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Visualization tools¶

In this example we will look at some external tools that can be used with clifford to help visualize geometric objects.

The two tools available are:

pyganja (github)
mpl_toolkits.clifford (github)

Both of these can be installed with pip install followed by the package name above.

G2C¶

Let’s start by creating some objects in 2d Conformal Geometric Algebra to visualize:

[1]:

from clifford.g2c import *

[2]:

point = up(2*e1+e2)
line = up(3*e1 + 2*e2) ^ up(3*e1 - 2*e2) ^ einf
circle = up(e1) ^ up(-e1 + 2*e2) ^ up(-e1 - 2*e2)

We’ll create copies of the point and line reflected in the circle, using \(X = C\hat X\tilde C\), where \(\hat X\) is the grade involution.

[3]:

point_refl = circle * point.gradeInvol() * ~circle
line_refl = circle * line.gradeInvol() * ~circle

`pyganja`¶

pyganja is a python interface to the ganja.js (github) library. To use it, typically we need to import two names from the library:

[4]:

from pyganja import GanjaScene, draw
import pyganja; pyganja.__version__

/home/docs/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/pyganja/__init__.py:2: UserWarning: Failed to import cef_gui, cef functions will be unavailable
  from .script_api import *

[4]:

'0.0.15'

GanjaScene lets us build scenes out of geometric objects, with attached labels and RGB colors:

[5]:

sc = GanjaScene()
sc.add_object(point, color=(255, 0, 0), label='point')
sc.add_object(line, color=(0, 255, 0), label='line')
sc.add_object(circle, color=(0, 0, 255), label='circle')

[6]:

sc_refl = GanjaScene()
sc_refl.add_object(point_refl, color=(128, 0, 0), label='point_refl')
sc_refl.add_object(line_refl, color=(0, 128, 0), label='line_refl')

Once we’ve built our scene, we can draw it, specifying a scale (which here we use to zoom out), and the signature of our algebra (which defaults to conformal 3D):

[7]:

draw(sc, sig=layout.sig, scale=0.5)

A cool feature of GanjaScene is the ability to use + to draw both scenes together:

[8]:

draw(sc + sc_refl, sig=layout.sig, scale=0.5)

`mpl_toolkits.clifford`¶

While ganja.js produces great diagrams, it’s hard to combine them with other plotting tools. mpl_toolkits.clifford works within matplotlib.

[9]:

from matplotlib import pyplot as plt
plt.ioff()  # we'll ask for plotting when we want it

# if you're editing this locally, you'll get an interactive UI if you uncomment the following
#
#    %matplotlib notebook

from mpl_toolkits.clifford import plot
import mpl_toolkits.clifford; mpl_toolkits.clifford.__version__

[9]:

'0.0.3'

Assembling the plot is a lot more work, but we also get much more control:

[10]:

# standard matplotlib stuff - construct empty plots side-by-side, and set the scaling
fig, (ax_before, ax_both) = plt.subplots(1, 2, sharex=True, sharey=True)
ax_before.set(xlim=[-4, 4], ylim=[-4, 4], aspect='equal')
ax_both.set(xlim=[-4, 4], ylim=[-4, 4], aspect='equal')

# plot the objects before reflection on both plots
for ax in (ax_before, ax_both):
    plot(ax, [point], color='tab:blue', label='point', marker='x', linestyle=' ')
    plot(ax, [line], color='tab:green', label='line')
    plot(ax, [circle], color='tab:red', label='circle')

# plot the objects after reflection, with thicker lines
plot(ax_both, [point_refl], color='tab:blue', label='point_refl',  marker='x', linestyle=' ', markeredgewidth=2)
plot(ax_both, [line_refl], color='tab:green', label='line_refl', linewidth=2)

fig.tight_layout()
ax_both.legend()

# show the figure
fig

/tmp/ipykernel_2174/3992657548.py:17: UserWarning: Legend does not support handles for PatchCollection instances.
See: https://matplotlib.org/stable/tutorials/intermediate/legend_guide.html#implementing-a-custom-legend-handler
  ax_both.legend()

[10]:

../../_images/tutorials_cga_visualization-tools_19_1.svg

../../_images/tutorials_cga_visualization-tools_19_2.svg

G3C¶

Let’s repeat the above, but with 3D Conformal Geometric Algebra. Note that if you’re viewing these docs in a jupyter notebook, the lines below will replace all your 2d variables with 3d ones

