from spb.defaults import cfg, TWO_D_B, THREE_D_B
from spb.functions import (
_set_labels, _plot3d_wireframe_helper
)
from spb.series import (
LineOver1DRangeSeries, ComplexSurfaceBaseSeries,
ComplexPointSeries, SurfaceOver2DRangeSeries,
Parametric2DLineSeries, ComplexDomainColoringSeries,
Parametric2DLineSeries, List2DSeries, GenericDataSeries,
RiemannSphereSeries
)
from spb.interactive import create_interactive_plot
from spb.utils import (
_unpack_args, _instantiate_backend, _plot_sympify, _check_arguments,
_is_range, prange, _get_free_symbols
)
from spb.vectors import plot_vector
from spb.plotgrid import plotgrid
from sympy import (latex, Tuple, sqrt, re, im, arg, Expr, Dummy, symbols, I,
sin, cos, pi)
import warnings
# NOTE:
# * `abs` refers to the absolute value;
# * `arg` refers to the argument;
# * `absarg` refers to the absolute value and argument, which will be used to
# create "domain coloring" plots.
def _build_complex_point_series(*args, allow_lambda=False, pc=False, **kwargs):
"""The following types of arguments are supported by plot_complex_list:
* plot_complex_list(n1, n2, ...) where `n-ith` is a complex number
* plot_complex_list((n1, label1, rend_kw1), (n2, label2, rend_kw2), ...)
where `n-ith` is a complex number
* plot_complex_list(l1, l2, ...) where `l-ith` is a list of complex numbers
* plot_complex_list((l1, label1, rend_kw1), (l2, label2, rend_kw2), ...)
where `l-ith` is a list of complex numbers
This function implements the above logic.
NOTE: this logic needs to be separated from the logic behind plot_complex,
plot_real_imag, otherwise there will be ambiguities.
"""
series = []
global_labels = kwargs.pop("label", [])
global_rendering_kw = kwargs.pop("rendering_kw", None)
if all([isinstance(a, Expr) for a in args]):
# args is a list of complex numbers
for a in args:
series.append(ComplexPointSeries([a], "", **kwargs))
elif (
(len(args) > 0)
and all([isinstance(a, (list, tuple, Tuple)) for a in args])
and all([len(a) > 0 for a in args])
and all([isinstance(a[0], (list, tuple, Tuple)) for a in args])
):
# args is a list of tuples of the form (list, label, rendering_kw)
# where list contains complex points
for a in args:
expr, ranges, label, rkw = _unpack_args(*a)
# Complex points do not require ranges. However, if 3 complex
# points are given inside a list, _unpack_args will see them as a
# range.
expr = expr or ranges
kw = kwargs.copy()
kw["rendering_kw"] = rkw
series.append(ComplexPointSeries(expr[0], label, **kw))
elif (
(len(args) > 0)
and all([isinstance(a, (list, tuple, Tuple)) for a in args])
and all([all([isinstance(t, Expr) and t.is_complex for t in a]) for a in args])
):
# args is a list of lists
for a in args:
series.append(ComplexPointSeries(a, "", **kwargs))
elif (
(len(args) > 0)
and all([isinstance(a, (list, tuple, Tuple)) for a in args])
and all([len(a) > 0 for a in args])
):
# args is a list of tuples of the form (number, label, rendering_kw)
# where list contains complex points
for a in args:
expr, ranges, label, rkw = _unpack_args(*a)
# Complex points do not require ranges. However, if 3 complex
# points are given inside a list, _unpack_args will see them as a
# range.
expr = expr or ranges
kw = kwargs.copy()
kw["rendering_kw"] = rkw
series.append(ComplexPointSeries([expr[0]], label, **kw))
else:
expr, ranges, label, rkw = _unpack_args(*args)
if isinstance(expr, (list, tuple, Tuple)):
expr = expr[0]
kw = kwargs.copy()
kw["rendering_kw"] = rkw
series.append(ComplexPointSeries(expr, label, **kw))
_set_labels(series, global_labels, global_rendering_kw)
return series
def _build_series(*args, interactive=False, allow_lambda=False, **kwargs):
series = []
new_args = []
global_labels = kwargs.pop("label", [])
global_rendering_kw = kwargs.pop("rendering_kw", None)
# apply the proper label.
# NOTE: the label is going to wrap the string representation of the
# expression. This design choice precludes the ability of setting latex
# labels, but this is not a problem as the user has the ability to set
# a custom alias for the function to be plotted.
mapping = {
"real": "Re(%s)",
"imag": "Im(%s)",
"abs": "Abs(%s)",
# NOTE: absarg is used to plot the absolute value colored by the
# argument. The colorbar indicates the argument, hence the following
# label is "Arg"
"absarg": "Arg(%s)",
"arg": "Arg(%s)",
}
# option to be used with lambdify with complex functions
kwargs.setdefault("modules", cfg["complex"]["modules"])
def add_series(argument):
nexpr, npar = 1, 1
if len([b for b in argument if _is_range(b)]) > 1:
# function of two variables
npar = 2
new_args.append(_check_arguments([argument], nexpr, npar, **kwargs)[0])
if all(isinstance(a, (list, tuple, Tuple)) for a in args):
# deals with the case:
# plot_complex((expr1, "name1"), (expr2, "name2"), range)
# Modify the tuples (expr, "name") to (expr, range, "name")
npar = len([b for b in args if _is_range(b)])
tmp = []
for i in range(len(args) - npar):
a = args[i]
tmp.append(a)
if len(a) == 2 and isinstance(a[-1], str):
tmp[i] = (a[0], *args[len(args) - npar:], a[-1])
# plotting multiple expressions
for a in tmp:
add_series(a)
else:
exprs, r, label, rkw = _unpack_args(*args)
for e in exprs:
add_series([e, *r, label, rkw])
params = kwargs.get("params", dict())
for a in new_args:
expr, ranges, label, rend_kw = a[0], a[1:-2], a[-2], a[-1]
if label is None:
label = str(expr)
kw = kwargs.copy()
kw["rendering_kw"] = rend_kw
if (not allow_lambda) and callable(expr):
raise TypeError("expr must be a symbolic expression.")
# NOTE:
# 1. as a design choice, a complex function will create one
# or more data series, depending on the keyword arguments
# (one for the real part, one for the imaginary part, etc.).
# This is undoubtely inefficient as we must evaluate the same
# expression multiple times. On the other hand, it allows to
# maintain a one-to-one correspondance between Plot.series
# and backend.data, making it easier to work with interactive
# widgets plot.
# 2. The expression used on each data series is the same one
# provided by the user. Each data series will receive the `return`
# keyword argument, which specify what data must be returned.
# So, if return="real", the series will return the real part
# of the function, and so on.
# Why not applying SymPy's re(), im(), arg(), ..., to the original
# expression and get rid of `return`? Because `re()` and `im()`
# evaluate the expression, usually creating new expressions
# containing many more terms, hence much slower evaluation. Instead,
# the series are going to evaluate the complex function and then
# extract the required data.
absarg = kw.pop("absarg", True)
real = kw.pop("real", False)
imag = kw.pop("imag", False)
_abs = kw.pop("abs", False)
_arg = kw.pop("arg", False)
# if ranges[0][1].imag == ranges[0][2].imag:
if im(ranges[0][1]) == im(ranges[0][2]):
# dealing with lines
def add_series(flag, key):
if flag:
kw2 = kw.copy()
kw2[key] = True
kw2["return"] = key
lbl_wrapper = mapping[key]
series.append(LineOver1DRangeSeries(expr, *ranges,
lbl_wrapper % label, **kw2))
else:
# 2D domain coloring or 3D plots
kw.setdefault("coloring", cfg["complex"]["coloring"])
def add_series(flag, key):
if flag:
kw2 = kw.copy()
kw2[key] = True
kw2["return"] = key
lbl_wrapper = mapping[key]
if key == "absarg":
lbl_wrapper = "%s"
series.append(ComplexSurfaceBaseSeries(expr, *ranges,
lbl_wrapper % label, **kw2))
add_series(absarg, "absarg")
add_series(real, "real")
add_series(imag, "imag")
add_series(_abs, "abs")
add_series(_arg, "arg")
_set_labels(series, global_labels, global_rendering_kw)
series += _plot3d_wireframe_helper(series, **kwargs)
return series
def _plot_complex(*args, allow_lambda=False, pcl=False, **kwargs):
"""Create the series and setup the backend."""
args = _plot_sympify(args)
if not pcl:
series = _build_series(*args, allow_lambda=allow_lambda, **kwargs)
else:
series = _build_complex_point_series(*args, allow_lambda=allow_lambda, pcl=True, **kwargs)
if len(series) == 0:
warnings.warn("No series found. Check your keyword arguments.")
