Functions for plotting polyhedra#

class sage.geometry.polyhedron.plot.Projection(polyhedron, proj=<function projection_func_identity>)#

Bases: SageObject

The projection of a Polyhedron.

This class keeps track of the necessary data to plot the input polyhedron.

coord_index_of(v)#

Convert a coordinate vector to its internal index.

EXAMPLES:

sage: p = polytopes.hypercube(3)
sage: proj = p.projection()
sage: proj.coord_index_of(vector((1,1,1)))
2
coord_indices_of(v_list)#

Convert list of coordinate vectors to the corresponding list of internal indices.

EXAMPLES:

sage: p = polytopes.hypercube(3)
sage: proj = p.projection()
sage: proj.coord_indices_of([vector((1,1,1)),vector((1,-1,1))])
[2, 3]
coordinates_of(coord_index_list)#

Given a list of indices, return the projected coordinates.

EXAMPLES:

sage: p = polytopes.simplex(4, project=True).projection()
sage: p.coordinates_of([1])
[[-0.7071067812, 0.4082482905, 0.2886751346, 0.2236067977]]
identity()#

Return the identity projection of the polyhedron.

EXAMPLES:

sage: p = polytopes.icosahedron(exact=False)
sage: from sage.geometry.polyhedron.plot import Projection
sage: pproj = Projection(p)
sage: ppid = pproj.identity()
sage: ppid.dimension
3
render_0d(point_opts=None, line_opts=None, polygon_opts=None)#

Return 0d rendering of the projection of a polyhedron into 2-dimensional ambient space.

INPUT:

See plot().

OUTPUT:

A 2-d graphics object.

EXAMPLES:

sage: print(Polyhedron([]).projection().render_0d().description())           # optional - sage.plot

sage: print(Polyhedron(ieqs=[(1,)]).projection().render_0d().description())  # optional - sage.plot
Point set defined by 1 point(s):    [(0.0, 0.0)]
render_1d(point_opts=None, line_opts=None, polygon_opts=None)#

Return 1d rendering of the projection of a polyhedron into 2-dimensional ambient space.

INPUT:

See plot().

OUTPUT:

A 2-d graphics object.

EXAMPLES:

sage: Polyhedron([(0,), (1,)]).projection().render_1d()  # optional - sage.plot
Graphics object consisting of 2 graphics primitives
render_2d(point_opts=None, line_opts=None, polygon_opts=None)#

Return 2d rendering of the projection of a polyhedron into 2-dimensional ambient space.

EXAMPLES:

sage: p1 = Polyhedron(vertices=[[1,1]], rays=[[1,1]])
sage: q1 = p1.projection()
sage: p2 = Polyhedron(vertices=[[1,0], [0,1], [0,0]])
sage: q2 = p2.projection()
sage: p3 = Polyhedron(vertices=[[1,2]])
sage: q3 = p3.projection()
sage: p4 = Polyhedron(vertices=[[2,0]], rays=[[1,-1]], lines=[[1,1]])
sage: q4 = p4.projection()
sage: q1.plot() + q2.plot() + q3.plot() + q4.plot()  # optional - sage.plot
Graphics object consisting of 18 graphics primitives
render_3d(point_opts=None, line_opts=None, polygon_opts=None)#

Return 3d rendering of a polyhedron projected into 3-dimensional ambient space.

EXAMPLES:

sage: p1 = Polyhedron(vertices=[[1,1,1]], rays=[[1,1,1]])
sage: p2 = Polyhedron(vertices=[[2,0,0], [0,2,0], [0,0,2]])
sage: p3 = Polyhedron(vertices=[[1,0,0], [0,1,0], [0,0,1]], rays=[[-1,-1,-1]])
sage: p1.projection().plot() + p2.projection().plot() + p3.projection().plot()  # optional - sage.plot
Graphics3d Object

It correctly handles various degenerate cases:

sage: Polyhedron(lines=[[1,0,0],[0,1,0],[0,0,1]]).plot()           # whole space              # optional - sage.plot
Graphics3d Object
sage: Polyhedron(vertices=[[1,1,1]], rays=[[1,0,0]],                                          # optional - sage.plot
....:            lines=[[0,1,0],[0,0,1]]).plot()                   # half space
Graphics3d Object
sage: Polyhedron(vertices=[[1,1,1]],                                                          # optional - sage.plot
....:            lines=[[0,1,0],[0,0,1]]).plot()                   # R^2 in R^3
Graphics3d Object
sage: Polyhedron(rays=[[0,1,0],[0,0,1]], lines=[[1,0,0]]).plot()   # quadrant wedge in R^2    # optional - sage.plot
Graphics3d Object
sage: Polyhedron(rays=[[0,1,0]], lines=[[1,0,0]]).plot()           # upper half plane in R^3  # optional - sage.plot
Graphics3d Object
sage: Polyhedron(lines=[[1,0,0]]).plot()                           # R^1 in R^2               # optional - sage.plot
Graphics3d Object
sage: Polyhedron(rays=[[0,1,0]]).plot()                            # Half-line in R^3         # optional - sage.plot
Graphics3d Object
sage: Polyhedron(vertices=[[1,1,1]]).plot()                        # point in R^3             # optional - sage.plot
Graphics3d Object

The origin is not included, if it is not in the polyhedron (trac ticket #23555):

sage: Q = Polyhedron([[100],[101]])
sage: P = Q*Q*Q; P
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 8 vertices
sage: p = P.plot()                                                                            # optional - sage.plot
sage: p.bounding_box()                                                                        # optional - sage.plot
((100.0, 100.0, 100.0), (101.0, 101.0, 101.0))

Plot 3d polytope with rainbow colors:

sage: polytopes.hypercube(3).plot(polygon='rainbow', alpha=0.4)                               # optional - sage.plot
Graphics3d Object
render_fill_2d(**kwds)#

Return the filled interior (a polygon) of a polyhedron in 2d.

EXAMPLES:

sage: cps = [i^3 for i in srange(-2,2,1/5)]
sage: p = Polyhedron(vertices = [[(t^2-1)/(t^2+1),2*t/(t^2+1)] for t in cps])
sage: proj = p.projection()
sage: filled_poly = proj.render_fill_2d()  # optional - sage.plot
sage: filled_poly.axes_width()             # optional - sage.plot
0.8
render_line_1d(**kwds)#

Return the line of a polyhedron in 1d.

INPUT:

  • **kwds – options passed through to line2d().

OUTPUT:

A 2-d graphics object.

EXAMPLES:

sage: outline = polytopes.hypercube(1).projection().render_line_1d()  # optional - sage.plot
sage: outline._objects[0]                                             # optional - sage.plot
Line defined by 2 points
render_outline_2d(**kwds)#

Return the outline (edges) of a polyhedron in 2d.

EXAMPLES:

sage: penta = polytopes.regular_polygon(5)
sage: outline = penta.projection().render_outline_2d()  # optional - sage.plot
sage: outline._objects[0]                               # optional - sage.plot
Line defined by 2 points
render_points_1d(**kwds)#

Return the points of a polyhedron in 1d.

INPUT:

  • **kwds – options passed through to point2d().

OUTPUT:

A 2-d graphics object.

EXAMPLES:

sage: cube1 = polytopes.hypercube(1)
sage: proj = cube1.projection()
sage: points = proj.render_points_1d()  # optional - sage.plot
sage: points._objects                   # optional - sage.plot
[Point set defined by 2 point(s)]
render_points_2d(**kwds)#

Return the points of a polyhedron in 2d.

EXAMPLES:

sage: hex = polytopes.regular_polygon(6)
sage: proj = hex.projection()
sage: hex_points = proj.render_points_2d()  # optional - sage.plot
sage: hex_points._objects                   # optional - sage.plot
[Point set defined by 6 point(s)]
render_solid_3d(**kwds)#

Return solid 3d rendering of a 3d polytope.

EXAMPLES:

sage: p = polytopes.hypercube(3).projection()
sage: p_solid = p.render_solid_3d(opacity=.7)  # optional - sage.plot
sage: type(p_solid)                            # optional - sage.plot
<class 'sage.plot.plot3d.index_face_set.IndexFaceSet'>
render_vertices_3d(**kwds)#

Return the 3d rendering of the vertices.

EXAMPLES:

sage: p = polytopes.cross_polytope(3)
sage: proj = p.projection()
sage: verts = proj.render_vertices_3d()  # optional - sage.plot
sage: verts.bounding_box()               # optional - sage.plot
((-1.0, -1.0, -1.0), (1.0, 1.0, 1.0))
render_wireframe_3d(**kwds)#

Return the 3d wireframe rendering.

EXAMPLES:

sage: cube = polytopes.hypercube(3)
sage: cube_proj = cube.projection()
sage: wire = cube_proj.render_wireframe_3d()                # optional - sage.plot
sage: print(wire.tachyon().split('\n')[77])  # for testing  # optional - sage.plot
FCylinder base 1.0 1.0 -1.0 apex -1.0 1.0 -1.0 rad 0.005 texture...
schlegel(facet=None, position=None)#

Return the Schlegel projection.

