Functions for plotting polyhedra¶

class sage.geometry.polyhedron.plot.Projection(polyhedron, proj=<function projection_func_identity>)[source]¶

Bases: SageObject

The projection of a Polyhedron.

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

coord_index_of(v)[source]¶

Convert a coordinate vector to its internal index.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> p = polytopes.hypercube(Integer(3))
>>> proj = p.projection()
>>> proj.coord_index_of(vector((Integer(1),Integer(1),Integer(1))))
2

Sage Live

p = polytopes.hypercube(3)
proj = p.projection()
proj.coord_index_of(vector((1,1,1)))

coord_indices_of(v_list)[source]¶

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

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> p = polytopes.hypercube(Integer(3))
>>> proj = p.projection()
>>> proj.coord_indices_of([vector((Integer(1),Integer(1),Integer(1))), vector((Integer(1),-Integer(1),Integer(1)))])
[2, 3]

Sage Live

p = polytopes.hypercube(3)
proj = p.projection()
proj.coord_indices_of([vector((1,1,1)), vector((1,-1,1))])

coordinates_of(coord_index_list)[source]¶

Given a list of indices, return the projected coordinates.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> p = polytopes.simplex(Integer(4), project=True).projection()
>>> p.coordinates_of([Integer(1)])
[[-0.7071067812, 0.4082482905, 0.2886751346, 0.2236067977]]

Sage Live

p = polytopes.simplex(4, project=True).projection()
p.coordinates_of([1])

identity()[source]¶

Return the identity projection of the polyhedron.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> p = polytopes.icosahedron(exact=False)
>>> from sage.geometry.polyhedron.plot import Projection
>>> pproj = Projection(p)
>>> ppid = pproj.identity()
>>> ppid.dimension
3

Sage Live

p = polytopes.icosahedron(exact=False)
from sage.geometry.polyhedron.plot import Projection
pproj = Projection(p)
ppid = pproj.identity()
ppid.dimension

render_0d(point_opts=None, line_opts=None, polygon_opts=None)[source]¶

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

sage: print(Polyhedron([]).projection().render_0d().description())

sage: P = Polyhedron(ieqs=[(1,)])
sage: print(P.projection().render_0d().description())
Point set defined by 1 point(s):    [(0.0, 0.0)]

Python

>>> from sage.all import *
>>> print(Polyhedron([]).projection().render_0d().description())
<BLANKLINE>
>>> P = Polyhedron(ieqs=[(Integer(1),)])
>>> print(P.projection().render_0d().description())
Point set defined by 1 point(s):    [(0.0, 0.0)]

Sage Live

print(Polyhedron([]).projection().render_0d().description())
P = Polyhedron(ieqs=[(1,)])
print(P.projection().render_0d().description())

render_1d(point_opts=None, line_opts=None, polygon_opts=None)[source]¶

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

sage: Polyhedron([(0,), (1,)]).projection().render_1d()
Graphics object consisting of 2 graphics primitives

Python

>>> from sage.all import *
>>> Polyhedron([(Integer(0),), (Integer(1),)]).projection().render_1d()
Graphics object consisting of 2 graphics primitives

Sage Live

Polyhedron([(0,), (1,)]).projection().render_1d()

render_2d(point_opts=None, line_opts=None, polygon_opts=None)[source]¶

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

EXAMPLES:

Sage

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()
Graphics object consisting of 18 graphics primitives

Python

>>> from sage.all import *
>>> p1 = Polyhedron(vertices=[[Integer(1),Integer(1)]], rays=[[Integer(1),Integer(1)]])
>>> q1 = p1.projection()
>>> p2 = Polyhedron(vertices=[[Integer(1),Integer(0)], [Integer(0),Integer(1)], [Integer(0),Integer(0)]])
>>> q2 = p2.projection()
>>> p3 = Polyhedron(vertices=[[Integer(1),Integer(2)]])
>>> q3 = p3.projection()
>>> p4 = Polyhedron(vertices=[[Integer(2),Integer(0)]], rays=[[Integer(1),-Integer(1)]], lines=[[Integer(1),Integer(1)]])
>>> q4 = p4.projection()
>>> q1.plot() + q2.plot() + q3.plot() + q4.plot()
Graphics object consisting of 18 graphics primitives

