r"""
Toric plotter
This module provides a helper class :class:`ToricPlotter` for producing plots
of objects related to toric geometry. Default plotting objects can be adjusted
using :func:`options` and reset using :func:`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: fan = toric_varieties.dP6().fan()
sage: print fan.plot()
Graphics object consisting of 31 graphics primitives
You may change default plotting options as follows::
sage: toric_plotter.options("show_rays")
True
sage: toric_plotter.options(show_rays=False)
sage: toric_plotter.options("show_rays")
False
sage: print fan.plot()
Graphics object consisting of 19 graphics primitives
sage: toric_plotter.reset_options()
sage: toric_plotter.options("show_rays")
True
sage: print fan.plot()
Graphics object consisting of 31 graphics primitives
"""
from copy import copy
from math import pi
from sage.functions.all import arccos, arctan2, ceil, floor
from sage.geometry.polyhedron.constructor import Polyhedron
from sage.modules.all import vector
from sage.plot.all import (Color, Graphics,
arrow, disk, line, point, polygon, rainbow, text)
from sage.plot.plot3d.all import text3d
from sage.rings.all import RDF
from sage.structure.sage_object import SageObject
_default_options = dict()
_default_options["mode"] = "round"
_default_options["show_lattice"] = None
_default_options["show_rays"] = True
_default_options["show_generators"] = True
_default_options["show_walls"] = True
_default_options["generator_color"] = "blue"
_default_options["label_color"] = "black"
_default_options["point_color"] = "black"
_default_options["ray_color"] = "purple"
_default_options["wall_color"] = "rainbow"
_default_options["wall_alpha"] = 0.4
_default_options["point_size"] = None
_default_options["ray_thickness"] = 3
_default_options["generator_thickness"] = None
_default_options["font_size"] = 14
_default_options["ray_label"] = "u"
_default_options["wall_label"] = r"\sigma"
_default_options["radius"] = None
_default_options["xmin"] = None
_default_options["xmax"] = None
_default_options["ymin"] = None
_default_options["ymax"] = None
_default_options["zmin"] = None
_default_options["zmax"] = None
_default_options["lattice_filter"] = None
_default_options["wall_zorder"] = -5
_default_options["ray_zorder"] = -4
_default_options["generator_zorder"] = -3
_default_options["point_zorder"] = -2
_default_options["label_zorder"] = -1
_options = copy(_default_options)
class ToricPlotter(SageObject):
r"""
Create a toric plotter.
INPUT:
- ``all_options`` -- a :class:`dictionary <dict>`, containing any of the
options related to toric objects (see :func:`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 :class:`ToricPlotter`
directly. Instead, use plotting method of the object which you want to
plot, e.g. ::
sage: fan = toric_varieties.dP6().fan()
sage: fan.plot()
sage: print fan.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 :class:`ToricPlotter`;
- call :meth:`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 :meth:`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 :meth:`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: 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: print 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: tp = ToricPlotter(options, fan.lattice().degree(), fan.rays())
All options are exposed as attributes of toric plotters and can be modified
after constructions, however you will have to manually call
:meth:`adjust_options` and :meth:`set_rays` again if you decide to change
the plotting mode and/or cut-off bounds. Otherwise plots maybe invalid.
"""
def __init__(self, all_options, dimension, generators=None):
r"""
See :class:`ToricPlotter` for documentation.
TESTS::
sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2)
sage: TestSuite(tp).run()
"""
super(ToricPlotter, self).__init__()
sd = self.__dict__
extra_options = dict()
self.extra_options = extra_options
toric_options = options()
for option, value in all_options.iteritems():
if option in toric_options:
sd[option] = value
else:
extra_options[option] = value
for option, value in toric_options.iteritems():
if option not in sd:
sd[option] = value
if dimension not in [1, 2, 3]:
raise ValueError("toric objects can be plotted only for "
"dimensions 1, 2, and 3, not %s!" % dimension)
self.dimension = dimension
self.origin = vector(RDF, max(dimension, 2))
if self.mode not in ["box", "generators", "round"]:
raise ValueError("unrecognized plotting mode: %s!" % mode)
if sd["radius"] is not None:
for key in ["xmin", "ymin", "zmin"]:
if sd[key] is None:
sd[key] = - sd["radius"]
for key in ["xmax", "ymax", "zmax"]:
if sd[key] is None:
sd[key] = sd["radius"]
if "axes" not in extra_options:
extra_options["axes"] = False
if "frame" not in extra_options:
extra_options["frame"] = False
if "aspect_ratio" not in extra_options:
extra_options["aspect_ratio"] = 1
if generators is not None:
self.include_points(generators)
self.adjust_options()
self.set_rays(generators)
def __eq__(self, other):
r"""
Check if ``self`` is equal to ``other``.