_set_axis_labels(series, kwargs)
if any(s.is_3Dsurface for s in series):
Backend = kwargs.get("backend", THREE_D_B)
else:
Backend = kwargs.get("backend", TWO_D_B)
if any(s.is_domain_coloring for s in series):
kwargs.setdefault("legend", True)
dc_2d_series = [s for s in series if s.is_domain_coloring and not s.is_3D]
if ((len(dc_2d_series) > 0) and kwargs.get("riemann_mask", False)):
# ask the backend to add annotations on unit circle in the complex
# plane. We can't do it here because each backend requires
# different data format.
kwargs.setdefault("aouc", any(s.annotate for s in dc_2d_series))
# add unit circle: hide it from legend and requests its color
# to be black
t = symbols("t")
series.append(
Parametric2DLineSeries(cos(t), sin(t), (t, 0, 2*pi), "__k__",
adaptive=False, n=1000, use_cm=False,
show_in_legend=False))
if kwargs.get("params", None):
return create_interactive_plot(*series, **kwargs)
return _instantiate_backend(Backend, *series, **kwargs)
def _set_axis_labels(series, kwargs):
"""Set the axis labels for the plot, depending on the series being
visualized.
"""
if all(s.is_parametric for s in series):
if kwargs.get("xlabel", None) is None:
kwargs["xlabel"] = "Real"
if kwargs.get("ylabel", None) is None:
kwargs["ylabel"] = "Abs"
elif all(s.is_domain_coloring or s.is_3Dsurface or
isinstance(s, ComplexPointSeries) or
s.is_parametric for s in series):
# when plotting real/imaginary or domain coloring/3D plots, the
# horizontal axis is the real, the vertical axis is the imaginary
if kwargs.get("xlabel", None) is None:
kwargs["xlabel"] = "Re"
if kwargs.get("ylabel", None) is None:
kwargs["ylabel"] = "Im"
if kwargs.get("zlabel", None) is None and any(s.is_domain_coloring for s in series):
kwargs["zlabel"] = "Abs"
else:
var = series[0].var
if kwargs.get("xlabel", None) is None:
fx = lambda use_latex: var.name if not use_latex else latex(var)
kwargs.setdefault("xlabel", fx)
if kwargs.get("ylabel", None) is None:
wrap = lambda use_latex: "f(%s)" if not use_latex else r"f\left(%s\right)"
x = kwargs["xlabel"] if callable(kwargs["xlabel"]) else lambda use_latex: kwargs["xlabel"]
fy = lambda use_latex: wrap(use_latex) % x(use_latex)
kwargs.setdefault("ylabel", fy)
if (kwargs.get("aspect", None) is None) and any(
(s.is_complex and s.is_domain_coloring and (not s.is_3D)) or s.is_point
for s in series):
# set aspect equal for 2D domain coloring or complex points
kwargs.setdefault("aspect", "equal")
[docs]def plot_real_imag(*args, **kwargs):
"""Plot the real part, the imaginary parts, the absolute value and the
argument of a complex function. By default, only the real and imaginary
parts will be plotted. Use keyword argument to be more specific.
By default, the aspect ratio of 2D plots is set to ``aspect="equal"``.
Depending on the provided expression, this function will produce different
types of plots:
1. line plot over the reals.
2. surface plot over the complex plane if `threed=True`.
3. contour plot over the complex plane if `threed=False`.
Typical usage examples are in the followings:
- Plotting a single expression with a single range.
`plot_real_imag(expr, range, **kwargs)`
- Plotting a single expression with the default range (-10, 10).
`plot_real_imag(expr, **kwargs)`
- Plotting multiple expressions with a single range.
`plot_real_imag(expr1, expr2, ..., range, **kwargs)`
- Plotting multiple expressions with multiple ranges.
`plot_real_imag((expr1, range1), (expr2, range2), ..., **kwargs)`
- Plotting multiple expressions with custom labels and rendering options.
`plot_real_imag((expr1, range1, label1, rendering_kw1), (expr2, range2, label2, rendering_kw2), ..., **kwargs)`
Parameters
==========
args :
expr : Expr
Represent the complex function to be plotted.
range : 3-element tuple
Denotes the range of the variables. For example:
* ``(z, -5, 5)``: plot a line over the reals from point `-5` to
`5`
* ``(z, -5 + 2*I, 5 + 2*I)``: plot a line from complex point
``(-5 + 2*I)`` to ``(5 + 2 * I)``. Note the same imaginary part
for the start/end point. Also note that we can specify the
ranges by using standard Python complex numbers, for example
``(z, -5+2j, 5+2j)``.
* ``(z, -5 - 3*I, 5 + 3*I)``: surface or contour plot of the
complex function over the specified domain using a rectangular
discretization.
label : str, optional
The name of the complex function to be eventually shown on the
legend. If none is provided, the string representation of the
function will be used.
rendering_kw : dict, optional
A dictionary of keywords/values which is passed to the backend's
function to customize the appearance of lines. Refer to the
plotting library (backend) manual for more informations. Note that
the same options will be applied to all series generated for the
specified expression.
abs : boolean, optional
If True, plot the modulus of the complex function. Default to False.
adaptive : bool, optional
If ``True``, creates line plots by using an adaptive algorithm.
Use ``adaptive_goal`` and ``loss_fn`` to further customize the output.
Image and surface plots do not use an adaptive algorithm.
Default to ``False``, which uses a uniform sampling strategy.
adaptive_goal : callable, int, float or None
Controls the "smoothness" of the evaluation. Possible values:
* ``None`` (default): it will use the following goal:
``lambda l: l.loss() < 0.01``
* number (int or float). The lower the number, the more
evaluation points. This number will be used in the following goal:
``lambda l: l.loss() < number``
* callable: a function requiring one input element, the learner. It
must return a float number. Refer to [#fn0]_ for more information.
arg : boolean, optional
If True, plot the argument of the complex function. Default to False.
aspect : (float, float) or str, optional
Set the aspect ratio of the plot. The value depends on the backend
being used. Read that backend's documentation to find out the
possible values.
backend : Plot, optional
A subclass of ``Plot``, which will perform the rendering.
Default to ``MatplotlibBackend``.
colorbar : boolean, optional
Show/hide the colorbar. Only works when ``use_cm=True`` and 3D plots.
Default to True (colorbar is visible).
detect_poles : boolean, optional
Chose whether to detect and correctly plot poles. Defaulto to False.
It only works with line plots. To improve detection, increase the
number of discretization points if ``adaptive=False`` and/or change
the value of ``eps``.
eps : float, optional
An arbitrary small value used by the ``detect_poles`` algorithm.
Default value to 0.1. Before changing this value, it is better to
increase the number of discretization points.
imag : boolean, optional
If True, plot the imaginary part of the complex function.
Default to True.
label : list/tuple, optional
The labels to be shown in the legend. If not provided, the string
representation of ``expr`` will be used. The number of labels must be
equal to the number of series generated by the plotting function.
loss_fn : callable or None
The loss function to be used by the adaptive learner.
Possible values:
* ``None`` (default): it will use the ``default_loss`` from the
``adaptive`` module.
* callable : Refer to [#fn0]_ for more information. Specifically,
look at ``adaptive.learner.learner1D`` to find more loss functions.
modules : str, optional
Specify the modules to be used for the numerical evaluation. Refer to
``lambdify`` to visualize the available options. Default to None,
meaning Numpy/Scipy will be used. Note that other modules might
produce different results, based on the way they deal with branch
cuts.
n1, n2 : int, optional
Number of discretization points in the real/imaginary-directions,
respectively, when `adaptive=False`. For line plots, default to 1000.
For surface/contour plots (2D and 3D), default to 300.
n : int or two-elements tuple (n1, n2), optional
If an integer is provided, set the same number of discretization
points in all directions. If a tuple is provided, it overrides
``n1`` and ``n2``. It only works when ``adaptive=False``.
params : dict
A dictionary mapping symbols to parameters. This keyword argument
enables the interactive-widgets plot, which doesn't support the
adaptive algorithm (meaning it will use ``adaptive=False``).