Sage Live

p1 = Polyhedron(vertices=[[1,1]], rays=[[1,1]])
q1 = p1.projection()
p2 = Polyhedron(vertices=[[1,0], [0,1], [0,0]])
q2 = p2.projection()
p3 = Polyhedron(vertices=[[1,2]])
q3 = p3.projection()
p4 = Polyhedron(vertices=[[2,0]], rays=[[1,-1]], lines=[[1,1]])
q4 = p4.projection()
q1.plot() + q2.plot() + q3.plot() + q4.plot()

render_3d(point_opts=None, line_opts=None, polygon_opts=None)[source]¶

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

EXAMPLES:

Sage

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())
Graphics3d Object

Python

>>> from sage.all import *
>>> p1 = Polyhedron(vertices=[[Integer(1),Integer(1),Integer(1)]], rays=[[Integer(1),Integer(1),Integer(1)]])
>>> p2 = Polyhedron(vertices=[[Integer(2),Integer(0),Integer(0)], [Integer(0),Integer(2),Integer(0)], [Integer(0),Integer(0),Integer(2)]])
>>> p3 = Polyhedron(vertices=[[Integer(1),Integer(0),Integer(0)], [Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(1)]],
...                 rays=[[-Integer(1),-Integer(1),-Integer(1)]])
>>> (p1.projection().plot() + p2.projection().plot()
...   + p3.projection().plot())
Graphics3d Object

Sage Live

p1 = Polyhedron(vertices=[[1,1,1]], rays=[[1,1,1]])
p2 = Polyhedron(vertices=[[2,0,0], [0,2,0], [0,0,2]])
p3 = Polyhedron(vertices=[[1,0,0], [0,1,0], [0,0,1]],
                rays=[[-1,-1,-1]])
(p1.projection().plot() + p2.projection().plot()
  + p3.projection().plot())

It correctly handles various degenerate cases:

Sage

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

Python

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

Sage Live

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

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

Sage

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()
sage: p.bounding_box()
((100.0, 100.0, 100.0), (101.0, 101.0, 101.0))

Python

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

Sage Live

Q = Polyhedron([[100],[101]])
P = Q*Q*Q; P
p = P.plot()
p.bounding_box()

Plot 3d polytope with rainbow colors:

Sage

sage: polytopes.hypercube(3).plot(polygon='rainbow', alpha=0.4)
Graphics3d Object

Python

>>> from sage.all import *
>>> polytopes.hypercube(Integer(3)).plot(polygon='rainbow', alpha=RealNumber('0.4'))
Graphics3d Object

Sage Live

polytopes.hypercube(3).plot(polygon='rainbow', alpha=0.4)

render_fill_2d(**kwds)[source]¶

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

EXAMPLES:

Sage

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()
sage: filled_poly.axes_width()
0.8

Python

>>> from sage.all import *
>>> cps = [i**Integer(3) for i in srange(-Integer(2), Integer(2), Integer(1)/Integer(5))]
>>> p = Polyhedron(vertices=[[(t**Integer(2)-Integer(1))/(t**Integer(2)+Integer(1)), Integer(2)*t/(t**Integer(2)+Integer(1))] for t in cps])
>>> proj = p.projection()
>>> filled_poly = proj.render_fill_2d()
>>> filled_poly.axes_width()
0.8

Sage Live

cps = [i^3 for i in srange(-2, 2, 1/5)]
p = Polyhedron(vertices=[[(t^2-1)/(t^2+1), 2*t/(t^2+1)] for t in cps])
proj = p.projection()
filled_poly = proj.render_fill_2d()
filled_poly.axes_width()

render_line_1d(**kwds)[source]¶

Return the line of a polyhedron in 1d.