INPUT:
- ``other`` -- anything.
OUTPUT:
- ``True`` if ``self`` is equal to ``other``, ``False`` otherwise.
TESTS::
sage: from sage.geometry.toric_plotter import ToricPlotter
sage: ToricPlotter(dict(), 2) == ToricPlotter(dict(), 2)
True
sage: ToricPlotter(dict(), 2) == 0
False
"""
return self.__dict__ == other.__dict__
def adjust_options(self):
r"""
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
:class:`ToricPlotter` for a detailed example.
OUTPUT:
- none.
TESTS::
sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2)
sage: print tp.show_lattice
None
sage: tp.adjust_options()
sage: print tp.show_lattice
True
"""
d = self.dimension
if d <= 2:
if self.point_size is None:
self.point_size = 50
elif d == 3:
if self.point_size is None:
self.point_size = 10
if self.generator_thickness is None:
self.generator_thickness = self.ray_thickness
sd = self.__dict__
bounds = ["radius", "xmin", "xmax", "ymin", "ymax", "zmin", "zmax"]
bounds = [abs(sd[bound]) for bound in bounds if sd[bound] is not None]
r = RDF(max(bounds + [0.5]) if bounds else 2.5)
self.radius = r
round = self.mode == "round"
for key in ["xmin", "ymin", "zmin"]:
if round or sd[key] is None:
sd[key] = - r
if sd[key] > - 0.5:
sd[key] = - 0.5
sd[key] = RDF(sd[key])
for key in ["xmax", "ymax", "zmax"]:
if round or sd[key] is None:
sd[key] = r
if sd[key] < 0.5:
sd[key] = 0.5
sd[key] = RDF(sd[key])
if self.show_lattice is None:
self.show_lattice = (r <= 5) if d <= 2 else r <= 3
def include_points(self, points, force=False):
r"""
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: 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...
"""
if not points:
return
points = [vector(RDF, pt) for pt in points]
sd = self.__dict__
def update(bound, new_value, points):
if force or sd[bound] is None:
new_value = eval(new_value)
if sd[bound] is None:
sd[bound] = new_value
elif abs(sd[bound]) < abs(new_value):
sd[bound] = new_value
update("radius", "max(pt.norm() for pt in points) + 0.5", points)
try:
update("xmin", "min(pt[0] for pt in points) - 0.5", points)
update("xmax", "max(pt[0] for pt in points) + 0.5", points)
update("ymin", "min(pt[1] for pt in points) - 0.5", points)
update("ymax", "max(pt[1] for pt in points) + 0.5", points)
update("zmin", "min(pt[2] for pt in points) - 0.5", points)
update("zmax", "max(pt[2] for pt in points) + 0.5", points)
except IndexError:
pass
def plot_generators(self):
r"""
Plot ray generators.
Ray generators must be specified during construction or using
:meth:`set_rays` before calling this method.
OUTPUT:
- a plot.
EXAMPLES::
sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2, [(3,4)])
sage: print tp.plot_generators()
Graphics object consisting of 1 graphics primitive
"""
generators = self.generators
result = Graphics()
if not generators or not self.show_generators:
return result
colors = color_list(self.generator_color, len(generators))
d = self.dimension
extra_options = self.extra_options
origin = self.origin
thickness = self.generator_thickness
zorder = self.generator_zorder
for generator, ray, color in zip(generators, self.rays, colors):
if ray.norm() < generator.norm():
result += line([origin, ray],
color=color, thickness=thickness,
zorder=zorder, **extra_options)
else:
if d <= 2:
result += arrow(origin, generator,
color=color, width=thickness,
arrowsize=thickness + 1,
zorder=zorder, **extra_options)
else:
result += line([origin, generator], arrow_head=True,
color=color, thickness=thickness,
zorder=zorder, **extra_options)
return result
def plot_labels(self, labels, positions):
r"""
Plot ``labels`` at specified ``positions``.