  • The facet is orthonormally transformed into its affine hull.

  • The position specifies a point coming out of the barycenter of the facet from which the other vertices will be projected into the facet.

INPUT:

  • facet – a PolyhedronFace. The facet into which the Schlegel diagram is created. The default is the first facet.

  • position – a positive number. Determines a relative distance from the barycenter of facet. A value close to 0 will place the projection point close to the facet and a large value further away. If the given value is too large, an error is returned. If no position is given, it takes the midpoint of the possible point of views along a line spanned by the barycenter of the facet and a valid point outside the facet.

EXAMPLES:

sage: cube4 = polytopes.hypercube(4)
sage: from sage.geometry.polyhedron.plot import Projection
sage: Projection(cube4).schlegel()
The projection of a polyhedron into 3 dimensions
sage: _.plot()  # optional - sage.plot
Graphics3d Object

The 4-cube with a truncated vertex seen into the resulting tetrahedron facet:

sage: tcube4 = cube4.face_truncation(cube4.faces(0)[0])
sage: tcube4.facets()[4]
A 3-dimensional face of a Polyhedron in QQ^4 defined as the convex hull of 4 vertices
sage: into_tetra = Projection(tcube4).schlegel(tcube4.facets()[4])
sage: into_tetra.plot()  # optional - sage.plot
Graphics3d Object

Taking a larger value for the position changes the image:

sage: into_tetra_far = Projection(tcube4).schlegel(tcube4.facets()[4],4)
sage: into_tetra_far.plot()  # optional - sage.plot
Graphics3d Object

A value which is too large or negative give a projection point that sees more than one facet resulting in a error:

sage: Projection(tcube4).schlegel(tcube4.facets()[4],5)
Traceback (most recent call last):
...
ValueError: the chosen position is too large
sage: Projection(tcube4).schlegel(tcube4.facets()[4],-1)
Traceback (most recent call last):
...
ValueError: 'position' should be a positive number
stereographic(projection_point=None)#

Return the stereographic projection.

INPUT:

  • projection_point - The projection point. This must be distinct from the polyhedron’s vertices. Default is \((1,0,\dots,0)\)

EXAMPLES:

sage: from sage.geometry.polyhedron.plot import Projection
sage: proj = Projection(polytopes.buckyball())  #long time
sage: proj                                      #long time
The projection of a polyhedron into 3 dimensions
sage: proj.stereographic([5,2,3]).plot()        #long time  # optional - sage.plot
Graphics object consisting of 123 graphics primitives
sage: Projection( polytopes.twenty_four_cell() ).stereographic([2,0,0,0])
The projection of a polyhedron into 3 dimensions
tikz(view=[0, 0, 1], angle=0, scale=1, edge_color='blue!95!black', facet_color='blue!95!black', opacity=0.8, vertex_color='green', axis=False, output_type=None)#

Return a tikz picture of self as a string or as a TikzPicture according to a projection view and an angle angle obtained via the threejs viewer.

INPUT:

  • view - list (default: [0,0,1]) representing the rotation axis (see note below).

  • angle - integer (default: 0) angle of rotation in degree from 0 to 360 (see note below).

  • scale - integer (default: 1) specifying the scaling of the tikz picture.

  • edge_color - string (default: ‘blue!95!black’) representing colors which tikz recognize.

  • facet_color - string (default: ‘blue!95!black’) representing colors which tikz recognize.

  • vertex_color - string (default: ‘green’) representing colors which tikz recognize.

  • opacity - real number (default: 0.8) between 0 and 1 giving the opacity of the front facets.

  • axis - Boolean (default: False) draw the axes at the origin or not.

  • output_type - string (default: None), valid values are None (deprecated), 'LatexExpr' and 'TikzPicture', whether to return a LatexExpr object (which inherits from Python str) or a TikzPicture object from module sage.misc.latex_standalone

OUTPUT:

  • LatexExpr object or TikzPicture object

Note

The inputs view and angle can be obtained by visualizing it using .show(aspect_ratio=1). This will open an interactive view in your default browser, where you can rotate the polytope. Once the desired view angle is found, click on the information icon in the lower right-hand corner and select Get Viewpoint. This will copy a string of the form ‘[x,y,z],angle’ to your local clipboard. Go back to Sage and type Img = P.projection().tikz([x,y,z],angle).