[11]:

from clifford.g3c import *

[12]:

point = up(2*e1+e2)
line = up(3*e1 + 2*e2) ^ up(3*e1 - 2*e2) ^ einf
circle = up(e1) ^ up(-e1 + 1.6*e2 + 1.2*e3) ^ up(-e1 - 1.6*e2 - 1.2*e3)
sphere = up(3*e1) ^ up(e1) ^ up(2*e1 + e2) ^ up(2*e1 + e3)

[13]:

# note that due to floating point rounding, we need to truncate back to a single grade here, with ``(grade)``
point_refl = homo((circle * point.gradeInvol() * ~circle)(1))
line_refl = (circle * line.gradeInvol() * ~circle)(3)
sphere_refl = (circle * sphere.gradeInvol() * ~circle)(4)

`pyganja`¶

Once again, we can create a pair of scenes exactly as before

[14]:

sc = GanjaScene()
sc.add_object(point, color=(255, 0, 0), label='point')
sc.add_object(line, color=(0, 255, 0), label='line')
sc.add_object(circle, color=(0, 0, 255), label='circle')
sc.add_object(sphere, color=(0, 255, 255), label='sphere')

[15]:

sc_refl = GanjaScene()
sc_refl.add_object(point_refl, color=(128, 0, 0), label='point_refl')
sc_refl.add_object(line_refl.normal(), color=(0, 128, 0), label='line_refl')
sc_refl.add_object(sphere_refl.normal(), color=(0, 128, 128), label='sphere_refl')

But this time, when we draw them we don’t need to pass sig. Better yet, we can rotate the 3D world around using left click, pan with right click, and zoom with the scroll wheel.

[16]:

draw(sc + sc_refl, scale=0.5)

Some more example of using pyganja to visualize 3D CGA can be found in the interpolation and clustering notebooks.

`mpl_toolkits.clifford`¶

The 3D approach for matplotlib is much the same. Note that due to poor handling of rounding errors in clifford.tools.classify, a call to .normal() is needed. Along with explicit grade selection, this is a useful trick to try and get something to render which otherwise would not.

[17]:

# standard matplotlib stuff - construct empty plots side-by-side, and set the scaling
fig, (ax_before, ax_both) = plt.subplots(1, 2, subplot_kw=dict(projection='3d'), figsize=(8, 4))
ax_before.set(xlim=[-4, 4], ylim=[-4, 4], zlim=[-4, 4])
ax_both.set(xlim=[-4, 4], ylim=[-4, 4], zlim=[-4, 4])

# plot the objects before reflection on both plots
for ax in (ax_before, ax_both):
    plot(ax, [point], color='tab:red', label='point', marker='x', linestyle=' ')
    plot(ax, [line], color='tab:green', label='line')
    plot(ax, [circle], color='tab:blue', label='circle')
    plot(ax, [sphere], color='tab:cyan')  # labels do not work for spheres: pygae/mpl_toolkits.clifford#5

# plot the objects after reflection
plot(ax_both, [point_refl], color='tab:red', label='point_refl', marker='x', linestyle=' ', markeredgewidth=2)
plot(ax_both, [line_refl.normal()], color='tab:green', label='line_refl', linewidth=2)
plot(ax_both, [sphere_refl], color='tab:cyan')

fig.tight_layout()
ax_both.legend()

# show the figure
fig

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/IPython/core/formatters.py:340, in BaseFormatter.__call__(self, obj)
    338     pass
    339 else:
--> 340     return printer(obj)
    341 # Finally look for special method names
    342 method = get_real_method(obj, self.print_method)

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
    149     from matplotlib.backend_bases import FigureCanvasBase
    150     FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
    153 data = bytes_io.getvalue()
    154 if fmt == 'svg':

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/backend_bases.py:2175, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
   2172     # we do this instead of `self.figure.draw_without_rendering`
   2173     # so that we can inject the orientation
   2174     with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2175         self.figure.draw(renderer)
   2176 if bbox_inches:
   2177     if bbox_inches == "tight":

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
     93 @wraps(draw)
     94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95     result = draw(artist, renderer, *args, **kwargs)
     96     if renderer._rasterizing:
     97         renderer.stop_rasterizing()

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
   3159             # ValueError can occur when resizing a window.
   3161     self.patch.draw(renderer)
-> 3162     mimage._draw_list_compositing_images(
   3163         renderer, self, artists, self.suppressComposite)
   3165     renderer.close_group('figure')
   3166 finally:

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
    130 if not_composite or not has_images:
    131     for a in artists:
--> 132         a.draw(renderer)
    133 else:
    134     # Composite any adjacent images together
    135     image_group = []

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/mpl_toolkits/mplot3d/axes3d.py:441, in Axes3D.draw(self, renderer)
    437 zorder_offset = max(axis.get_zorder()
    438                     for axis in self._axis_map.values()) + 1
    439 collection_zorder = patch_zorder = zorder_offset
--> 441 for artist in sorted(collections_and_patches,
    442                      key=lambda artist: artist.do_3d_projection(),
    443                      reverse=True):
    444     if isinstance(artist, mcoll.Collection):
    445         artist.zorder = collection_zorder

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/mpl_toolkits/mplot3d/axes3d.py:442, in Axes3D.draw.<locals>.<lambda>(artist)
    437 zorder_offset = max(axis.get_zorder()
    438                     for axis in self._axis_map.values()) + 1
    439 collection_zorder = patch_zorder = zorder_offset
    441 for artist in sorted(collections_and_patches,
--> 442                      key=lambda artist: artist.do_3d_projection(),
    443                      reverse=True):
    444     if isinstance(artist, mcoll.Collection):
    445         artist.zorder = collection_zorder

TypeError: do_3d_projection() missing 1 required positional argument: 'renderer'

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/IPython/core/formatters.py:340, in BaseFormatter.__call__(self, obj)
    338     pass
    339 else:
--> 340     return printer(obj)
    341 # Finally look for special method names
    342 method = get_real_method(obj, self.print_method)

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
    149     from matplotlib.backend_bases import FigureCanvasBase
    150     FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
    153 data = bytes_io.getvalue()
    154 if fmt == 'svg':

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/backend_bases.py:2175, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
   2172     # we do this instead of `self.figure.draw_without_rendering`
   2173     # so that we can inject the orientation
   2174     with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2175         self.figure.draw(renderer)
   2176 if bbox_inches:
   2177     if bbox_inches == "tight":

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
     93 @wraps(draw)
     94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95     result = draw(artist, renderer, *args, **kwargs)
     96     if renderer._rasterizing:
     97         renderer.stop_rasterizing()

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
   3159             # ValueError can occur when resizing a window.
   3161     self.patch.draw(renderer)
-> 3162     mimage._draw_list_compositing_images(
   3163         renderer, self, artists, self.suppressComposite)
   3165     renderer.close_group('figure')
   3166 finally:

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
    130 if not_composite or not has_images:
    131     for a in artists:
--> 132         a.draw(renderer)
    133 else:
    134     # Composite any adjacent images together
    135     image_group = []

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/mpl_toolkits/mplot3d/axes3d.py:441, in Axes3D.draw(self, renderer)
    437 zorder_offset = max(axis.get_zorder()
    438                     for axis in self._axis_map.values()) + 1
    439 collection_zorder = patch_zorder = zorder_offset
--> 441 for artist in sorted(collections_and_patches,
    442                      key=lambda artist: artist.do_3d_projection(),
    443                      reverse=True):
    444     if isinstance(artist, mcoll.Collection):
    445         artist.zorder = collection_zorder

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/mpl_toolkits/mplot3d/axes3d.py:442, in Axes3D.draw.<locals>.<lambda>(artist)
    437 zorder_offset = max(axis.get_zorder()
    438                     for axis in self._axis_map.values()) + 1
    439 collection_zorder = patch_zorder = zorder_offset
    441 for artist in sorted(collections_and_patches,
--> 442                      key=lambda artist: artist.do_3d_projection(),
    443                      reverse=True):
    444     if isinstance(artist, mcoll.Collection):
    445         artist.zorder = collection_zorder

TypeError: do_3d_projection() missing 1 required positional argument: 'renderer'

[17]:

<Figure size 768x384 with 2 Axes>

Error in callback <function _draw_all_if_interactive at 0x79a251e50b80> (for post_execute), with arguments args (),kwargs {}:

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/pyplot.py:268, in _draw_all_if_interactive()
    266 def _draw_all_if_interactive() -> None:
    267     if matplotlib.is_interactive():
--> 268         draw_all()

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/_pylab_helpers.py:131, in Gcf.draw_all(cls, force)
    129 for manager in cls.get_all_fig_managers():
    130     if force or manager.canvas.figure.stale:
--> 131         manager.canvas.draw_idle()

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/backend_bases.py:1905, in FigureCanvasBase.draw_idle(self, *args, **kwargs)
   1903 if not self._is_idle_drawing:
   1904     with self._idle_draw_cntx():
-> 1905         self.draw(*args, **kwargs)