INPUT:
- ``labels`` -- a string or a list of strings;
- ``positions`` -- a list of points.
OUTPUT:
- a plot.
EXAMPLES::
sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2)
sage: print tp.plot_labels("u", [(1.5,0)])
Graphics object consisting of 1 graphics primitive
"""
result = Graphics()
color = self.label_color
extra_options = self.extra_options
zorder = self.label_zorder
font_size = self.font_size
twod = self.dimension <= 2
labels = label_list(labels, len(positions), twod)
for label, position in zip(labels, positions):
if label is None:
continue
if twod:
result += text(label, position,
color=color, fontsize=font_size,
zorder=zorder, **extra_options)
else:
result += text3d(label, position, color=color, **extra_options)
return result
def plot_lattice(self):
r"""
Plot the lattice (i.e. its points in the cut-off bounds of ``self``).
OUTPUT:
- a plot.
EXAMPLES::
sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2)
sage: print tp.plot_lattice()
Graphics object consisting of 1 graphics primitive
"""
if not self.show_lattice:
return self.plot_points([self.origin])
d = self.dimension
extra_options = self.extra_options
if d == 1:
points = ((x, 0)
for x in range(ceil(self.xmin), floor(self.xmax) + 1))
elif d == 2:
points = ((x, y)
for x in range(ceil(self.xmin), floor(self.xmax) + 1)
for y in range(ceil(self.ymin), floor(self.ymax) + 1))
elif d == 3:
points = ((x, y, z)
for x in range(ceil(self.xmin), floor(self.xmax) + 1)
for y in range(ceil(self.ymin), floor(self.ymax) + 1)
for z in range(ceil(self.zmin), floor(self.zmax) + 1))
if self.mode == "round":
r = 1.01 * self.radius
points = (pt for pt in points if vector(pt).norm() <= r)
f = self.lattice_filter
if f is not None:
points = (pt for pt in points if f(pt))
return self.plot_points(tuple(points))
def plot_points(self, points):
r"""
Plot given ``points``.
INPUT:
- ``points`` -- a list of points.
OUTPUT:
- a plot.
EXAMPLES::
sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2)
sage: print tp.plot_points([(1,0), (0,1)])
Graphics object consisting of 1 graphics primitive
"""
return point(points, color=self.point_color, size=self.point_size,
zorder=self.point_zorder, **self.extra_options)
def plot_ray_labels(self):
r"""
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
:meth:`set_rays` before calling this method.
OUTPUT:
- a plot.
EXAMPLES::
sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2, [(3,4)])
sage: print tp.plot_ray_labels()
Graphics object consisting of 1 graphics primitive
"""
return self.plot_labels(self.ray_label,
[1.1 * ray for ray in self.rays])
def plot_rays(self):
r"""
Plot rays and their labels.
Ray generators must be specified during construction or using
:meth:`set_rays` before calling this method.
OUTPUT:
- a plot.
EXAMPLES::
sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2, [(3,4)])
sage: print tp.plot_rays()
Graphics object consisting of 2 graphics primitives
"""
result = Graphics()
rays = self.rays
if not rays or not self.show_rays:
return result
extra_options = self.extra_options
origin = self.origin
colors = color_list(self.ray_color, len(rays))
thickness = self.ray_thickness
zorder = self.ray_zorder
for end, color in zip(rays, colors):
result += line([origin, end],
color=color, thickness=thickness,
zorder=zorder, **extra_options)
result += self.plot_ray_labels()
return result
def plot_walls(self, walls):
r"""
Plot ``walls``, i.e. 2-d cones, and their labels.
Ray generators must be specified during construction or using
:meth:`set_rays` before calling this method and these specified
ray generators will be used in conjunction with
:meth:`~sage.geometry.cone.ConvexRationalPolyhedralCone.ambient_ray_indices`
of ``walls``.
INPUT:
- ``walls`` -- a list of 2-d cones.
OUTPUT:
- a plot.