Learn more by reading the documentation of the interactive sub-module.
rendering_kw : dict or list of dicts, optional
A dictionary of keywords/values which is passed to the backend's
function to customize the appearance of lines. Refer to the
plotting library (backend) manual for more informations.
If a list of dictionaries is provided, the number of dictionaries must
be equal to the number of series generated by the plotting function.
real : boolean, optional
If True, plot the real part of the complex function. Default to True.
show : boolean, optional
Default to True, in which case the plot will be shown on the screen.
size : (float, float), optional
A tuple in the form (width, height) to specify the size of
the overall figure. The default value is set to `None`, meaning
the size will be set by the backend.
surface_kw : dict, optional
A dictionary of keywords/values which is passed to the backend's
function to customize the appearance of surfaces. Refer to the
plotting library (backend) manual for more informations.
threed : boolean, optional
It only applies to a complex function over a complex range. If False,
contour plots will be shown. If True, 3D surfaces will be shown.
Default to False.
use_cm : boolean, optional
If False, surfaces will be rendered with a solid color.
If True, a color map highlighting the elevation will be used.
Default to True.
use_latex : boolean, optional
Turn on/off the rendering of latex labels. If the backend doesn't
support latex, it will render the string representations instead.
title : str, optional
Title of the plot. It is set to the latex representation of
the expression, if the plot has only one expression.
wireframe : boolean, optional
Enable or disable a wireframe over the 3D surface. Depending on the
number of wireframe lines (see ``wf_n1`` and ``wf_n2``), activating
thisoption might add a considerable overhead during the plot's
creation. Default to False (disabled).
wf_n1, wf_n2 : int, optional
Number of wireframe lines along the x and y ranges, respectively.
Default to 10. Note that increasing this number might considerably
slow down the plot's creation.
wf_npoint : int or None, optional
Number of discretization points for the wireframe lines. Default to
None, meaning that each wireframe line will have ``n1`` or ``n2``
number of points, depending on the line direction.
wf_rendering_kw : dict, optional
A dictionary of keywords/values which is passed to the backend's
function to customize the appearance of wireframe lines.
xlabel, ylabel, zlabel : str, optional
Labels for the x-axis, y-axis or z-axis, respectively.
``zlabel`` is only available for 3D plots.
xscale, yscale : 'linear' or 'log', optional
Sets the scaling of the x-axis or y-axis, respectively.
Default to ``'linear'``.
xlim, ylim, zlim : (float, float), optional
Denotes the x-axis limits, y-axis limits or z-axis limits,
respectively, ``(min, max)``. ``zlim`` is only available for 3D plots.
Examples
========
.. plot::
:context: reset
:format: doctest
:include-source: True
>>> from sympy import I, symbols, exp, sqrt, cos, sin, pi, gamma
>>> from spb import plot_real_imag
>>> x, y, z = symbols('x, y, z')
Plot the real and imaginary parts of a function over reals:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_real_imag(sqrt(x), (x, -3, 3))
Plot object containing:
[0]: cartesian line: re(sqrt(x)) for x over (-3.0, 3.0)
[1]: cartesian line: im(sqrt(x)) for x over (-3.0, 3.0)
Plot only the real part:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_real_imag(sqrt(x), (x, -3, 3), imag=False)
Plot object containing:
[0]: cartesian line: re(sqrt(x)) for x over (-3.0, 3.0)
Plot only the imaginary part:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_real_imag(sqrt(x), (x, -3, 3), real=False)
Plot object containing:
[0]: cartesian line: im(sqrt(x)) for x over (-3.0, 3.0)
Plot only the absolute value and argument:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_real_imag(sqrt(x), (x, -3, 3), real=False, imag=False, abs=True, arg=True)
Plot object containing:
[0]: cartesian line: abs(sqrt(x)) for x over (-3.0, 3.0)
[1]: cartesian line: arg(sqrt(x)) for x over (-3.0, 3.0)
Interactive-widget plot. Refer to the interactive sub-module documentation
to learn more about the ``params`` dictionary. This plot illustrates:
* the use of ``prange`` (parametric plotting range).
* for 1D ``plot_real_imag``, symbols going into ``prange`` must be real.
* the use of the ``params`` dictionary to specify sliders in
their basic form: (default, min, max).
.. panel-screenshot::
:small-size: 800, 600
from sympy import *
from spb import *
x, u = symbols("x, u")
a = symbols("a", real=True)
plot_real_imag(sqrt(x) * exp(-u * x**2), prange(x, -3*a, 3*a),
params={u: (1, 0, 2), a: (1, 0, 2)},
ylim=(-0.25, 2), use_latex=False)
3D plot of the real and imaginary part of the principal branch of a
function over a complex range. Note the jump in the imaginary part: that's
a branch cut. The rectangular discretization is unable to properly capture
it, hence the near vertical wall. Refer to ``plot3d_parametric_surface``
for an example about plotting Riemann surfaces and properly capture
the branch cuts.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_real_imag(sqrt(x), (x, -3-3j, 3+3j), n=100, threed=True,
... use_cm=True)
Plot object containing:
[0]: complex cartesian surface: re(sqrt(x)) for re(x) over (-3.0, 3.0) and im(x) over (-3.0, 3.0)
[1]: complex cartesian surface: im(sqrt(x)) for re(x) over (-3.0, 3.0) and im(x) over (-3.0, 3.0)
3D plot of the absolute value of a function over a complex range:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_real_imag(sqrt(x), (x, -3-3j, 3+3j),
... n=100, real=False, imag=False, abs=True, threed=True)
Plot object containing:
[0]: complex cartesian surface: abs(sqrt(x)) for re(x) over (-3.0, 3.0) and im(x) over (-3.0, 3.0)
Interactive-widget plot. Refer to the interactive sub-module documentation
to learn more about the ``params`` dictionary. This plot illustrates:
* the use of ``prange`` (parametric plotting range).
* the use of the ``params`` dictionary to specify sliders in
their basic form: (default, min, max).
.. panel-screenshot::
:small-size: 800, 600
from sympy import *
from spb import *
x, u, a, b = symbols("x, u, a, b")
plot_real_imag(
sqrt(x) * exp(u * x), prange(x, -3*a-b*3j, 3*a+b*3j),
backend=PB, aspect="cube",
wireframe=True, wf_rendering_kw={"line_width": 1},
params={
u: (0.25, 0, 1),
a: (1, 0, 2),
b: (1, 0, 2)
}, n=25, threed=True, use_latex=False, use_cm=True)
References
==========
.. [#fn0] https://github.com/python-adaptive/adaptive
See Also
========
plot_complex, plot_complex_list, plot_complex_vector
"""
kwargs["absarg"] = False
kwargs.setdefault("abs", False)
kwargs.setdefault("arg", False)
kwargs.setdefault("real", True)
kwargs.setdefault("imag", True)
return _plot_complex(*args, **kwargs)
[docs]def plot_complex(*args, **kwargs):
"""Plot the absolute value of a complex function colored by its argument.
By default, the aspect ratio of 2D plots is set to ``aspect="equal"``.
Depending on the provided range, this function will produce different
types of plots:
1. Line plot over the reals.
2. Image plot over the complex plane if ``threed=False``. This is also
known as Domain Coloring. Use the ``coloring`` keyword argument to
select a different coloring strategy and ``cmap`` to set a custom
color map (default to HSV).
3. If ``threed=True``, plot a 3D surface of the absolute value over the
complex plane, colored by its argument. Use the ``coloring`` keyword
argument to select a different coloring strategy and ``cmap`` to set
a custom color map (default to HSV).
Read the `Notes` section to learn more about colormaps.
Typical usage examples are in the followings:
- Plotting a single expression with a single range.
`plot_complex(expr, range, **kwargs)`
- Plotting a single expression with the default range (-10, 10).
`plot_complex(expr, **kwargs)`
- Plotting multiple expressions with a single range.
`plot_complex(expr1, expr2, ..., range, **kwargs)`
- Plotting multiple expressions with multiple ranges.
`plot_complex((expr1, range1), (expr2, range2), ..., **kwargs)`
- Plotting multiple expressions with custom labels and rendering options.
`plot_complex((expr1, range1, label1, rendering_kw1), (expr2, range2, label2, rendering_kw2), ..., **kwargs)`
Parameters
==========
args :
expr : Expr or callable
Represent the complex function to be plotted. It can be a:
* Symbolic expression.
* Numerical function of one variable, supporting vectorization.