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/backends/backend_agg.py:387, in FigureCanvasAgg.draw(self)
    384 # Acquire a lock on the shared font cache.
    385 with (self.toolbar._wait_cursor_for_draw_cm() if self.toolbar
    386       else nullcontext()):
--> 387     self.figure.draw(self.renderer)
    388     # A GUI class may be need to update a window using this draw, so
    389     # don't forget to call the superclass.
    390     super().draw()

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
     93 @wraps(draw)
     94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95     result = draw(artist, renderer, *args, **kwargs)
     96     if renderer._rasterizing:
     97         renderer.stop_rasterizing()

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
   3159             # ValueError can occur when resizing a window.
   3161     self.patch.draw(renderer)
-> 3162     mimage._draw_list_compositing_images(
   3163         renderer, self, artists, self.suppressComposite)
   3165     renderer.close_group('figure')
   3166 finally:

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
    130 if not_composite or not has_images:
    131     for a in artists:
--> 132         a.draw(renderer)
    133 else:
    134     # Composite any adjacent images together
    135     image_group = []

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/mpl_toolkits/mplot3d/axes3d.py:441, in Axes3D.draw(self, renderer)
    437 zorder_offset = max(axis.get_zorder()
    438                     for axis in self._axis_map.values()) + 1
    439 collection_zorder = patch_zorder = zorder_offset
--> 441 for artist in sorted(collections_and_patches,
    442                      key=lambda artist: artist.do_3d_projection(),
    443                      reverse=True):
    444     if isinstance(artist, mcoll.Collection):
    445         artist.zorder = collection_zorder

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/mpl_toolkits/mplot3d/axes3d.py:442, in Axes3D.draw.<locals>.<lambda>(artist)
    437 zorder_offset = max(axis.get_zorder()
    438                     for axis in self._axis_map.values()) + 1
    439 collection_zorder = patch_zorder = zorder_offset
    441 for artist in sorted(collections_and_patches,
--> 442                      key=lambda artist: artist.do_3d_projection(),
    443                      reverse=True):
    444     if isinstance(artist, mcoll.Collection):
    445         artist.zorder = collection_zorder

TypeError: do_3d_projection() missing 1 required positional argument: 'renderer'

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/IPython/core/formatters.py:340, in BaseFormatter.__call__(self, obj)
    338     pass
    339 else:
--> 340     return printer(obj)
    341 # Finally look for special method names
    342 method = get_real_method(obj, self.print_method)

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
    149     from matplotlib.backend_bases import FigureCanvasBase
    150     FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
    153 data = bytes_io.getvalue()
    154 if fmt == 'svg':

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/backend_bases.py:2175, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
   2172     # we do this instead of `self.figure.draw_without_rendering`
   2173     # so that we can inject the orientation
   2174     with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2175         self.figure.draw(renderer)
   2176 if bbox_inches:
   2177     if bbox_inches == "tight":

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
     93 @wraps(draw)
     94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95     result = draw(artist, renderer, *args, **kwargs)
     96     if renderer._rasterizing:
     97         renderer.stop_rasterizing()

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
   3159             # ValueError can occur when resizing a window.
   3161     self.patch.draw(renderer)
-> 3162     mimage._draw_list_compositing_images(
   3163         renderer, self, artists, self.suppressComposite)
   3165     renderer.close_group('figure')
   3166 finally:

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
    130 if not_composite or not has_images:
    131     for a in artists:
--> 132         a.draw(renderer)
    133 else:
    134     # Composite any adjacent images together
    135     image_group = []

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/mpl_toolkits/mplot3d/axes3d.py:441, in Axes3D.draw(self, renderer)
    437 zorder_offset = max(axis.get_zorder()
    438                     for axis in self._axis_map.values()) + 1
    439 collection_zorder = patch_zorder = zorder_offset
--> 441 for artist in sorted(collections_and_patches,
    442                      key=lambda artist: artist.do_3d_projection(),
    443                      reverse=True):
    444     if isinstance(artist, mcoll.Collection):
    445         artist.zorder = collection_zorder

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/mpl_toolkits/mplot3d/axes3d.py:442, in Axes3D.draw.<locals>.<lambda>(artist)
    437 zorder_offset = max(axis.get_zorder()
    438                     for axis in self._axis_map.values()) + 1
    439 collection_zorder = patch_zorder = zorder_offset
    441 for artist in sorted(collections_and_patches,
--> 442                      key=lambda artist: artist.do_3d_projection(),
    443                      reverse=True):
    444     if isinstance(artist, mcoll.Collection):
    445         artist.zorder = collection_zorder

