Toric plotter#

This module provides a helper class ToricPlotter for producing plots of objects related to toric geometry. Default plotting objects can be adjusted using options() and reset using reset_options().

AUTHORS:

Andrey Novoseltsev (2010-10-03): initial version, using some code bits by Volker Braun.

EXAMPLES:

In most cases, this module is used indirectly, e.g.

Sage

sage: fan = toric_varieties.dP6().fan()                                             # needs palp sage.graphs
sage: fan.plot()                                                                    # needs palp sage.graphs sage.plot
Graphics object consisting of 31 graphics primitives

Python

>>> from sage.all import *
>>> fan = toric_varieties.dP6().fan()                                             # needs palp sage.graphs
>>> fan.plot()                                                                    # needs palp sage.graphs sage.plot
Graphics object consisting of 31 graphics primitives

You may change default plotting options as follows:

Sage

sage: toric_plotter.options("show_rays")
True
sage: toric_plotter.options(show_rays=False)
sage: toric_plotter.options("show_rays")
False
sage: fan.plot()                                                                    # needs palp sage.graphs sage.plot
Graphics object consisting of 19 graphics primitives
sage: toric_plotter.reset_options()
sage: toric_plotter.options("show_rays")
True
sage: fan.plot()                                                                    # needs palp sage.graphs sage.plot
Graphics object consisting of 31 graphics primitives

Python

>>> from sage.all import *
>>> toric_plotter.options("show_rays")
True
>>> toric_plotter.options(show_rays=False)
>>> toric_plotter.options("show_rays")
False
>>> fan.plot()                                                                    # needs palp sage.graphs sage.plot
Graphics object consisting of 19 graphics primitives
>>> toric_plotter.reset_options()
>>> toric_plotter.options("show_rays")
True
>>> fan.plot()                                                                    # needs palp sage.graphs sage.plot
Graphics object consisting of 31 graphics primitives

class sage.geometry.toric_plotter.ToricPlotter(all_options, dimension, generators=None)[source]#

Bases: SageObject

Create a toric plotter.

INPUT:

all_options – a dictionary, containing any of the options related to toric objects (see options()) and any other options that will be passed to lower level plotting functions;
dimension – an integer (1, 2, or 3), dimension of toric objects to be plotted;
generators – (optional) a list of ray generators, see examples for a detailed explanation of this argument.

OUTPUT:

a toric plotter.

EXAMPLES:

In most cases there is no need to create and use ToricPlotter directly. Instead, use plotting method of the object which you want to plot, e.g.

Sage

sage: fan = toric_varieties.dP6().fan()                                         # needs palp sage.graphs
sage: fan.plot()                                                                # needs palp sage.graphs sage.plot
Graphics object consisting of 31 graphics primitives
sage: print(fan.plot())                                                         # needs palp sage.graphs sage.plot
Graphics object consisting of 31 graphics primitives

Python

>>> from sage.all import *
>>> fan = toric_varieties.dP6().fan()                                         # needs palp sage.graphs
>>> fan.plot()                                                                # needs palp sage.graphs sage.plot
Graphics object consisting of 31 graphics primitives
>>> print(fan.plot())                                                         # needs palp sage.graphs sage.plot
Graphics object consisting of 31 graphics primitives

If you do want to create your own plotting function for some toric structure, the anticipated usage of toric plotters is the following:

collect all necessary options in a dictionary;
pass these options and dimension to ToricPlotter;
call include_points() on ray generators and any other points that you want to be present on the plot (it will try to set appropriate cut-off bounds);
call adjust_options() to choose “nice” default values for all options that were not set yet and ensure consistency of rectangular and spherical cut-off bounds;
call set_rays() on ray generators to scale them to the cut-off bounds of the plot;
call appropriate plot_* functions to actually construct the plot.