The inputs view and angle can also be obtained from the viewer Jmol:

1) Right click on the image
2) Select ``Console``
3) Select the tab ``State``
4) Scroll to the line ``moveto``

It reads something like:

moveto 0.0 {x y z angle} Scale

The view is then [x,y,z] and angle is angle. The following number is the scale.

Jmol performs a rotation of angle degrees along the vector [x,y,z] and show the result from the z-axis.

EXAMPLES:

sage: P1 = polytopes.small_rhombicuboctahedron()
sage: Image1 = P1.projection().tikz([1,3,5], 175, scale=4, output_type='TikzPicture')
sage: type(Image1)
<class 'sage.misc.latex_standalone.TikzPicture'>
sage: Image1
\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}%
        [x={(-0.939161cm, 0.244762cm)},
        y={(0.097442cm, -0.482887cm)},
        z={(0.329367cm, 0.840780cm)},
        scale=4.000000,
...
Use print to see the full content.
...
\node[vertex] at (-2.41421, 1.00000, -1.00000)     {};
\node[vertex] at (-2.41421, -1.00000, 1.00000)     {};
%%
%%
\end{tikzpicture}
\end{document}
sage: _ = Image1.tex('polytope-tikz1.tex')          # not tested
sage: _ = Image1.png('polytope-tikz1.png')          # not tested
sage: _ = Image1.pdf('polytope-tikz1.pdf')          # not tested
sage: _ = Image1.svg('polytope-tikz1.svg')          # not tested

A second example:

sage: P2 = Polyhedron(vertices=[[1, 1],[1, 2],[2, 1]])
sage: Image2 = P2.projection().tikz(scale=3, edge_color='blue!95!black', facet_color='orange!95!black', opacity=0.4, vertex_color='yellow', axis=True, output_type='TikzPicture')
sage: Image2
\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}%
        [scale=3.000000,
        back/.style={loosely dotted, thin},
        edge/.style={color=blue!95!black, thick},
        facet/.style={fill=orange!95!black,fill opacity=0.400000},
...
Use print to see the full content.
...
\node[vertex] at (1.00000, 2.00000)     {};
\node[vertex] at (2.00000, 1.00000)     {};
%%
%%
\end{tikzpicture}
\end{document}

The second example using a LatexExpr as output type:

sage: Image2 = P2.projection().tikz(scale=3, edge_color='blue!95!black', facet_color='orange!95!black', opacity=0.4, vertex_color='yellow', axis=True, output_type='LatexExpr')
sage: type(Image2)
<class 'sage.misc.latex.LatexExpr'>
sage: print('\n'.join(Image2.splitlines()[:4]))
\begin{tikzpicture}%
    [scale=3.000000,
    back/.style={loosely dotted, thin},
    edge/.style={color=blue!95!black, thick},
sage: with open('polytope-tikz2.tex', 'w') as f:  # not tested
....:     _ = f.write(Image2)

A third example:

sage: P3 = Polyhedron(vertices=[[-1, -1, 2],[-1, 2, -1],[2, -1, -1]])
sage: P3
A 2-dimensional polyhedron in ZZ^3 defined as the convex hull of 3 vertices
sage: Image3 = P3.projection().tikz([0.5,-1,-0.1], 55, scale=3, edge_color='blue!95!black',facet_color='orange!95!black', opacity=0.7, vertex_color='yellow', axis=True, output_type='TikzPicture')
sage: Image3
\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}%
        [x={(0.658184cm, -0.242192cm)},
        y={(-0.096240cm, 0.912008cm)},
        z={(-0.746680cm, -0.331036cm)},
        scale=3.000000,
...
Use print to see the full content.
...
\node[vertex] at (-1.00000, 2.00000, -1.00000)     {};
\node[vertex] at (2.00000, -1.00000, -1.00000)     {};
%%
%%
\end{tikzpicture}
\end{document}
sage: _ = Image3.tex('polytope-tikz3.tex')          # not tested
sage: _ = Image3.png('polytope-tikz3.png')          # not tested
sage: _ = Image3.pdf('polytope-tikz3.pdf')          # not tested
sage: _ = Image3.svg('polytope-tikz3.svg')          # not tested

A fourth example:

sage: P = Polyhedron(vertices=[[1,1,0,0],[1,2,0,0],[2,1,0,0],[0,0,1,0],[0,0,0,1]])
sage: P
A 4-dimensional polyhedron in ZZ^4 defined as the convex hull of 5 vertices
sage: P.projection().tikz(output_type='TikzPicture')
Traceback (most recent call last):
...
NotImplementedError: The polytope has to live in 2 or 3 dimensions.