TypeError: do_3d_projection() missing 1 required positional argument: 'renderer'

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/IPython/core/formatters.py:340, in BaseFormatter.__call__(self, obj)
    338     pass
    339 else:
--> 340     return printer(obj)
    341 # Finally look for special method names
    342 method = get_real_method(obj, self.print_method)

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/IPython/core/pylabtools.py:152, in print_figure(fig, fmt, bbox_inches, base64, **kwargs)
    149     from matplotlib.backend_bases import FigureCanvasBase
    150     FigureCanvasBase(fig)
--> 152 fig.canvas.print_figure(bytes_io, **kw)
    153 data = bytes_io.getvalue()
    154 if fmt == 'svg':

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/backend_bases.py:2175, in FigureCanvasBase.print_figure(self, filename, dpi, facecolor, edgecolor, orientation, format, bbox_inches, pad_inches, bbox_extra_artists, backend, **kwargs)
   2172     # we do this instead of `self.figure.draw_without_rendering`
   2173     # so that we can inject the orientation
   2174     with getattr(renderer, "_draw_disabled", nullcontext)():
-> 2175         self.figure.draw(renderer)
   2176 if bbox_inches:
   2177     if bbox_inches == "tight":

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/artist.py:95, in _finalize_rasterization.<locals>.draw_wrapper(artist, renderer, *args, **kwargs)
     93 @wraps(draw)
     94 def draw_wrapper(artist, renderer, *args, **kwargs):
---> 95     result = draw(artist, renderer, *args, **kwargs)
     96     if renderer._rasterizing:
     97         renderer.stop_rasterizing()

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/figure.py:3162, in Figure.draw(self, renderer)
   3159             # ValueError can occur when resizing a window.
   3161     self.patch.draw(renderer)
-> 3162     mimage._draw_list_compositing_images(
   3163         renderer, self, artists, self.suppressComposite)
   3165     renderer.close_group('figure')
   3166 finally:

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/image.py:132, in _draw_list_compositing_images(renderer, parent, artists, suppress_composite)
    130 if not_composite or not has_images:
    131     for a in artists:
--> 132         a.draw(renderer)
    133 else:
    134     # Composite any adjacent images together
    135     image_group = []

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/matplotlib/artist.py:72, in allow_rasterization.<locals>.draw_wrapper(artist, renderer)
     69     if artist.get_agg_filter() is not None:
     70         renderer.start_filter()
---> 72     return draw(artist, renderer)
     73 finally:
     74     if artist.get_agg_filter() is not None:

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/mpl_toolkits/mplot3d/axes3d.py:441, in Axes3D.draw(self, renderer)
    437 zorder_offset = max(axis.get_zorder()
    438                     for axis in self._axis_map.values()) + 1
    439 collection_zorder = patch_zorder = zorder_offset
--> 441 for artist in sorted(collections_and_patches,
    442                      key=lambda artist: artist.do_3d_projection(),
    443                      reverse=True):
    444     if isinstance(artist, mcoll.Collection):
    445         artist.zorder = collection_zorder

File ~/checkouts/readthedocs.org/user_builds/clifford/envs/stable/lib/python3.9/site-packages/mpl_toolkits/mplot3d/axes3d.py:442, in Axes3D.draw.<locals>.<lambda>(artist)
    437 zorder_offset = max(axis.get_zorder()
    438                     for axis in self._axis_map.values()) + 1
    439 collection_zorder = patch_zorder = zorder_offset
    441 for artist in sorted(collections_and_patches,
--> 442                      key=lambda artist: artist.do_3d_projection(),
    443                      reverse=True):
    444     if isinstance(artist, mcoll.Collection):
    445         artist.zorder = collection_zorder

TypeError: do_3d_projection() missing 1 required positional argument: 'renderer'

<Figure size 768x384 with 2 Axes>

Some more example of using mpl_toolkits.clifford to visualize 3D CGA can be found in the examples folder of the mpl_toolkits.clifford repositiory, here.

Visualization tools¶

G2C¶

pyganja¶

mpl_toolkits.clifford¶

G3C¶

pyganja¶

mpl_toolkits.clifford¶

`pyganja`¶

`mpl_toolkits.clifford`¶

`pyganja`¶

`mpl_toolkits.clifford`¶