NOTE:
For technical reasons, all interactive-widgets plots in this documentation are
created using Holoviz’s Panel. Often, they will ran just fine with ipywidgets
too. However, if a specific example uses the param library, then users
will have to modify the params dictionary in order to make it work with
ipywidgets. Refer to Interactive module for more information.
Symbolic expression representing the function of one variable
to be plotted.
Numerical function of one variable, supporting vectorization.
In this case the following keyword arguments are not supported:
params, sum_bound.
range(symbol, min, max)
A 3-tuple denoting the range of the x variable. Default values:
min=-10 and max=10.
labelstr, optional
The label to be shown in the legend. If not provided, the string
representation of expr will be used.
rendering_kwdict, 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.
adaptivebool, optional
Setting adaptive=True activates the adaptive algorithm
implemented in [1] to create smooth plots. Use adaptive_goal
and loss_fn to further customize the output.
The default value is False, which uses an uniform sampling
strategy, where the number of discretization points is specified by
the n keyword argument.
adaptive_goalcallable, int, float or None
Controls the “smoothness” of the evaluation. Possible values:
None (default): it will use the following goal:
lambdal: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:
lambdal:l.loss()<number
callable: a function requiring one input element, the learner. It
must return a float number. Refer to [1] 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.
axis_center(float, float), optional
Tuple of two floats denoting the coordinates of the center or
{‘center’, ‘auto’}. Only available with MatplotlibBackend.
backendPlot, optional
A subclass of Plot, which will perform the rendering.
Default to MatplotlibBackend.
color_funccallable or Expr, optional
Define the line color mapping. It can either be:
A numerical function of 2 variables, x, y (the points computed by
the internal algorithm) supporting vectorization.
A symbolic expression having at most as many free symbols as
expr.
None: the default value (no color mapping).
detect_polesboolean or str, optional
Chose whether to detect and correctly plot poles. There are two
algorithms at work:
based on the gradient of the numerical data, it introduces NaN
values at locations where the steepness is greater than some
threshold. This splits the line into multiple segments. To improve
detection, increase the number of discretization points n
and/or change the value of eps.
a symbolic approach based on the continuous_domain function
from the sympy.calculus.util module, which computes the
locations of discontinuities. If any is found, vertical lines
will be shown.
Possible options:
True: activate poles detection computed with the numerical
gradient.
False: no poles detection.
"symbolic": use both numerical and symbolic algorithms.
Default to False.
epsfloat, optional
An arbitrary small value used by the detect_poles algorithm.
Default value to 0.1. Before changing this value, it is recommended to
increase the number of discretization points.
excludelist, optional
A list of numerical values in the horizontal coordinate which are
going to be excluded from the plot. In practice, it introduces
discontinuities in the resulting line.
force_real_evalboolean, optional
Default to False, with which the numerical evaluation is attempted
over a complex domain, which is slower but produces correct results.
Set this to True if performance is of paramount importance, but be
aware that it might produce wrong results. It only works with
adaptive=False.
is_pointboolean, optional
Default to False, which will render a line connecting all the points.
If True, a scatter plot will be generated.
is_filledboolean, 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.
labelstr or list/tuple, optional
The label 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 expressions.
legendbool, optional
Show/hide the legend. Default to None (the backend determines when
it is appropriate to show it).
loss_fncallable 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 [1] for more information. Specifically,
look at adaptive.learner.learner1D to find more loss functions.
nint, optional
Used when the adaptive=False: the function is uniformly
sampled at n number of points. Default value to 1000.
If the adaptive=True, this parameter will be ignored.
only_integersboolean, optional
Default to False. If True, discretize the domain with integer
numbers. It only works when adaptive=False.
When only_integers=True, the number of discretization points is
choosen by the algorithm.
paramsdict
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_kwdict 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 expressions.
is_polarboolean, optional
Default to False. If True, requests the backend to use a 2D polar
chart.
showbool, optional
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.
stepsboolean, optional
Default to False. If True, connects consecutive points with
steps rather than straight segments.
sum_boundint, optional
When plotting sums, the expression will be pre-processed in order
to replace lower/upper bounds set to +/- infinity with this +/-
numerical value. Default value to 1000. Note: the higher this number,
the slower the evaluation.
titlestr, optional
Title of the plot.
tx, tycallable, optional
Apply a numerical function to the discretized x-direction or to the
output of the numerical evaluation, the y-direction.
use_latexboolean, optional
Turn on/off the rendering of latex labels. If the backend doesn’t
support latex, it will render the string representations instead.
xlabel, ylabelstr, 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(float, float), optional
Denotes the x-axis limits, (min,max), visible in the chart.