In this case the following keyword arguments are not supported:
``params``.
range : 3-element tuple
Denotes the range of the variables. For example:
* ``(z, -5, 5)``: plot a line over the reals from point `-5` to
`5`
* ``(z, -5 + 2*I, 5 + 2*I)``: plot a line from complex point
``(-5 + 2*I)`` to ``(5 + 2 * I)``. Note the same imaginary part
for the start/end point. Also note that we can specify the
ranges by using standard Python complex numbers, for example
``(z, -5+2j, 5+2j)``.
* ``(z, -5 - 3*I, 5 + 3*I)``: surface or contour plot of the
complex function over the specified domain.
label : str, optional
The name of the complex function to be eventually shown on the
legend. If none is provided, the string representation of the
function will be used.
rendering_kw : dict, optional
A dictionary of keywords/values which is passed to the backend's
function to customize the appearance of lines, surfaces or images.
Refer to the plotting library (backend) manual for more
informations.
adaptive : bool, optional
If ``True``, creates line plots by using an adaptive algorithm.
Use ``adaptive_goal`` and ``loss_fn`` to further customize the output.
Image and surface plots do not use an adaptive algorithm.
Default to ``False``, which uses a uniform sampling strategy.
adaptive_goal : callable, int, float or None
Controls the "smoothness" of the evaluation. Possible values:
* ``None`` (default): it will use the following goal:
``lambda l: l.loss() < 0.01``
* number (int or float). The lower the number, the more
evaluation points. This number will be used in the following goal:
``lambda l: l.loss() < number``
* callable: a function requiring one input element, the learner. It
must return a float number. Refer to [#fn2]_ for more information.
aspect : (float, float) or str, optional
Set the aspect ratio of the plot. The value depends on the backend
being used. Read that backend's documentation to find out the
possible values.
at_infinity : boolean, optional
Apply the transformation $z \\rightarrow \\frac{1}{z}$ in order to
study the behaviour of the function around the point at infinity.
It is recommended to also set ``show_axis=False`` in order to avoid
confusion.
backend : Plot, optional
A subclass of ``Plot``, which will perform the rendering.
Default to ``MatplotlibBackend``.
blevel : float, optional
Controls the black level of ehanced domain coloring plots. It must be
`0 (black) <= blevel <= 1 (white)`. Default to 0.75.
cmap : str, iterable, optional
Specify the colormap to be used on enhanced domain coloring plots
(both images and 3d plots). Default to ``"hsv"``. Can be any colormap
from matplotlib or colorcet.
colorbar : boolean, optional
Show/hide the colorbar. Default to True (colorbar is visible).
coloring : str or callable, optional
Choose between different domain coloring options. Default to ``"a"``.
Refer to [#fn1]_ for more information.
- ``"a"``: standard domain coloring showing the argument of the
complex function.
- ``"b"``: enhanced domain coloring showing iso-modulus and iso-phase
lines.
- ``"c"``: enhanced domain coloring showing iso-modulus lines.
- ``"d"``: enhanced domain coloring showing iso-phase lines.
- ``"e"``: alternating black and white stripes corresponding to
modulus.
- ``"f"``: alternating black and white stripes corresponding to
phase.
- ``"g"``: alternating black and white stripes corresponding to
real part.
- ``"h"``: alternating black and white stripes corresponding to
imaginary part.
- ``"i"``: cartesian chessboard on the complex points space. The
result will hide zeros.
- ``"j"``: polar Chessboard on the complex points space. The result
will show conformality.
- ``"k"``: black and white magnitude of the complex function.
Zeros are black, poles are white.
- ``"l"``:enhanced domain coloring showing iso-modulus and iso-phase
lines, blended with the magnitude: white regions indicates greater
magnitudes. Can be used to distinguish poles from zeros.
- ``"m"``: enhanced domain coloring showing iso-modulus lines, blended
with the magnitude: white regions indicates greater magnitudes.
Can be used to distinguish poles from zeros.
- ``"n"``: enhanced domain coloring showing iso-phase lines, blended
with the magnitude: white regions indicates greater magnitudes.
Can be used to distinguish poles from zeros.
- ``"o"``: enhanced domain coloring showing iso-phase lines, blended
with the magnitude: white regions indicates greater magnitudes.
Can be used to distinguish poles from zeros.
The user can also provide a callable, ``f(w)``, where ``w`` is an
[n x m] Numpy array (provided by the plotting module) containing
the results (complex numbers) of the evaluation of the complex
function. The callable should return:
- img : ndarray [n x m x 3]
An array of RGB colors (0 <= R,G,B <= 255)
- colorscale : ndarray [N x 3] or None
An array with N RGB colors, (0 <= R,G,B <= 255).
If ``colorscale=None``, no color bar will be shown on the plot.
label : str or list/tuple, optional
The label to be shown in the legend or colorbar in case of a line plot.
If not provided, the string representation of ``expr`` will be used.
The number of labels must be equal to the number of expressions.
loss_fn : callable or None
The loss function to be used by the adaptive learner.
Possible values:
* ``None`` (default): it will use the ``default_loss`` from the
``adaptive`` module.
* callable : Refer to [#fn2]_ for more information. Specifically,
look at `adaptive.learner.learner1D` to find more loss functions.
modules : str, optional
Specify the modules to be used for the numerical evaluation. Refer to
``lambdify`` to visualize the available options. Default to None,
meaning Numpy/Scipy will be used. Note that other modules might
produce different results, based on the way they deal with branch
cuts.
n1, n2 : int, optional
Number of discretization points in the real/imaginary-directions,
respectively, when ``adaptive=False``. For line plots, default to 1000.
For surface/contour plots (2D and 3D), default to 300.
n : int or two-elements tuple (n1, n2), optional
If an integer is provided, set the same number of discretization
points in all directions. If a tuple is provided, it overrides
``n1`` and ``n2``. It only works when ``adaptive=False``.
params : dict, optional
A dictionary mapping symbols to parameters. This keyword argument
enables the interactive-widgets plot, which doesn't support the
adaptive algorithm (meaning it will use ``adaptive=False``).
Learn more by reading the documentation of the interactive sub-module.
phaseres : int, optional
Default value to 20. It controls the number of iso-phase and/or
iso-modulus lines in domain coloring plots.
phaseoffset : float, optional
Controls the phase offset of the colormap in domain coloring plots.
Default to 0.
rendering_kw : dict or list of dicts, optional
A dictionary of keywords/values which is passed to the backend's
function to customize the appearance of lines, surfaces or images.
Refer to the plotting library (backend) manual for more informations.
If a list of dictionaries is provided, the number of dictionaries must
be equal to the number of series generated by the plotting function.
show : boolean, optional
Default to True, in which case the plot will be shown on the screen.
show_axis : boolean, optional
Turn on/off the axis of the plot. Default to True (axis are visible).
size : (float, float), optional
A tuple in the form (width, height) to specify the size of
the overall figure. The default value is set to ``None``, meaning
the size will be set by the backend.
threed : boolean, optional
It only applies to a complex function over a complex range. If False,
a 2D image plot will be shown. If True, 3D surfaces will be shown.
Default to False.
title : str, optional
Title of the plot. It is set to the latex representation of
the expression, if the plot has only one expression.
use_latex : boolean, optional
Turn on/off the rendering of latex labels. If the backend doesn't
support latex, it will render the string representations instead.
xlabel, ylabel, zlabel : str, optional
Labels for the x-axis, y-axis or z-axis, respectively.
``zlabel`` is only available for 3D plots.
xscale, yscale : 'linear' or 'log', optional
Sets the scaling of the x-axis or y-axis, respectively.
Default to ``'linear'``.
xlim, ylim, zlim : (float, float), optional
Denotes the x-axis limits, y-axis limits or z-axis limits,
respectively, ``(min, max)``. ``zlim`` is only available for 3D plots.
Examples
========
.. plot::
:context: reset
:format: doctest
:include-source: True
>>> from sympy import I, symbols, exp, sqrt, cos, sin, pi, gamma
>>> from spb import plot_complex
>>> x, y, z = symbols('x, y, z')
Plot the modulus of a complex function colored by its magnitude:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_complex(cos(x) + sin(I * x), "f", (x, -2, 2))
Plot object containing:
[0]: cartesian abs-arg line: cos(x) + I*sinh(x) for x over ((-2+0j), (2+0j))
Interactive-widget plot of a Fourier Transform. Refer to the interactive
sub-module documentation to learn more about the ``params`` dictionary.