EXAMPLES::
sage: quadrant = Cone([(1,0), (0,1)])
sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2, quadrant.rays())
sage: print tp.plot_walls([quadrant])
Graphics object consisting of 2 graphics primitives
"""
result = Graphics()
if not walls or not self.show_walls:
return result
rays = self.rays
extra_options = self.extra_options
mode = self.mode
alpha = self.wall_alpha
colors = color_list(self.wall_color, len(walls))
zorder = self.wall_zorder
if mode == "box":
if self.dimension <= 2:
ieqs = [(self.xmax, -1, 0), (- self.xmin, 1, 0),
(self.ymax, 0, -1), (- self.ymin, 0, 1)]
else:
ieqs = [(self.xmax, -1, 0, 0), (- self.xmin, 1, 0, 0),
(self.ymax, 0, -1, 0), (- self.ymin, 0, 1, 0),
(self.zmax, 0, 0, -1), (- self.zmin, 0, 0, 1)]
box = Polyhedron(ieqs=ieqs, field=RDF)
for wall, color in zip(walls, colors):
result += box.intersection(wall.polyhedron()).render_solid(
alpha=alpha, color=color, zorder=zorder, **extra_options)
elif mode == "generators":
origin = self.origin
for wall, color in zip(walls, colors):
vertices = [rays[i] for i in wall.ambient_ray_indices()]
vertices.append(origin)
result += Polyhedron(vertices=vertices, field=RDF).render_solid(
alpha=alpha, color=color, zorder=zorder, **extra_options)
label_sectors = []
round = mode == "round"
for wall, color in zip(walls, colors):
S = wall.linear_subspace()
lsd = S.dimension()
if lsd == 0:
r1, r2 = (rays[i] for i in wall.ambient_ray_indices())
elif lsd == 1:
for i, ray in zip(wall.ambient_ray_indices(), wall.rays()):
if ray in S:
r1 = rays[i]
else:
r2 = rays[i]
if round:
result += sector(- r1, r2,
alpha=alpha, color=color, zorder=zorder, **extra_options)
else:
r1, r2 = S.basis()
r1 = vector(RDF, r1)
r1 = r1 / r1.norm() * self.radius
r2 = vector(RDF, r2)
r2 = r2 / r2.norm() * self.radius
if round:
result += sector(r1, - r2,
alpha=alpha, color=color, zorder=zorder, **extra_options)
result += sector(- r1, r2,
alpha=alpha, color=color, zorder=zorder, **extra_options)
result += sector(- r1, - r2,
alpha=alpha, color=color, zorder=zorder, **extra_options)
label_sectors.append([r1, r2])
if round:
result += sector(r1, r2,
alpha=alpha, color=color, zorder=zorder, **extra_options)
result += self.plot_labels(self.wall_label,
[sum(label_sector) / 3 for label_sector in label_sectors])
return result
def set_rays(self, generators):
r"""
Set up rays and their ``generators`` to be used by plotting functions.
As an alternative to using this method, you can pass ``generators`` to
:class:`ToricPlotter` constructor.
INPUT:
- ``generators`` - a list of primitive non-zero ray generators.
OUTPUT:
- none.
EXAMPLES::
sage: from sage.geometry.toric_plotter import ToricPlotter
sage: tp = ToricPlotter(dict(), 2)
sage: tp.adjust_options()
sage: print tp.plot_rays()
Traceback (most recent call last):
...
AttributeError: 'ToricPlotter' object has no attribute 'rays'
sage: tp.set_rays([(0,1)])
sage: print tp.plot_rays()
Graphics object consisting of 2 graphics primitives
"""
d = self.dimension
if d == 1:
generators = [vector(RDF, 2, (gen[0], 0)) for gen in generators]
else:
generators = [vector(RDF, d, gen) for gen in generators]
self.generators = generators
if self.mode == "box":
rays = []
bounds = [self.__dict__[bound]
for bound in ["xmin", "xmax", "ymin", "ymax", "zmin", "zmax"]]
bounds = bounds[:2 * d]
for gen in generators:
factors = []
for i, gen_i in enumerate(gen):
factors.append(gen_i / bounds[2 * i])
factors.append(gen_i / bounds[2 * i + 1])
rays.append(gen / max(factors))
elif self.mode == "generators":
rays = generators
elif self.mode == "round":
r = self.radius
rays = [r * gen / gen.norm() for gen in generators]
self.rays = rays
def _unrecognized_option(option):
r"""
Raise an exception about wrong ``option``.