For example, the plot from the previous example can be obtained as follows:

Sage

sage: # needs palp sage.graphs sage.plot
sage: from sage.geometry.toric_plotter import ToricPlotter
sage: options = dict()  # use default for everything
sage: tp = ToricPlotter(options, fan.lattice().degree())
sage: tp.include_points(fan.rays())
sage: tp.adjust_options()
sage: tp.set_rays(fan.rays())
sage: result = tp.plot_lattice()
sage: result += tp.plot_rays()
sage: result += tp.plot_generators()
sage: result += tp.plot_walls(fan(2))
sage: result
Graphics object consisting of 31 graphics primitives

Python

>>> from sage.all import *
>>> # needs palp sage.graphs sage.plot
>>> from sage.geometry.toric_plotter import ToricPlotter
>>> options = dict()  # use default for everything
>>> tp = ToricPlotter(options, fan.lattice().degree())
>>> tp.include_points(fan.rays())
>>> tp.adjust_options()
>>> tp.set_rays(fan.rays())
>>> result = tp.plot_lattice()
>>> result += tp.plot_rays()
>>> result += tp.plot_generators()
>>> result += tp.plot_walls(fan(Integer(2)))
>>> result
Graphics object consisting of 31 graphics primitives

In most situations it is only necessary to include generators of rays, in this case they can be passed to the constructor as an optional argument. In the example above, the toric plotter can be completely set up using

Sage

sage: tp = ToricPlotter(options, fan.lattice().degree(), fan.rays())            # needs palp sage.graphs sage.plot

Python

>>> from sage.all import *
>>> tp = ToricPlotter(options, fan.lattice().degree(), fan.rays())            # needs palp sage.graphs sage.plot

All options are exposed as attributes of toric plotters and can be modified after constructions, however you will have to manually call adjust_options() and set_rays() again if you decide to change the plotting mode and/or cut-off bounds. Otherwise plots maybe invalid.

adjust_options()[source]#

Adjust plotting options.

This function determines appropriate default values for those options, that were not specified by the user, based on the other options. See ToricPlotter for a detailed example.

OUTPUT:

none.

include_points(points, force=False)[source]#

Try to include points into the bounding box of self.

INPUT:

points – a list of points;
force – boolean (default: False). by default, only bounds that were not set before will be chosen to include points. Use force=True if you don’t mind increasing existing bounding box.

OUTPUT:

none.

EXAMPLES:

Sage

sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2)
sage: print(tp.radius)
None
sage: tp.include_points([(3, 4)])
sage: print(tp.radius)
5.5...
sage: tp.include_points([(5, 12)])
sage: print(tp.radius)
5.5...
sage: tp.include_points([(5, 12)], force=True)
sage: print(tp.radius)
13.5...

Python

>>> from sage.all import *
>>> from sage.geometry.toric_plotter import ToricPlotter
>>> tp = ToricPlotter(dict(), Integer(2))
>>> print(tp.radius)
None
>>> tp.include_points([(Integer(3), Integer(4))])
>>> print(tp.radius)
5.5...
>>> tp.include_points([(Integer(5), Integer(12))])
>>> print(tp.radius)
5.5...
>>> tp.include_points([(Integer(5), Integer(12))], force=True)
>>> print(tp.radius)
13.5...

plot_generators()[source]#

Plot ray generators.

Ray generators must be specified during construction or using set_rays() before calling this method.

OUTPUT:

a plot.

EXAMPLES:

Sage

sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2, [(3,4)])
sage: tp.plot_generators()                                                  # needs sage.plot
Graphics object consisting of 1 graphics primitive

Python

>>> from sage.all import *
>>> from sage.geometry.toric_plotter import ToricPlotter
>>> tp = ToricPlotter(dict(), Integer(2), [(Integer(3),Integer(4))])
>>> tp.plot_generators()                                                  # needs sage.plot
Graphics object consisting of 1 graphics primitive

plot_labels(labels, positions)[source]#

Plot labels at specified positions.

INPUT:

labels – a string or a list of strings;
positions – a list of points.

OUTPUT:

a plot.