INPUT:

**kwds – options passed through to line2d()

OUTPUT: a 2-d graphics object

EXAMPLES:

Sage

sage: outline = polytopes.hypercube(1).projection().render_line_1d()
sage: outline._objects[0]
Line defined by 2 points

Python

>>> from sage.all import *
>>> outline = polytopes.hypercube(Integer(1)).projection().render_line_1d()
>>> outline._objects[Integer(0)]
Line defined by 2 points

Sage Live

outline = polytopes.hypercube(1).projection().render_line_1d()
outline._objects[0]

render_outline_2d(**kwds)[source]¶

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

EXAMPLES:

Sage

sage: penta = polytopes.regular_polygon(5)
sage: outline = penta.projection().render_outline_2d()
sage: outline._objects[0]
Line defined by 2 points

Python

>>> from sage.all import *
>>> penta = polytopes.regular_polygon(Integer(5))
>>> outline = penta.projection().render_outline_2d()
>>> outline._objects[Integer(0)]
Line defined by 2 points

Sage Live

penta = polytopes.regular_polygon(5)
outline = penta.projection().render_outline_2d()
outline._objects[0]

render_points_1d(**kwds)[source]¶

Return the points of a polyhedron in 1d.

INPUT:

**kwds – options passed through to point2d()

OUTPUT: a 2-d graphics object

EXAMPLES:

Sage

sage: cube1 = polytopes.hypercube(1)
sage: proj = cube1.projection()
sage: points = proj.render_points_1d()
sage: points._objects
[Point set defined by 2 point(s)]

Python

>>> from sage.all import *
>>> cube1 = polytopes.hypercube(Integer(1))
>>> proj = cube1.projection()
>>> points = proj.render_points_1d()
>>> points._objects
[Point set defined by 2 point(s)]

Sage Live

cube1 = polytopes.hypercube(1)
proj = cube1.projection()
points = proj.render_points_1d()
points._objects

render_points_2d(**kwds)[source]¶

Return the points of a polyhedron in 2d.

EXAMPLES:

Sage

sage: hex = polytopes.regular_polygon(6)
sage: proj = hex.projection()
sage: hex_points = proj.render_points_2d()
sage: hex_points._objects
[Point set defined by 6 point(s)]

Python

>>> from sage.all import *
>>> hex = polytopes.regular_polygon(Integer(6))
>>> proj = hex.projection()
>>> hex_points = proj.render_points_2d()
>>> hex_points._objects
[Point set defined by 6 point(s)]

Sage Live

hex = polytopes.regular_polygon(6)
proj = hex.projection()
hex_points = proj.render_points_2d()
hex_points._objects

render_solid_3d(**kwds)[source]¶

Return solid 3d rendering of a 3d polytope.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> p = polytopes.hypercube(Integer(3)).projection()
>>> p_solid = p.render_solid_3d(opacity=RealNumber('.7'))
>>> type(p_solid)
<class 'sage.plot.plot3d.index_face_set.IndexFaceSet'>

Sage Live

p = polytopes.hypercube(3).projection()
p_solid = p.render_solid_3d(opacity=.7)
type(p_solid)

render_vertices_3d(**kwds)[source]¶

Return the 3d rendering of the vertices.

EXAMPLES:

Sage

sage: p = polytopes.cross_polytope(3)
sage: proj = p.projection()
sage: verts = proj.render_vertices_3d()
sage: verts.bounding_box()
((-1.0, -1.0, -1.0), (1.0, 1.0, 1.0))

Python

>>> from sage.all import *
>>> p = polytopes.cross_polytope(Integer(3))
>>> proj = p.projection()
>>> verts = proj.render_vertices_3d()
>>> verts.bounding_box()
((-1.0, -1.0, -1.0), (1.0, 1.0, 1.0))

Sage Live

p = polytopes.cross_polytope(3)
proj = p.projection()
verts = proj.render_vertices_3d()
verts.bounding_box()

render_wireframe_3d(**kwds)[source]¶

Return the 3d wireframe rendering.