INPUT:
- ``option`` -- a string.
OUTPUT:
- none, a ``KeyError`` exception is raised.
TESTS::
sage: from sage.geometry.toric_plotter import _unrecognized_option
sage: _unrecognized_option("nontoric")
Traceback (most recent call last):
...
KeyError: "unrecognized toric plot option: 'nontoric'!
Type 'toric_plotter.options?' to see available options."
"""
raise KeyError("unrecognized toric plot option: '%s'! " % option
+ "Type 'toric_plotter.options?' to see available options.")
def color_list(color, n):
r"""
Normalize a list of ``n`` colors.
INPUT:
- ``color`` -- anything specifying a :class:`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: 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: color_list("rainbow", 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)]
"""
try:
color = Color(color)
return [color] * n
except ValueError, TypeError:
if isinstance(color, (list, tuple)):
if len(color) != n:
raise ValueError("expected %d colors, got %d!"
% (n, len(label)))
return color
if color == "rainbow":
return [Color(c) for c in rainbow(n, "rgbtuple")]
raise TypeError("cannot interpret %s as a color!" % color)
def label_list(label, n, math_mode, index_set=None):
r"""
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: 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)
['$u$']
"""
if label is None:
return [None] * n
if isinstance(label, (list, tuple)):
if len(label) != n:
raise ValueError("expected %d labels, got %d!" % (n, len(label)))
return label
if index_set is None:
if n == 1:
return ["$%s$" % label.strip("$")] if math_mode else [label]
index_set = range(n)
if math_mode:
label = label.strip("$")
return list("$%s_{%d}$" % (label, i) for i in index_set)
else:
return list("%s_%d" % (label, i) for i in index_set)
def options(option=None, **kwds):
r"""
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 :func:`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
(:meth:`~ToricPlotter.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 :class:`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;
- ``wall_label`` -- a string or a list of strings used for wall 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: 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 :meth:`reset_options`.)
EXAMPLES:
The following line will make all subsequent toric plotting commands to draw
"rainbows" from walls::
sage: 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: toric_plotter.options(wall_color="grey")
"""
global _options
if option is None and not kwds:
return copy(_options)
elif option is not None and not kwds:
try:
return _options[option]
except KeyError:
_unrecognized_option(option)
elif option is None and kwds:
for option in kwds:
try:
_options[option] = kwds[option]
except KeyError:
_unrecognized_option(option)
else:
raise ValueError("you cannot specify 'option' and other arguments at "
"the same time!")
def reset_options():
r"""
Reset options for plots of toric geometry objects.
OUTPUT:
- none.
EXAMPLES::
sage: toric_plotter.options("show_rays")
True
sage: toric_plotter.options(show_rays=False)
sage: 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: toric_plotter.reset_options()
sage: toric_plotter.options("show_rays")
True
"""
global _options
_options = copy(_default_options)
def sector(ray1, ray2, **extra_options):
r"""
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: from sage.geometry.toric_plotter import sector
sage: print sector((1,0), (0,1))
Graphics object consisting of 1 graphics primitive
sage: print sector((3,2,1), (1,2,3))
Graphics3d Object
"""
ray1 = vector(RDF, ray1)
ray2 = vector(RDF, ray2)
r = ray1.norm()
if len(ray1) == 2:
phi1 = arctan2(ray1[1], ray1[0])
phi2 = arctan2(ray2[1], ray2[0])
if phi1 > phi2:
phi1, phi2 = phi2, phi1
if phi2 - phi1 > pi:
phi1, phi2 = phi2, phi1 + 2 * pi
return disk((0,0), r, (phi1, phi2), **extra_options)
else:
vertices_per_radian = 30
n = ceil(arccos(ray1 * ray2 / r**2) * vertices_per_radian)
dr = (ray2 - ray1) / n
points = (ray1 + i * dr for i in range(n + 1))
points = [r / pt.norm() * pt for pt in points]
points.append(vector(RDF, 3))
return polygon(points, **extra_options)