EXAMPLES:

Sage

sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2)
sage: tp.plot_labels("u", [(1.5,0)])                                        # needs sage.plot
Graphics object consisting of 1 graphics primitive

Python

>>> from sage.all import *
>>> from sage.geometry.toric_plotter import ToricPlotter
>>> tp = ToricPlotter(dict(), Integer(2))
>>> tp.plot_labels("u", [(RealNumber('1.5'),Integer(0))])                                        # needs sage.plot
Graphics object consisting of 1 graphics primitive

plot_lattice()[source]#

Plot the lattice (i.e. its points in the cut-off bounds of self).

OUTPUT:

a plot.

EXAMPLES:

Sage

sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2)
sage: tp.adjust_options()
sage: tp.plot_lattice()                                                     # needs sage.plot
Graphics object consisting of 1 graphics primitive

Python

>>> from sage.all import *
>>> from sage.geometry.toric_plotter import ToricPlotter
>>> tp = ToricPlotter(dict(), Integer(2))
>>> tp.adjust_options()
>>> tp.plot_lattice()                                                     # needs sage.plot
Graphics object consisting of 1 graphics primitive

plot_points(points)[source]#

Plot given points.

INPUT:

points – a list of points.

OUTPUT:

a plot.

EXAMPLES:

Sage

sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2)
sage: tp.adjust_options()
sage: tp.plot_points([(1,0), (0,1)])                                        # needs sage.plot
Graphics object consisting of 1 graphics primitive

Python

>>> from sage.all import *
>>> from sage.geometry.toric_plotter import ToricPlotter
>>> tp = ToricPlotter(dict(), Integer(2))
>>> tp.adjust_options()
>>> tp.plot_points([(Integer(1),Integer(0)), (Integer(0),Integer(1))])                                        # needs sage.plot
Graphics object consisting of 1 graphics primitive

plot_ray_labels()[source]#

Plot ray labels.

Usually ray labels are plotted together with rays, but in some cases it is desirable to output them separately.

Ray generators must be specified during construction or using set_rays() before calling this method.

OUTPUT:

a plot.

EXAMPLES:

Sage

sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2, [(3,4)])
sage: tp.plot_ray_labels()                                                  # needs sage.plot
Graphics object consisting of 1 graphics primitive

Python

>>> from sage.all import *
>>> from sage.geometry.toric_plotter import ToricPlotter
>>> tp = ToricPlotter(dict(), Integer(2), [(Integer(3),Integer(4))])
>>> tp.plot_ray_labels()                                                  # needs sage.plot
Graphics object consisting of 1 graphics primitive

plot_rays()[source]#

Plot rays and their labels.

Ray generators must be specified during construction or using set_rays() before calling this method.

OUTPUT:

a plot.

EXAMPLES:

Sage

sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2, [(3,4)])
sage: tp.plot_rays()                                                        # needs sage.plot
Graphics object consisting of 2 graphics primitives

Python

>>> from sage.all import *
>>> from sage.geometry.toric_plotter import ToricPlotter
>>> tp = ToricPlotter(dict(), Integer(2), [(Integer(3),Integer(4))])
>>> tp.plot_rays()                                                        # needs sage.plot
Graphics object consisting of 2 graphics primitives

plot_walls(walls)[source]#

Plot walls, i.e. 2-d cones, and their labels.

Ray generators must be specified during construction or using set_rays() before calling this method and these specified ray generators will be used in conjunction with ambient_ray_indices() of walls.

INPUT:

walls – a list of 2-d cones.

OUTPUT:

a plot.