EXAMPLES:

Sage

sage: cube = polytopes.hypercube(3)
sage: cube_proj = cube.projection()
sage: wire = cube_proj.render_wireframe_3d()
sage: print(wire.tachyon().split('\n')[77])  # for testing
FCylinder base 1.0 1.0 -1.0 apex -1.0 1.0 -1.0 rad 0.005 texture...

Python

>>> from sage.all import *
>>> cube = polytopes.hypercube(Integer(3))
>>> cube_proj = cube.projection()
>>> wire = cube_proj.render_wireframe_3d()
>>> print(wire.tachyon().split('\n')[Integer(77)])  # for testing
FCylinder base 1.0 1.0 -1.0 apex -1.0 1.0 -1.0 rad 0.005 texture...

Sage Live

cube = polytopes.hypercube(3)
cube_proj = cube.projection()
wire = cube_proj.render_wireframe_3d()
print(wire.tachyon().split('\n')[77])  # for testing

schlegel(facet=None, position=None)[source]¶

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

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()
Graphics3d Object

Python

>>> from sage.all import *
>>> cube4 = polytopes.hypercube(Integer(4))
>>> from sage.geometry.polyhedron.plot import Projection
>>> Projection(cube4).schlegel()
The projection of a polyhedron into 3 dimensions
>>> _.plot()
Graphics3d Object

Sage Live

cube4 = polytopes.hypercube(4)
from sage.geometry.polyhedron.plot import Projection
Projection(cube4).schlegel()
_.plot()

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

Sage

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()
Graphics3d Object

Python

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

Sage Live

tcube4 = cube4.face_truncation(cube4.faces(0)[0])
tcube4.facets()[4]
into_tetra = Projection(tcube4).schlegel(tcube4.facets()[4])
into_tetra.plot()

Taking a larger value for the position changes the image:

Sage

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

Python

>>> from sage.all import *
>>> into_tetra_far = Projection(tcube4).schlegel(tcube4.facets()[Integer(4)], Integer(4))
>>> into_tetra_far.plot()
Graphics3d Object

Sage Live

into_tetra_far = Projection(tcube4).schlegel(tcube4.facets()[4], 4)
into_tetra_far.plot()

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

Sage

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

Python

>>> from sage.all import *
>>> Projection(tcube4).schlegel(tcube4.facets()[Integer(4)], Integer(5))
Traceback (most recent call last):
...
ValueError: the chosen position is too large
>>> Projection(tcube4).schlegel(tcube4.facets()[Integer(4)], -Integer(1))
Traceback (most recent call last):
...
ValueError: 'position' should be a positive number

Sage Live

Projection(tcube4).schlegel(tcube4.facets()[4], 5)
Projection(tcube4).schlegel(tcube4.facets()[4], -1)

stereographic(projection_point=None)[source]¶

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

sage: from sage.geometry.polyhedron.plot import Projection
sage: proj = Projection(polytopes.buckyball()); proj  # long time
The projection of a polyhedron into 3 dimensions
sage: proj.stereographic([5,2,3]).plot()    # long time
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

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.plot import Projection
>>> proj = Projection(polytopes.buckyball()); proj  # long time
The projection of a polyhedron into 3 dimensions
>>> proj.stereographic([Integer(5),Integer(2),Integer(3)]).plot()    # long time
Graphics object consisting of 123 graphics primitives
>>> Projection(polytopes.twenty_four_cell()).stereographic([Integer(2),Integer(0),Integer(0),Integer(0)])
The projection of a polyhedron into 3 dimensions

Sage Live

from sage.geometry.polyhedron.plot import Projection
proj = Projection(polytopes.buckyball()); proj  # long time
proj.stereographic([5,2,3]).plot()    # long time
Projection(polytopes.twenty_four_cell()).stereographic([2,0,0,0])

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='TikzPicture')[source]¶

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); the scaling of the tikz picture
edge_color – string (default: 'blue!95!black'); representing colors which tikz recognizes
facet_color – string (default: 'blue!95!black'); representing colors which tikz recognizes
vertex_color – string (default: 'green'); representing colors which tikz recognizes
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: 'TikzPicture'); valid values are '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