This plot illustrates:
* the use of ``prange`` (parametric plotting range).
* for ``plot_complex``, symbols going into ``prange`` must be real.
* the use of the ``params`` dictionary to specify sliders in
their basic form: (default, min, max).
.. panel-screenshot::
:small-size: 800, 600
from sympy import *
from spb import *
x, k, a, b = symbols("x, k, a, b")
c = symbols("c", real=True)
f = exp(-x**2) * (Heaviside(x + a) - Heaviside(x - b))
fs = fourier_transform(f, x, k)
plot_complex(fs, prange(k, -c, c),
params={a: (1, -2, 2), b: (-2, -2, 2), c: (4, 0.5, 4)},
label="Arg(fs)", xlabel="k", yscale="log", ylim=(1e-03, 10),
use_latex=False)
Domain coloring plot. To improve the smoothness of the results, increase
the number of discretization points and/or apply an interpolation (if the
backend supports it):
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_complex(gamma(z), (z, -3-3j, 3+3j),
... coloring="b", n=500, grid=False)
Plot object containing:
[0]: complex domain coloring: gamma(z) for re(z) over (-3.0, 3.0) and im(z) over (-3.0, 3.0)
Domain coloring of the same function evaluated near the point
$z=\\infty$:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_complex(gamma(z), (z, -1-1j, 1+1j), coloring="b", n=500,
... grid=False, at_infinity=True, show_axis=False)
Plot object containing:
[0]: complex domain coloring: gamma(1/z) for re(z) over (-1.0, 1.0) and im(z) over (-1.0, 1.0)
Interactive-widget domain coloring plot. Refer to the interactive
sub-module documentation to learn more about the ``params`` dictionary.
This plot illustrates:
* setting a custom colormap and adjusting the black-level of the enhanced
visualization.
* the use of ``prange`` (parametric plotting range).
* the use of the ``params`` dictionary to specify sliders in
their basic form: (default, min, max).
.. panel-screenshot::
:small-size: 800, 600
from sympy import *
from spb import *
import colorcet
z, u, a, b = symbols("z, u, a, b")
plot_complex(
sin(u * z), prange(z, -a - b*I, a + b*I),
cmap=colorcet.colorwheel, blevel=0.85, use_latex=False,
coloring="b", n=250, grid=False,
params={
u: (0.5, 0, 2),
a: (pi, 0, 2*pi),
b: (pi, 0, 2*pi),
})
The analytic landscape is 3D plot of the absolute value of a complex
function colored by its argument:
.. plotly::
:context: reset
from sympy import symbols, gamma, I
from spb import plot_complex, PB
z = symbols('z')
plot_complex(gamma(z), (z, -3 - 3*I, 3 + 3*I), threed=True,
backend=PB, zlim=(-1, 6), use_cm=True)
Because the function goes to infinity at poles, sometimes it might be
beneficial to visualize the logarithm of the absolute value in order to
easily identify zeros:
.. k3d-screenshot::
:camera: -4.28, 6.55, 4.83, 0.13, -0.20, 1.9, 0.16, -0.24, 0.96
from sympy import symbols, I
from spb import plot_complex, KB
z = symbols("z")
expr = (z**3 - 5) / z
plot_complex(expr, (z, -3-3j, 3+3j), coloring="b", threed=True,
use_cm=True, grid=False, n=500, backend=KB, tz=np.log)
Notes
=====
By default, a domain coloring plot will show the phase portrait: each point
of the complex plane is color-coded according to its argument. The default
colormap is HSV, which is characterized by 2 important problems:
* It is not friendly to people affected by color deficiencies.
* Because it isn't perceptually uniform, it might be misleading: features
disappear at points of low perceptual contrast, or false features appear
that are in the colormap but not in the data (refer to colorcet [#fn3]_
for more information).
Hence, it might be helpful to chose a perceptually uniform colormap.
Domaing coloring plots are naturally suited to be represented by cyclic
colormaps, but sequential colormaps can be used too. In the following
example we illustrate the phase portrait of `f(z) = z` using different
colormaps:
.. plot::
:context: close-figs
:include-source: True
from sympy import symbols, pi
import colorcet
from spb import *
z = symbols("z")
cmaps = {
"hsv": "hsv",
"twilight": "twilight",
"colorwheel": colorcet.colorwheel,
"CET-C7": colorcet.CET_C7,
"viridis": "viridis"
}
plots = []
for k, v in cmaps.items():
plots.append(
plot_complex(z, (z, -2-2j, 2+2j), coloring="a",
grid=False, show=False, legend=True, cmap=v, title=k))
plotgrid(*plots, nc=2, size=(6.5, 8))
In the above figure, when using the HSV colormap the eye is drawn to
the yellow, cyan and magenta colors, where there is a lightness gradient:
those are false features caused by the colormap. Indeed, there is nothing
going on these regions when looking with a perceptually uniform colormap.
Phase is computed with Numpy: it lies between [-pi, pi]. Then, phase is
normalized between [0, 1] using `(arg / (2 * pi)) % 1`. The figure
below shows the mapping between phase in radians and normalized phase.
A phase of 0 radians corresponds to a normalized phase of 0, which gets
mapped to the beginning of a colormap.
.. plot:: ./modules/plot_complex_explanation.py
:context: close-figs
:include-source: False
The zero radians phase is then located in the middle of the colorbar.
Hence, the colorbar might feel "weird" if a sequential colormap is chosen,
because there is a color-discontinuity in the middle of it, as can be seen
in the previous example.
The ``phaseoffset`` keyword argument allows to adjust the position of
the colormap:
.. plot::
:context: close-figs
:include-source: True
p1 = plot_complex(
z, (z, -2-2j, 2+2j), grid=False, show=False, legend=True,
coloring="a", cmap="viridis", phaseoffset=0,
title="phase offset = 0", show_axis=False)
p2 = plot_complex(
z, (z, -2-2j, 2+2j), grid=False, show=False, legend=True,
coloring="a", cmap="viridis", phaseoffset=pi,
title=r"phase offset = $\pi$", show_axis=False)
plotgrid(p1, p2, nc=2, size=(6, 2))
A pure phase portrait is rarely useful, as it conveys too little
information. Let's now quickly visualize the different ``coloring``
schemes. In the following, `arg` is the argument (phase), `mag` is the
magnitude (absolute value) and `contour` is a line of constant value.
Refer to [#fn1]_ for more information.
.. plot::
:context: close-figs
:include-source: True
from matplotlib import rcParams
rcParams["font.size"] = 8
colorings = "abcdlmnoefghijk"
titles = [
"phase portrait", "mag + arg contours", "mag contours", "arg contours",
"'a' + poles", "'b' + poles", "'c' + poles", "'d' + poles",
"mag stripes", "arg stripes", "real part stripes", "imag part stripes",
"hide zeros", "conformality", "magnitude"]
plots = []
expr = (z - 1) / (z**2 + z + 1)
for c, t in zip(colorings, titles):
plots.append(
plot_complex(expr, (z, -2-2j, 2+2j), coloring=c,
grid=False, show=False, legend=False, cmap=colorcet.CET_C7,
title=("'%s'" % c) + ": " + t, xlabel="", ylabel=""))
plotgrid(*plots, nc=4, size=(8, 8.5))
From the above picture, we can see that:
* Some enhancements decrese the lighness of the colors: depending on the
colormap, it might be difficult to distinguish features in darker
regions.
* Other enhancements increases the lightness in proximity of poles. Hence,
colormaps with very light colors might not convey enough information.
The selection of a proper colormap is left to the user because not only
it depends on the target audience of the visualization, but also on the
function being visualized.
References
==========
.. [#fn1] Domain Coloring is based on Elias Wegert's book
`"Visual Complex Functions" <https://www.springer.com/de/book/9783034801799>`_.
The book provides the background to better understand the images.
.. [#fn2] https://github.com/python-adaptive/adaptive
.. [#fn3] https://colorcet.com/
See Also
========
plot_riemann_sphere, plot_real_imag, plot_complex_list, plot_complex_vector
"""
kwargs["absarg"] = True
kwargs["real"] = False
kwargs["imag"] = False
kwargs["abs"] = False
kwargs["arg"] = False
return _plot_complex(*args, allow_lambda=True, **kwargs)
[docs]def plot_complex_list(*args, **kwargs):
"""Plot lists of complex points. By default, the aspect ratio of the plot
is set to ``aspect="equal"``.