EXAMPLES:

Sage

sage: quadrant = Cone([(1,0), (0,1)])
sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2, quadrant.rays())
sage: tp.plot_walls([quadrant])                                             # needs sage.plot
Graphics object consisting of 2 graphics primitives

Python

>>> from sage.all import *
>>> quadrant = Cone([(Integer(1),Integer(0)), (Integer(0),Integer(1))])
>>> from sage.geometry.toric_plotter import ToricPlotter
>>> tp = ToricPlotter(dict(), Integer(2), quadrant.rays())
>>> tp.plot_walls([quadrant])                                             # needs sage.plot
Graphics object consisting of 2 graphics primitives

Let’s also check that the truncating polyhedron is functioning correctly:

Sage

sage: tp = ToricPlotter({"mode": "box"}, 2, quadrant.rays())
sage: tp.plot_walls([quadrant])                                             # needs sage.plot
Graphics object consisting of 2 graphics primitives

Python

>>> from sage.all import *
>>> tp = ToricPlotter({"mode": "box"}, Integer(2), quadrant.rays())
>>> tp.plot_walls([quadrant])                                             # needs sage.plot
Graphics object consisting of 2 graphics primitives

set_rays(generators)[source]#

Set up rays and their generators to be used by plotting functions.

As an alternative to using this method, you can pass generators to ToricPlotter constructor.

INPUT:

generators – a list of primitive non-zero ray generators.

OUTPUT:

none.

EXAMPLES:

Sage

sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2)
sage: tp.adjust_options()
sage: tp.plot_rays()                                                        # needs sage.plot
Traceback (most recent call last):
...
AttributeError: 'ToricPlotter' object has no attribute 'rays'...
sage: tp.set_rays([(0,1)])
sage: tp.plot_rays()                                                        # needs sage.plot
Graphics object consisting of 2 graphics primitives

Python

>>> from sage.all import *
>>> from sage.geometry.toric_plotter import ToricPlotter
>>> tp = ToricPlotter(dict(), Integer(2))
>>> tp.adjust_options()
>>> tp.plot_rays()                                                        # needs sage.plot
Traceback (most recent call last):
...
AttributeError: 'ToricPlotter' object has no attribute 'rays'...
>>> tp.set_rays([(Integer(0),Integer(1))])
>>> tp.plot_rays()                                                        # needs sage.plot
Graphics object consisting of 2 graphics primitives

sage.geometry.toric_plotter.color_list(color, n)[source]#

Normalize a list of n colors.

INPUT:

color – anything specifying a Color, a list of such specifications, or the string “rainbow”;
n – an integer.

OUTPUT:

a list of n colors.

If color specified a single color, it is repeated n times. If it was a list of n colors, it is returned without changes. If it was “rainbow”, the rainbow of n colors is returned.

EXAMPLES:

Sage

sage: # needs sage.plot
sage: from sage.geometry.toric_plotter import color_list
sage: color_list("grey", 1)
[RGB color (0.5019607843137255, 0.5019607843137255, 0.5019607843137255)]
sage: len(color_list("grey", 3))
3
sage: L = color_list("rainbow", 3)
sage: L
[RGB color (1.0, 0.0, 0.0),
 RGB color (0.0, 1.0, 0.0),
 RGB color (0.0, 0.0, 1.0)]
sage: color_list(L, 3)
[RGB color (1.0, 0.0, 0.0),
 RGB color (0.0, 1.0, 0.0),
 RGB color (0.0, 0.0, 1.0)]
sage: color_list(L, 4)
Traceback (most recent call last):
...
ValueError: expected 4 colors, got 3!

Python

>>> from sage.all import *
>>> # needs sage.plot
>>> from sage.geometry.toric_plotter import color_list
>>> color_list("grey", Integer(1))
[RGB color (0.5019607843137255, 0.5019607843137255, 0.5019607843137255)]
>>> len(color_list("grey", Integer(3)))
3
>>> L = color_list("rainbow", Integer(3))
>>> L
[RGB color (1.0, 0.0, 0.0),
 RGB color (0.0, 1.0, 0.0),
 RGB color (0.0, 0.0, 1.0)]
>>> color_list(L, Integer(3))
[RGB color (1.0, 0.0, 0.0),
 RGB color (0.0, 1.0, 0.0),
 RGB color (0.0, 0.0, 1.0)]
>>> color_list(L, Integer(4))
Traceback (most recent call last):
...
ValueError: expected 4 colors, got 3!

sage.geometry.toric_plotter.label_list(label, n, math_mode, index_set=None)[source]#

Normalize a list of n labels.