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

Python

>>> from sage.all import *
>>> P1 = polytopes.small_rhombicuboctahedron()
>>> Image1 = P1.projection().tikz([Integer(1),Integer(3),Integer(5)], Integer(175), scale=Integer(4),
...                               output_type='TikzPicture')
>>> type(Image1)
<class 'sage.misc.latex_standalone.TikzPicture'>
>>> 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}
>>> _ = Image1.tex('polytope-tikz1.tex')          # not tested
>>> _ = Image1.png('polytope-tikz1.png')          # not tested
>>> _ = Image1.pdf('polytope-tikz1.pdf')          # not tested
>>> _ = Image1.svg('polytope-tikz1.svg')          # not tested

Sage Live

P1 = polytopes.small_rhombicuboctahedron()
Image1 = P1.projection().tikz([1,3,5], 175, scale=4,
                              output_type='TikzPicture')
type(Image1)
Image1
_ = Image1.tex('polytope-tikz1.tex')          # not tested
_ = Image1.png('polytope-tikz1.png')          # not tested
_ = Image1.pdf('polytope-tikz1.pdf')          # not tested
_ = Image1.svg('polytope-tikz1.svg')          # not tested

A second example:

Sage

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}

Python

>>> from sage.all import *
>>> P2 = Polyhedron(vertices=[[Integer(1), Integer(1)], [Integer(1), Integer(2)], [Integer(2), Integer(1)]])
>>> Image2 = P2.projection().tikz(scale=Integer(3), edge_color='blue!95!black',
...                               facet_color='orange!95!black', opacity=RealNumber('0.4'),
...                               vertex_color='yellow', axis=True,
...                               output_type='TikzPicture')
>>> 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}

Sage Live

P2 = Polyhedron(vertices=[[1, 1], [1, 2], [2, 1]])
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')
Image2

The second example using a LatexExpr as output type:

Sage

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)

Python

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

Sage Live

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')
type(Image2)
print('\n'.join(Image2.splitlines()[:4]))
with open('polytope-tikz2.tex', 'w') as f:    # not tested
    _ = f.write(Image2)

A third example:

Sage

sage: P3 = Polyhedron(vertices=[[-1, -1, 2], [-1, 2, -1], [2, -1, -1]]); 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)
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

Python

>>> from sage.all import *
>>> P3 = Polyhedron(vertices=[[-Integer(1), -Integer(1), Integer(2)], [-Integer(1), Integer(2), -Integer(1)], [Integer(2), -Integer(1), -Integer(1)]]); P3
A 2-dimensional polyhedron in ZZ^3 defined as the convex hull of 3 vertices
>>> Image3 = P3.projection().tikz([RealNumber('0.5'), -Integer(1), -RealNumber('0.1')], Integer(55), scale=Integer(3),
...                               edge_color='blue!95!black',
...                               facet_color='orange!95!black', opacity=RealNumber('0.7'),
...                               vertex_color='yellow', axis=True)
>>> 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}
>>> _ = Image3.tex('polytope-tikz3.tex')          # not tested
>>> _ = Image3.png('polytope-tikz3.png')          # not tested
>>> _ = Image3.pdf('polytope-tikz3.pdf')          # not tested
>>> _ = Image3.svg('polytope-tikz3.svg')          # not tested

Sage Live

P3 = Polyhedron(vertices=[[-1, -1, 2], [-1, 2, -1], [2, -1, -1]]); P3
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)
Image3
_ = Image3.tex('polytope-tikz3.tex')          # not tested
_ = Image3.png('polytope-tikz3.png')          # not tested
_ = Image3.pdf('polytope-tikz3.pdf')          # not tested
_ = Image3.svg('polytope-tikz3.svg')          # not tested

A fourth example:

Sage

sage: P = Polyhedron(vertices=[[1,1,0,0], [1,2,0,0],
....:                          [2,1,0,0], [0,0,1,0], [0,0,0,1]]); 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.

