickoff finding a prepare of triangles formed between adjacent points (recollect that x, y, and z here are one-dimensional arrays):
In [thirteen]:
ax = plt . axes ( projection = '3d' ) ax . plot_trisurf ( x , y , z , cmap = 'viridis' , edgecolor = 'none' );
The outcome is certainly not equally clean equally when it is plotted with a grid, just the flexibility of such a triangulation allows for some really interesting three-dimensional plots. For example, information technology is really possible to plot a three-dimensional Möbius strip using this, as nosotros'll see adjacent.
Example: Visualizing a Möbius strip¶
A Möbius strip is similar to a strip of newspaper glued into a loop with a half-twist. Topologically, it's quite interesting considering 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 information technology's parametrization: information technology'due south a two-dimensional strip, so we demand ii intrinsic dimensions. Let'south call them $\theta$, which ranges from $0$ to $2\pi$ effectually the loop, and $w$ which ranges from -1 to ane beyond the width of the strip:
In [xiv]:
theta = np . linspace ( 0 , 2 * np . pi , thirty ) westward = 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 nearly it, we might realize that in that location are ii rotations happening: one is the position of the loop about its center (what nosotros've called $\theta$), while the other is the twisting of the strip about its axis (we'll call this $\phi$). For a Möbius strip, nosotros must have the strip makes half a twist during a full loop, or $\Delta\phi = \Delta\theta/2$.
Now we utilise our recollection of trigonometry to derive the three-dimensional embedding. Nosotros'll ascertain $r$, the altitude of each point from the center, and utilise this to find the embedded $(ten, y, z)$ coordinates:
In [16]:
# radius in x-y airplane r = i + west * np . cos ( phi ) 10 = 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 certain the triangulation is right. The all-time way to practise 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 tin can exist accomplished as follows:
In [17]:
# triangulate in the underlying parametrization from matplotlib.tri import Triangulation 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 ( - ane , 1 ); ax . set_ylim ( - ane , 1 ); ax . set_zlim ( - 1 , ane );
Combining all of these techniques, information technology is possible to create and display a wide variety of three-dimensional objects and patterns in Matplotlib.
Source: https://jakevdp.github.io/PythonDataScienceHandbook/04.12-three-dimensional-plotting.html
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