INPUT:

label – None, a string, or a list of string;
n – an integer;
math_mode – boolean, if True, will produce LaTeX expressions for labels;
index_set – a list of integers (default: range(n)) that will be used as subscripts for labels.

OUTPUT:

a list of n labels.

If label was a list of n entries, it is returned without changes. If label is None, a list of n None’s is returned. If label is a string, a list of strings of the form $label_{i}$ is returned, where $i$ ranges over index_set. (If math_mode=False, the form “label_i” is used instead.) If n=1, there is no subscript added, unless index_set was specified explicitly.

EXAMPLES:

Sage

sage: from sage.geometry.toric_plotter import label_list
sage: label_list("u", 3, False)
['u_0', 'u_1', 'u_2']
sage: label_list("u", 3, True)
['$u_{0}$', '$u_{1}$', '$u_{2}$']
sage: label_list("u", 1, True)
['$u$']

Python

>>> from sage.all import *
>>> from sage.geometry.toric_plotter import label_list
>>> label_list("u", Integer(3), False)
['u_0', 'u_1', 'u_2']
>>> label_list("u", Integer(3), True)
['$u_{0}$', '$u_{1}$', '$u_{2}$']
>>> label_list("u", Integer(1), True)
['$u$']

sage.geometry.toric_plotter.options(option=None, **kwds)[source]#

Get or set options for plots of toric geometry objects.

Note

This function provides access to global default options. Any of these options can be overridden by passing them directly to plotting functions. See also reset_options().

INPUT:

None;

OR:

option – a string, name of the option whose value you wish to get;

OR:

keyword arguments specifying new values for one or more options.

OUTPUT:

if there was no input, the dictionary of current options for toric plots;
if option argument was given, the current value of option;
if other keyword arguments were given, none.

Name Conventions

To clearly distinguish parts of toric plots, in options and methods we use the following name conventions:

Generator: A primitive integral vector generating a 1-dimensional cone, plotted as an arrow from the origin (or a line, if the head of the arrow is beyond cut-off bounds for the plot).
Ray: A 1-dimensional cone, plotted as a line from the origin to the cut-off bounds for the plot.
Wall: A 2-dimensional cone, plotted as a region between rays (in the above sense). Its exact shape depends on the plotting mode (see below).
Chamber: A 3-dimensional cone, plotting is not implemented yet.

Plotting Modes

A plotting mode mostly determines the shape of the cut-off region (which is always relevant for toric plots except for trivial objects consisting of the origin only). The following options are available:

Box: The cut-off region is a box with edges parallel to coordinate axes.
Generators: The cut-off region is determined by primitive integral generators of rays. Note that this notion is well-defined only for rays and walls, in particular you should plot the lattice on your own (plot_lattice() will use box mode which is likely to be unsuitable). While this method may not be suitable for general fans, it is quite natural for fans of CPR-Fano toric varieties. <sage.schemes.toric.fano_variety.CPRFanoToricVariety_field
Round: The cut-off regions is a sphere centered at the origin.

Available Options

Default values for the following options can be set using this function:

mode – “box”, “generators”, or “round”, see above for descriptions;
show_lattice – boolean, whether to show lattice points in the cut-off region or not;
show_rays – boolean, whether to show rays or not;
show_generators – boolean, whether to show rays or not;
show_walls – boolean, whether to show rays or not;
generator_color – a color for generators;
label_color – a color for labels;
point_color – a color for lattice points;
ray_color – a color for rays, a list of colors (one for each ray), or the string “rainbow”;
wall_color – a color for walls, a list of colors (one for each wall), or the string “rainbow”;
wall_alpha – a number between 0 and 1, the alpha-value for walls (determining their transparency);
point_size – an integer, the size of lattice points;
ray_thickness – an integer, the thickness of rays;
generator_thickness – an integer, the thickness of generators;
font_size – an integer, the size of font used for labels;
ray_label – a string or a list of strings used for ray labels; use None to hide labels;
wall_label – a string or a list of strings used for wall labels; use None to hide labels;
radius – a positive number, the radius of the cut-off region for “round” mode;
xmin, xmax, ymin, ymax, zmin, zmax – numbers determining the cut-off region for “box” mode. Note that you cannot exclude the origin - if you try to do so, bounds will be automatically expanded to include it;
lattice_filter – a callable, taking as an argument a lattice point and returning True if this point should be included on the plot (useful, e.g. for plotting sublattices);
wall_zorder, ray_zorder, generator_zorder, point_zorder, label_zorder – integers, z-orders for different classes of objects. By default all values are negative, so that you can add other graphic objects on top of a toric plot. You may need to adjust these parameters if you want to put a toric plot on top of something else or if you want to overlap several toric plots.