Python

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

Sage Live

P = Polyhedron(vertices=[[1,1,0,0], [1,2,0,0],
                         [2,1,0,0], [0,0,1,0], [0,0,0,1]]); P
P.projection().tikz(output_type='TikzPicture')

Todo

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

Sage

sage: P = Polyhedron(vertices=[[1,1,0,0], [1,2,0,0],
....:                          [2,1,0,0], [0,0,1,0], [0,0,0,1]]); 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.

Python

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

Sage Live

P = Polyhedron(vertices=[[1,1,0,0], [1,2,0,0],
                         [2,1,0,0], [0,0,1,0], [0,0,0,1]]); P
P.projection().tikz(output_type='TikzPicture')

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

class sage.geometry.polyhedron.plot.ProjectionFuncSchlegel(facet, projection_point)[source]¶

Bases: object

The Schlegel projection from the given input point.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.plot import ProjectionFuncSchlegel
>>> fcube = polytopes.hypercube(Integer(4))
>>> facet = fcube.facets()[Integer(0)]
>>> proj = ProjectionFuncSchlegel(facet,[Integer(0),-RealNumber('1.5'),Integer(0),Integer(0)])
>>> proj([Integer(0),Integer(0),Integer(0),Integer(0)])[Integer(0)]
1.0

Sage Live

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

class sage.geometry.polyhedron.plot.ProjectionFuncStereographic(projection_point)[source]¶

Bases: object

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

EXAMPLES:

Sage

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...)

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.plot import ProjectionFuncStereographic
>>> cube = polytopes.hypercube(Integer(3)).vertices()
>>> proj = ProjectionFuncStereographic([RealNumber('1.2'), RealNumber('3.4'), RealNumber('5.6')])
>>> ppoints = [proj(vector(x)) for x in cube]
>>> ppoints[Integer(5)]
(-0.0918273..., -0.036375...)

Sage Live

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

sage.geometry.polyhedron.plot.cyclic_sort_vertices_2d(Vlist)[source]¶

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

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)]

Python

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

Sage Live

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

Rays are allowed, too:

Sage

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)]

Python

>>> from sage.all import *
>>> P = Polyhedron(vertices=[(Integer(0), Integer(1)), (Integer(1), Integer(0)), (Integer(2), Integer(0)), (Integer(3), Integer(0)), (Integer(4), Integer(1))], rays=[(Integer(0),Integer(1))])
>>> 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]
>>> 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)]

>>> P = Polyhedron(vertices=[(Integer(0), Integer(1)), (Integer(1), Integer(0)), (Integer(2), Integer(0)), (Integer(3), Integer(0)), (Integer(4), Integer(1))], rays=[(Integer(0),Integer(1)), (Integer(1),Integer(1))])
>>> 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]
>>> 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)]

>>> P = Polyhedron(vertices=[(Integer(1),Integer(2))], rays=[(Integer(0),Integer(1))], lines=[(Integer(1),Integer(0))])
>>> P.adjacency_matrix()
[0 0 1]
[0 0 0]
[1 0 0]
>>> 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 Live

P = Polyhedron(vertices=[(0, 1), (1, 0), (2, 0), (3, 0), (4, 1)], rays=[(0,1)])
P.adjacency_matrix()
cyclic_sort_vertices_2d(P.Vrepresentation())
P = Polyhedron(vertices=[(0, 1), (1, 0), (2, 0), (3, 0), (4, 1)], rays=[(0,1), (1,1)])
P.adjacency_matrix()
cyclic_sort_vertices_2d(P.Vrepresentation())
P = Polyhedron(vertices=[(1,2)], rays=[(0,1)], lines=[(1,0)])
P.adjacency_matrix()
cyclic_sort_vertices_2d(P.Vrepresentation())

sage.geometry.polyhedron.plot.projection_func_identity(x)[source]¶

The identity projection.

EXAMPLES:

Sage

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

Python

>>> from sage.all import *
>>> from sage.geometry.polyhedron.plot import projection_func_identity
>>> projection_func_identity((Integer(1),Integer(2),Integer(3)))
[1, 2, 3]

Sage Live

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