Typical usage examples are in the followings:
- Plotting a single list of complex numbers.
`plot_complex_list(l1, **kwargs)`
- Plotting multiple lists of complex numbers.
`plot_complex_list(l1, l2, **kwargs)`
- Plotting multiple lists of complex numbers each one with a custom label.
`plot_complex_list((l1, label1), (l2, label2), **kwargs)`
Parameters
==========
args :
numbers : list, tuple
A list of complex numbers.
label : str
The name associated to the list of the complex numbers to be
eventually shown on the legend. Default to empty string.
rendering_kw : dict, optional
A dictionary of keywords/values which is passed to the backend's
function to customize the appearance of lines. Refer to the
plotting library (backend) manual for more informations. Note that
the same options will be applied to all series generated for the
specified expression.
aspect : (float, float) or str, optional
Set the aspect ratio of the plot. The value depends on the backend
being used. Read that backend's documentation to find out the
possible values.
backend : Plot, optional
A subclass of ``Plot``, which will perform the rendering.
Default to ``MatplotlibBackend``.
is_point : boolean
If True, a scatter plot will be produced. Otherwise a line plot will
be created. Default to True.
is_filled : boolean, optional
Default to True, which will render empty circular markers. It only
works if ``is_point=True``.
If False, filled circular markers will be rendered.
label : str or list/tuple, optional
The name associated to the list of the complex numbers to be
eventually shown on the legend. The number of labels must be equal to
the number of series generated by the plotting function.
params : dict
A dictionary mapping symbols to parameters. This keyword argument
enables the interactive-widgets plot, which doesn't support the
adaptive algorithm (meaning it will use ``adaptive=False``).
Learn more by reading the documentation of the interactive sub-module.
rendering_kw : dict or list of dicts, optional
A dictionary of keywords/values which is passed to the backend's
function to customize the appearance of lines. Refer to the
plotting library (backend) manual for more informations.
If a list of dictionaries is provided, the number of dictionaries must
be equal to the number of series generated by the plotting function.
show : boolean
Default to True, in which case the plot will be shown on the screen.
size : (float, float), optional
A tuple in the form (width, height) to specify the size of
the overall figure. The default value is set to `None`, meaning
the size will be set by the backend.
title : str, optional
Title of the plot. It is set to the latex representation of
the expression, if the plot has only one expression.
use_latex : boolean, optional
Turn on/off the rendering of latex labels. If the backend doesn't
support latex, it will render the string representations instead.
xlabel, ylabel : str, optional
Labels for the x-axis or y-axis, respectively.
xscale, yscale : 'linear' or 'log', optional
Sets the scaling of the x-axis or y-axis, respectively.
Default to ``'linear'``.
xlim, ylim : (float, float), optional
Denotes the x-axis limits or y-axis limits, respectively,
``(min, max)``.
Examples
========
.. plot::
:context: reset
:format: doctest
:include-source: True
>>> from sympy import I, symbols, exp, sqrt, cos, sin, pi, gamma
>>> from spb import plot_complex_list
>>> x, y, z = symbols('x, y, z')
Plot individual complex points:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_complex_list(3 + 2 * I, 4 * I, 2)
Plot object containing:
[0]: complex points: (3 + 2*I,)
[1]: complex points: (4*I,)
[2]: complex points: (2,)
Plot two lists of complex points and assign to them custom labels:
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> expr1 = z * exp(2 * pi * I * z)
>>> expr2 = 2 * expr1
>>> n = 15
>>> l1 = [expr1.subs(z, t / n) for t in range(n)]
>>> l2 = [expr2.subs(z, t / n) for t in range(n)]
>>> plot_complex_list((l1, "f1"), (l2, "f2"))
Plot object containing:
[0]: complex points: (0.0, 0.0666666666666667*exp(0.133333333333333*I*pi), 0.133333333333333*exp(0.266666666666667*I*pi), 0.2*exp(0.4*I*pi), 0.266666666666667*exp(0.533333333333333*I*pi), 0.333333333333333*exp(0.666666666666667*I*pi), 0.4*exp(0.8*I*pi), 0.466666666666667*exp(0.933333333333333*I*pi), 0.533333333333333*exp(1.06666666666667*I*pi), 0.6*exp(1.2*I*pi), 0.666666666666667*exp(1.33333333333333*I*pi), 0.733333333333333*exp(1.46666666666667*I*pi), 0.8*exp(1.6*I*pi), 0.866666666666667*exp(1.73333333333333*I*pi), 0.933333333333333*exp(1.86666666666667*I*pi))
[1]: complex points: (0, 0.133333333333333*exp(0.133333333333333*I*pi), 0.266666666666667*exp(0.266666666666667*I*pi), 0.4*exp(0.4*I*pi), 0.533333333333333*exp(0.533333333333333*I*pi), 0.666666666666667*exp(0.666666666666667*I*pi), 0.8*exp(0.8*I*pi), 0.933333333333333*exp(0.933333333333333*I*pi), 1.06666666666667*exp(1.06666666666667*I*pi), 1.2*exp(1.2*I*pi), 1.33333333333333*exp(1.33333333333333*I*pi), 1.46666666666667*exp(1.46666666666667*I*pi), 1.6*exp(1.6*I*pi), 1.73333333333333*exp(1.73333333333333*I*pi), 1.86666666666667*exp(1.86666666666667*I*pi))
Interactive-widget plot. Refer to the interactive sub-module documentation
to learn more about the ``params`` dictionary.
.. panel-screenshot::
:small-size: 800, 600
from sympy import *
from spb import *
z, u = symbols("z u")
expr1 = z * exp(2 * pi * I * z)
expr2 = u * expr1
n = 15
l1 = [expr1.subs(z, t / n) for t in range(n)]
l2 = [expr2.subs(z, t / n) for t in range(n)]
plot_complex_list(
(l1, "f1"), (l2, "f2"),
params={u: (0.5, 0, 2)}, use_latex=False,
xlim=(-1.5, 2), ylim=(-2, 1))
See Also
========
plot_real_imag, plot_complex, plot_complex_vector
"""
kwargs["absarg"] = False
kwargs["abs"] = False
kwargs["arg"] = False
kwargs["real"] = False
kwargs["imag"] = False
kwargs["threed"] = False
return _plot_complex(*args, allow_lambda=False, pcl=True, **kwargs)
[docs]def plot_complex_vector(*args, **kwargs):
"""Plot the vector field `[re(f), im(f)]` for a complex function `f`
over the specified complex domain. By default, the aspect ratio of 2D
plots is set to ``aspect="equal"``.
Typical usage examples are in the followings:
- Plotting a vector field of a complex function.
`plot_complex_vector(expr, range, **kwargs)`
- Plotting multiple vector fields with different ranges and custom labels.
`plot_complex_vector((expr1, range1, label1 [optional]), (expr2, range2, label2 [optional]), **kwargs)`
Parameters
==========
args :
expr : Expr
Represent the complex function.
range : 3-element tuples
Denotes the range of the variables. For example
``(z, -5 - 3*I, 5 + 3*I)``. Note that we can specify the range
by using standard Python complex numbers, for example
``(z, -5-3j, 5+3j)``.
label : str, optional
The name of the complex expression to be eventually shown on the
legend. If none is provided, the string representation of the
expression will be used.
aspect : (float, float) or str, optional
Set the aspect ratio of the plot. The value depends on the backend
being used. Read that backend's documentation to find out the
possible values.
backend : Plot, optional
A subclass of `Plot`, which will perform the rendering.
Default to `MatplotlibBackend`.
colorbar : boolean, optional
Show/hide the colorbar. Default to True (colorbar is visible).
contours_kw : dict
A dictionary of keywords/values which is passed to the backend
contour function to customize the appearance. Refer to the plotting
library (backend) manual for more informations.
n1, n2 : int
Number of discretization points for the quivers or streamlines in the
x/y-direction, respectively. Default to 25.
n : int or two-elements tuple (n1, n2), optional
If an integer is provided, set the same number of discretization
points in all directions for quivers or streamlines. If a tuple is
provided, it overrides ``n1`` and ``n2``. It only works when
``adaptive=False``. Default to 25.
nc : int
Number of discretization points for the scalar contour plot.
Default to 100.
params : dict
A dictionary mapping symbols to parameters. This keyword argument
enables the interactive-widgets plot, which doesn't support the
adaptive algorithm (meaning it will use ``adaptive=False``).