Todo

Make it possible to draw Schlegel diagram for 4-polytopes.

sage: P=Polyhedron(vertices=[[1,1,0,0],[1,2,0,0],[2,1,0,0],[0,0,1,0],[0,0,0,1]])
sage: P
A 4-dimensional polyhedron in ZZ^4 defined as the convex hull of 5 vertices
sage: P.projection().tikz(output_type='TikzPicture')
Traceback (most recent call last):
...
NotImplementedError: The polytope has to live in 2 or 3 dimensions.

Make it possible to draw 3-polytopes living in higher dimension.

class sage.geometry.polyhedron.plot.ProjectionFuncSchlegel(facet, projection_point)#

Bases: object

The Schlegel projection from the given input point.

EXAMPLES:

sage: from sage.geometry.polyhedron.plot import ProjectionFuncSchlegel
sage: fcube = polytopes.hypercube(4)
sage: facet = fcube.facets()[0]
sage: proj = ProjectionFuncSchlegel(facet,[0,-1.5,0,0])
sage: proj([0,0,0,0])[0]
1.0
class sage.geometry.polyhedron.plot.ProjectionFuncStereographic(projection_point)#

Bases: object

The stereographic (or perspective) projection onto a codimension-1 linear subspace with respect to a sphere centered at the origin.

EXAMPLES:

sage: from sage.geometry.polyhedron.plot import ProjectionFuncStereographic
sage: cube = polytopes.hypercube(3).vertices()
sage: proj = ProjectionFuncStereographic([1.2, 3.4, 5.6])
sage: ppoints = [proj(vector(x)) for x in cube]
sage: ppoints[5]
(-0.0918273..., -0.036375...)
sage.geometry.polyhedron.plot.cyclic_sort_vertices_2d(Vlist)#

Return the vertices/rays in cyclic order if possible.

Note

This works if and only if each vertex/ray is adjacent to exactly two others. For example, any 2-dimensional polyhedron satisfies this.

See vertex_adjacency_matrix() for a discussion of “adjacent”.

EXAMPLES:

sage: from sage.geometry.polyhedron.plot import cyclic_sort_vertices_2d
sage: square = Polyhedron([[1,0],[-1,0],[0,1],[0,-1]])
sage: vertices = [v for v in square.vertex_generator()]
sage: vertices
[A vertex at (-1, 0),
 A vertex at (0, -1),
 A vertex at (0, 1),
 A vertex at (1, 0)]
sage: cyclic_sort_vertices_2d(vertices)
[A vertex at (1, 0),
 A vertex at (0, -1),
 A vertex at (-1, 0),
 A vertex at (0, 1)]

Rays are allowed, too:

sage: P = Polyhedron(vertices=[(0, 1), (1, 0), (2, 0), (3, 0), (4, 1)], rays=[(0,1)])
sage: P.adjacency_matrix()
[0 1 0 1 0]
[1 0 1 0 0]
[0 1 0 0 1]
[1 0 0 0 1]
[0 0 1 1 0]
sage: cyclic_sort_vertices_2d(P.Vrepresentation())
[A vertex at (3, 0),
 A vertex at (1, 0),
 A vertex at (0, 1),
 A ray in the direction (0, 1),
 A vertex at (4, 1)]

sage: P = Polyhedron(vertices=[(0, 1), (1, 0), (2, 0), (3, 0), (4, 1)], rays=[(0,1), (1,1)])
sage: P.adjacency_matrix()
[0 1 0 0 0]
[1 0 1 0 0]
[0 1 0 0 1]
[0 0 0 0 1]
[0 0 1 1 0]
sage: cyclic_sort_vertices_2d(P.Vrepresentation())
[A ray in the direction (1, 1),
 A vertex at (3, 0),
 A vertex at (1, 0),
 A vertex at (0, 1),
 A ray in the direction (0, 1)]

sage: P = Polyhedron(vertices=[(1,2)], rays=[(0,1)], lines=[(1,0)])
sage: P.adjacency_matrix()
[0 0 1]
[0 0 0]
[1 0 0]
sage: cyclic_sort_vertices_2d(P.Vrepresentation())
[A vertex at (0, 2),
 A line in the direction (1, 0),
 A ray in the direction (0, 1)]
sage.geometry.polyhedron.plot.projection_func_identity(x)#

The identity projection.

EXAMPLES:

sage: from sage.geometry.polyhedron.plot import projection_func_identity
sage: projection_func_identity((1,2,3))
[1, 2, 3]