You can see the current default value of any options by typing, e.g.

Sage

sage: toric_plotter.options("show_rays")
True

Python

>>> from sage.all import *
>>> toric_plotter.options("show_rays")
True

If the default value is None, it means that the actual default is determined later based on the known options. Note, that not all options can be determined in such a way, so you should not set options to None unless it was its original state. (You can always revert to this “original state” using reset_options().)

EXAMPLES:

The following line will make all subsequent toric plotting commands to draw “rainbows” from walls:

Sage

sage: toric_plotter.options(wall_color="rainbow")

Python

>>> from sage.all import *
>>> toric_plotter.options(wall_color="rainbow")

If you prefer a less colorful output (e.g. if you need black-and-white illustrations for a paper), you can use something like this:

Sage

sage: toric_plotter.options(wall_color="grey")

Python

>>> from sage.all import *
>>> toric_plotter.options(wall_color="grey")

sage.geometry.toric_plotter.reset_options()[source]#

Reset options for plots of toric geometry objects.

OUTPUT:

none.

EXAMPLES:

Sage

sage: toric_plotter.options("show_rays")
True
sage: toric_plotter.options(show_rays=False)
sage: toric_plotter.options("show_rays")
False

Python

>>> from sage.all import *
>>> toric_plotter.options("show_rays")
True
>>> toric_plotter.options(show_rays=False)
>>> toric_plotter.options("show_rays")
False

Now all toric plots will not show rays, unless explicitly requested. If you want to go back to “default defaults”, use this method:

Sage

sage: toric_plotter.reset_options()
sage: toric_plotter.options("show_rays")
True

Python

>>> from sage.all import *
>>> toric_plotter.reset_options()
>>> toric_plotter.options("show_rays")
True

sage.geometry.toric_plotter.sector(ray1, ray2, **extra_options)[source]#

Plot a sector between ray1 and ray2 centered at the origin.

Note

This function was intended for plotting strictly convex cones, so it plots the smaller sector between ray1 and ray2 and, therefore, they cannot be opposite. If you do want to use this function for bigger regions, split them into several parts.

Note

As of version 4.6 Sage does not have a graphic primitive for sectors in 3-dimensional space, so this function will actually approximate them using polygons (the number of vertices used depends on the angle between rays).

INPUT:

ray1, ray2 – rays in 2- or 3-dimensional space of the same length;
extra_options – a dictionary of options that should be passed to lower level plotting functions.

OUTPUT:

a plot.

EXAMPLES:

Sage

sage: from sage.geometry.toric_plotter import sector
sage: sector((1,0), (0,1))                                                      # needs sage.symbolic
Graphics object consisting of 1 graphics primitive
sage: sector((3,2,1), (1,2,3))                                                  # needs sage.plot
Graphics3d Object

Python

>>> from sage.all import *
>>> from sage.geometry.toric_plotter import sector
>>> sector((Integer(1),Integer(0)), (Integer(0),Integer(1)))                                                      # needs sage.symbolic
Graphics object consisting of 1 graphics primitive
>>> sector((Integer(3),Integer(2),Integer(1)), (Integer(1),Integer(2),Integer(3)))                                                  # needs sage.plot
Graphics3d Object