Learn more by reading the documentation of the interactive sub-module.
quiver_kw : dict
A dictionary of keywords/values which is passed to the backend
quivers-plotting function to customize the appearance. Refer to the
plotting library (backend) manual for more informations.
scalar : boolean, Expr, None or list/tuple of 2 elements
Represents the scalar field to be plotted in the background of a 2D
vector field plot. Can be:
- ``True``: plot the magnitude of the vector field. Only works when a
single vector field is plotted.
- ``False``/``None``: do not plot any scalar field.
- ``Expr``: a symbolic expression representing the scalar field.
- ``list``/``tuple``: [scalar_expr, label], where the label will be
shown on the colorbar.
Remember: the scalar function must return real data.
Default to True.
show : boolean
The default value is set to ``True``. Set show to ``False`` and
the function will not display the plot. The returned instance of
the ``Plot`` class can then be used to save or display the plot
by calling the ``save()`` and ``show()`` methods respectively.
size : (float, float), optional
A tuple in the form (width, height) to specify the size of
the overall figure. The default value is set to ``None``, meaning
the size will be set by the backend.
streamlines : boolean
Whether to plot the vector field using streamlines (True) or quivers
(False). Default to False.
stream_kw : dict
A dictionary of keywords/values which is passed to the backend
streamlines-plotting function to customize the appearance. Refer to
the plotting library (backend) manual for more informations.
title : str, optional
Title of the plot. It is set to the latex representation of
the expression, if the plot has only one expression.
use_latex : boolean, optional
Turn on/off the rendering of latex labels. If the backend doesn't
support latex, it will render the string representations instead.
xlabel, ylabel : str, optional
Labels for the x-axis or y-axis, respectively.
xscale, yscale : 'linear' or 'log', optional
Sets the scaling of the x-axis or y-axis, respectively.
Default to ``'linear'``.
xlim, ylim, zlim : (float, float), optional
Denotes the x-axis limits ory-axis limits, respectively,
``(min, max)``.
Examples
========
.. plot::
:context: reset
:format: doctest
:include-source: True
>>> from sympy import I, symbols, gamma, latex, log
>>> from spb import plot_complex_vector, plot_complex
>>> z = symbols('z')
Quivers plot with normalize lengths and a contour plot in background
representing the vector's magnitude (a scalar field).
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> expr = z**2 + 2
>>> plot_complex_vector(expr, (z, -5 - 5j, 5 + 5j),
... quiver_kw=dict(color="orange"), normalize=True, grid=False)
Plot object containing:
[0]: contour: sqrt(4*(re(_x) - im(_y))**2*(re(_y) + im(_x))**2 + ((re(_x) - im(_y))**2 - (re(_y) + im(_x))**2 + 2)**2) for _x over (-5.0, 5.0) and _y over (-5.0, 5.0)
[1]: 2D vector series: [(re(_x) - im(_y))**2 - (re(_y) + im(_x))**2 + 2, 2*(re(_x) - im(_y))*(re(_y) + im(_x))] over (_x, -5.0, 5.0), (_y, -5.0, 5.0)
Only quiver plot with normalized lengths and solid color.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_complex_vector(expr, (z, -5 - 5j, 5 + 5j),
... scalar=False, use_cm=False, normalize=True)
Plot object containing:
[0]: 2D vector series: [(re(_x) - im(_y))**2 - (re(_y) + im(_x))**2 + 2, 2*(re(_x) - im(_y))*(re(_y) + im(_x))] over (_x, -5.0, 5.0), (_y, -5.0, 5.0)
Only streamlines plot.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> plot_complex_vector(expr, (z, -5 - 5j, 5 + 5j),
... "Magnitude of $%s$" % latex(expr),
... scalar=False, streamlines=True)
Plot object containing:
[0]: 2D vector series: [(re(_x) - im(_y))**2 - (re(_y) + im(_x))**2 + 2, 2*(re(_x) - im(_y))*(re(_y) + im(_x))] over (_x, -5.0, 5.0), (_y, -5.0, 5.0)
Overlay the quiver plot to a domain coloring plot. By setting ``n=26``
(even number) in the complex vector plot, the quivers won't to cross
the branch cut.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> expr = z * log(2 * z) + 3
>>> p1 = plot_complex(expr, (z, -2-2j, 2+2j), grid=False, show=False)
>>> p2 = plot_complex_vector(expr, (z, -2-2j, 2+2j),
... n=26, grid=False, scalar=False, use_cm=False, normalize=True,
... quiver_kw={"color": "k", "pivot": "tip"}, show=False)
>>> (p1 + p2).show()
>>> (p1 + p2)
Plot object containing:
[0]: complex domain coloring: z*log(2*z) + 3 for re(z) over (-2.0, 2.0) and im(z) over (-2.0, 2.0)
[1]: 2D vector series: [(re(_x) - im(_y))*log(Abs(2*_x + 2*_y*I)) - (re(_y) + im(_x))*arg(_x + _y*I) + 3, (re(_x) - im(_y))*arg(_x + _y*I) + (re(_y) + im(_x))*log(Abs(2*_x + 2*_y*I))] over (_x, -2.0, 2.0), (_y, -2.0, 2.0)
Interactive-widget plot. Refer to the interactive sub-module documentation
to learn more about the ``params`` dictionary. This plot illustrates:
* the use of ``prange`` (parametric plotting range).
* the use of the ``params`` dictionary to specify sliders in
their basic form: (default, min, max).
.. panel-screenshot::
:small-size: 800, 600
from sympy import *
from spb import *
z, u, a, b = symbols("z u a b")
plot_complex_vector(
log(gamma(u * z)), prange(z, -5*a - b*5j, 5*a + b*5j),
params={
u: (1, 0, 2),
a: (1, 0, 2),
b: (1, 0, 2)
}, n=20, grid=False, use_latex=False,
quiver_kw=dict(color="orange", headwidth=4))
See Also
========
plot_real_imag, plot_complex, plot_complex_list, plot_vector
"""
# for each argument, generate one series. Those series will be used to
# generate the proper input arguments for plot_vector
kwargs["absarg"] = False
kwargs["abs"] = False
kwargs["arg"] = False
kwargs["real"] = True
kwargs["imag"] = False
kwargs["threed"] = False
kwargs.setdefault("xlabel", "Re")
kwargs.setdefault("ylabel", "Im")
global_labels = kwargs.pop("label", [])
args = _plot_sympify(args)
params = kwargs.get("params", None)
series = _build_series(*args, allow_lambda=False, **kwargs)
multiple_expr = len(series) > 1
def get_label(i):
_iterable = args[i] if multiple_expr else args
for t in _iterable:
if isinstance(t, str):
return t
return str(args[i][0] if multiple_expr else args[0])
# create new arguments to be used by plot_vector
new_args = []
x, y = symbols("x, y", cls=Dummy)
for i, s in enumerate(series):
expr1 = re(s.expr)
expr2 = im(s.expr)
free_symbols = s.expr.free_symbols
if params is not None:
free_symbols = free_symbols.difference(params.keys())
free_symbols = list(free_symbols)
if len(free_symbols) > 0:
fs = free_symbols[0]
expr1 = expr1.subs({fs: x + I * y})
expr2 = expr2.subs({fs: x + I * y})
r1 = prange(x, re(s.start), re(s.end))
r2 = prange(y, im(s.start), im(s.end))
label = get_label(i)
new_args.append(((expr1, expr2), r1, r2, label))
# substitute the complex variable in the scalar field
scalar = kwargs.get("scalar", None)
if scalar is not None:
if isinstance(scalar, Expr):
scalar = scalar.subs({fs: x + I * y})
elif isinstance(scalar, (list, tuple)):
scalar = list(scalar)
scalar[0] = scalar[0].subs({fs: x + I * y})
kwargs["scalar"] = scalar
kwargs["label"] = global_labels
kwargs.setdefault("xlabel", "x")
kwargs.setdefault("ylabel", "y")
return plot_vector(*new_args, **kwargs)
[docs]def plot_riemann_sphere(*args, **kwargs):
"""Visualize stereographic projections of the Riemann sphere.
Note:
1. Differently from other plot functions that return instances of
``BaseBackend``, this function returns a Matplotlib figure.
2. This function calls ``plot_complex``: refer to its documentation for
the full list of keyword arguments.
Parameters
==========
args :
expr : Expr
Represent the complex function to be plotted.
range : 3-element tuple, optional
Denotes the range of the variables. Only works for 2D plots.
Default to ``(z, -1.25 - 1.25*I, 1.25 + 1.25*I)``.
annotate : boolean, optional
Turn on/off the annotations on the 2D projections of the Riemann
sphere. Default to True (annotations are visible). They can only
be visible when ``riemann_mask=True``.
riemann_mask : boolean, optional
Turn on/off the unit disk mask representing the Riemann sphere on the
2D projections. Default to True (mask is active).
show_axis : boolean, optional
Turn on/off the axis of the 2D subplots. Default to False (axis not
visible).
size : (width, height)
Specify the size of the resulting figure.
title : str, list, optional
A list of two strings representing the titles for the two plots.
Notes
=====
The Riemann sphere is a model of the extented complex plane, comprised of
the complex plane plus a point at infinity. Let's consider a 3D space with
a sphere with radius 1 centered at the origin. The xy plane, representing
the complex plane, cut the sphere in half at the equator.
The stereographic projection of any point in the complex plane on the
sphere is given by the intersection point between a line connecting the
complex point with the north pole of the sphere.
Let's consider the magnitude of a complex point:
* if its lower than one (points inside the unit disk), then the point is
mapped to the Southern Hemisphere (the line connecting the complex point
to the north pole intersects the sphere in the Southern Hemisphere).
The origin of the complex plane is mapped to the south pole.
* if its equal to one (points in the unit circle), then the point is
already on the sphere, specifically in its equator.
* if its greater than one (points outside the unit disk), then the point
is mapped to the Northen Hemisphere. The north pole represents the point
at infinity.
Visualizing a 3D sphere is difficult (refer to Wegert [#fn4]_ for more
information): the most obvious problem is that only a part can be seen
from any location. A better way to fully visualize the sphere is with
two 2D charts depicting the sphere from the inside:
1. a stereographic projection of the sphere from the north pole, which
depict the Southern Hemisphere. It corresponds to an ordinary
(enhanced) domain coloring plot around the complex point $z=0$.
2. a stereographic projection of the sphere from the south pole, which
depict the Northen Hemisphere. It corresponds to an ordinary
(enhanced) domain coloring plot around the complex point $z=\\infty$
(infinity). Practically, it depicts the transformation
$z \\rightarrow \\frac{1}{z}$.
Let's look at an example:
.. plot::
:context: close-figs
:include-source: True
from sympy import symbols, pi
from spb import *
z = symbols("z")
expr = (z - 1) / (z**2 + z + 2)
plot_riemann_sphere(expr, coloring="b", n=800)
The saturated disks represents the hemispheres. The black circle is the
equator. Also, a few important points are displayed to make the plot
easier to understand.
Note the orientation of the Northen Hemisphere: it has been rotated
around the point at infinity by an angle `pi` and flipped about the real
axis. This is convenient because:
1. we can now imagine to fold the two charts so that the points 1, i, -i
are overlayed, glue the equator and blow it up to obtain a sphere.
2. imagine bringing the two discs closer so that they touch at the point 1.
Now, roll the two discs together: assuming there are no branch cuts,
there is continuity of argument and absolute value across the equator:
what is outside of the disc in the left plot, is inside of the disk in
the second plot, and vice-versa.
From the above plots, the zero located at $z=1$ is clearly visible, as
well as the two poles located at
$z = -\\frac{1}{2} - i \\frac{\\sqrt{7}}{2}$ and
$z = -\\frac{1}{2} + i \\frac{\\sqrt{7}}{2}$.
Not obvious at first, there is a zero located at $z=\\infty$. We can tell
its a zero by looking at ordering of colors around it in comparison to
the poles. Alternatively, we can use some enhanced color scheme, for
example one which brings poles to white:
.. plot::
:context: close-figs
:include-source: True
plot_riemann_sphere(expr, coloring="m", n=800)
Examples
========
Standard output:
.. plot::
:context: close-figs
:include-source: True
from sympy import symbols, Rational, I
from spb import *
z = symbols("z")
expr = 1 / (2 * z**2) + z
plot_riemann_sphere(expr, coloring="b", n=800)
Hide annotations:
.. plot::
:context: close-figs
:include-source: True
plot_riemann_sphere(expr, coloring="b", n=800, annotate=False)
Hiding Riemann disk mask and annotations, set a custom domain, show axis
(note that the right-most plot might be misleading because the center
represents infinity), custom colormap, set the black level of contours,
set titles.
.. plot::
:context: close-figs
:include-source: True
import colorcet
expr = z**5 + Rational(1, 10)
l = 2
plot_riemann_sphere(
expr, (z, -l-l*I, l+l*I), coloring="b", n=800,
riemann_mask=False, show_axis=True, grid=False,
cmap=colorcet.CET_C2, blevel=0.85,
title=["Around zero", "Around infinity"])
3D plot of a complex function on the Riemann sphere. Note, the higher the
number of discretization points, the better the final results, but the
higher memory consumption:
.. k3d-screenshot::
:camera: 1.87, 1.40, 1.96, 0, 0, 0, -0.45, -0.4, 0.8
from sympy import *
from spb import *
z = symbols("z")
expr = (z - 1) / (z**2 + z + 1)
plot_riemann_sphere(expr, threed=True, n=150,
coloring="b", backend=KB, legend=False, grid=False)
See Also
========
plot_complex
References
==========
.. [#fn4] Domain Coloring is based on Elias Wegert's book
`"Visual Complex Functions" <https://www.springer.com/de/book/9783034801799>`_.
The book provides the background to better understand the images.
"""
if kwargs.get("params", dict()):
raise NotImplementedError("Interactive widgets plots over the "
"Riemann sphere is not implemented.")
args = _plot_sympify(args)
if kwargs.get("threed", False):
kwargs.setdefault("xlabel", "Re")
kwargs.setdefault("ylabel", "Im")
kwargs.setdefault("zlabel", "")
colorbar = kwargs.pop("colorbar", True)
Backend = kwargs.get("backend", THREE_D_B)
t, p = symbols("theta phi")
# Northen and Southern hemispheres
s1 = RiemannSphereSeries(args[0], (t, 0, pi/2), (p, 0, 2*pi),
colorbar=False, **kwargs)
s2 = RiemannSphereSeries(args[0], (t, pi/2, pi), (p, 0, 2*pi),
colorbar=colorbar, **kwargs)
return _instantiate_backend(Backend, s1, s2, **kwargs)
# look for the range: if not given, set it to an appropriate domain
r, found_r, fs = None, False, set()
for a in args:
if _is_range(a):
r = a
found_r = True
elif isinstance(a, Expr):
fs = _get_free_symbols([a])
if not r:
s = fs.pop() if len(fs) > 0 else symbols("z")
args.append(Tuple(s, -1.25 - 1.25 * I, 1.25 + 1.25 * I))
# don't show the individual plots
show = kwargs.get("show", True)
kwargs["show"] = False
# set default options for Riemann sphere plots
kwargs.setdefault("show_axis", False)
kwargs.setdefault("riemann_mask", True)
kwargs.setdefault("annotate", True)
# size is applied to the final figure, not individual plots
size = kwargs.pop("size", None)
title = kwargs.pop("title", None)
get_title = lambda i: title[i] if isinstance(title, (tuple, list)) else title
# hide colorbar on first plot
legend = kwargs.get("legend", None)
kwargs["legend"] = False
kwargs["title"] = get_title(0) if title is not None else "Southern Hemisphere"
kwargs["at_infinity"] = False
p1 = plot_complex(*args, **kwargs)
test = (ComplexDomainColoringSeries, Parametric2DLineSeries,
List2DSeries, GenericDataSeries)
series = [s for s in p1.series if not isinstance(s, test)]
if len(series) > 1:
msg = "\n".join(str(s) for s in p1.series)
raise ValueError("Only one symbolic expression can be plotted. "
"Instead, the following have been received:\n" + msg)
kwargs["title"] = get_title(1) if title is not None else "Northen Hemisphere"
kwargs["at_infinity"] = True
kwargs["legend"] = True if legend or (legend is None) else False
p2 = plot_complex(*args, **kwargs)
if legend or (legend is None):
pg = plotgrid(p1, p2, nc=2, imagegrid=True, size=size, show=False)
else:
pg = plotgrid(p1, p2, nc=2, size=size, show=False)
if pg.is_matplotlib_fig:
if show:
pg.show()
return pg
return pg
if show:
return pg.show()
return pg