Note that the function is still being evaluated over the specified
range.
ylim(float, float), optional
Denotes the y-axis limits, (min,max), visible in the chart.
Multiple functions over the same range with custom rendering options:
>>> plot(x,log(x),exp(x),(x,-3,3),aspect="equal",ylim=(-3,3),... rendering_kw=[{},{"linestyle":"--"},{"linestyle":":"}])Plot object containing:[0]: cartesian line: x for x over (-3.0, 3.0)[1]: cartesian line: log(x) for x over (-3.0, 3.0)[2]: cartesian line: exp(x) for x over (-3.0, 3.0)
Plotting a summation in which the free symbol of the expression is
used in the lower/upper bounds. Here, the discretization variable must
assume integer values:
>>> expr=Sum(1/x,(x,1,y))>>> plot(expr,(y,2,10),adaptive=False,... is_point=True,is_filled=True,title="$%s$"%latex(expr))Plot object containing:[0]: cartesian line: Sum(1/x, (x, 1, y)) for y over (2.0, 10.0)
Using an adaptive algorithm, detect and plot vertical lines at
singularities. Also, apply a transformation function to the discretized
domain in order to convert radians to degrees:
>>> importnumpyasnp>>> plot(tan(x),(x,-1.5*pi,1.5*pi),... adaptive=True,adaptive_goal=0.001,... detect_poles="symbolic",tx=np.rad2deg,ylim=(-7,7),... xlabel="x [deg]",grid=False)Plot object containing:[0]: cartesian line: tan(x) for x over (-4.71238898038469, 4.71238898038469)
Introducing discontinuities by excluding specified points:
>>> plot(floor(x)/x,(x,-3.25,3.25),ylim=(-1,5),... exclude=list(range(-4,5)))Plot object containing:[0]: cartesian line: floor(x)/x for x over (-3.25, 3.25)
detect singularities by setting adaptive=False (better performance),
increasing the number of discretization points (in order to have
‘vertical’ segments on the lines) and reducing the threshold for the
singularity-detection algorithm.
application of color function.
>>> importnumpyasnp>>> expr=1/cos(10*x)+5*sin(x)>>> defcf(x,y):... # map a colormap to the distance from the origin... d=np.sqrt(x**2+y**2)... # visibility of the plot is limited: ylim=(-10, 10). However,... # some of the y-values computed by the function are much higher... # (or lower). Filter them out in order to have the entire... # colormap spectrum visible in the plot.... offset=12# 12 > 10 (safety margin)... d[(y>offset)|(y<-offset)]=0... returnd>>> p1=plot(expr,(x,-5,5),... "distance from (0, 0)",{"cmap":"plasma"},... ylim=(-10,10),adaptive=False,detect_poles=True,n=3e04,... eps=1e-04,color_func=cf,title="$%s$"%latex(expr))
Interactive-widget plot of an oscillator. Refer to the interactive
sub-module documentation to learn more about the params dictionary.
This plot illustrates:
plotting multiple expressions, each one with its own label and
rendering options.
the use of prange (parametric plotting range).
the use of the params dictionary to specify sliders in
their basic form: (default, min, max).
the use of a parametric title, specified with a tuple of the form:
(title_str,param_symbol1,...), where:
title_str must be a formatted string, for example:
"test={:.2f}".
param_symbol1,... must be a symbol or a symbolic expression
whose free symbols are contained in the params dictionary.
fromsympyimport*fromspbimport*x,a,b,c,n=symbols("x, a, b, c, n")plot((cos(a*x+b)*exp(-c*x),"oscillator"),(exp(-c*x),"upper limit",{"linestyle":":"}),(-exp(-c*x),"lower limit",{"linestyle":":"}),prange(x,0,n*pi),params={a:(1,0,10),# frequencyb:(0,0,2*pi),# phasec:(0.25,0,1),# dampingn:(2,0,4)# multiple of pi},ylim=(-1.25,1.25),use_latex=False,title=("Frequency = {:.2f} Hz",a))
plot_implicit, by default, generates a contour using a mesh grid of fixed
number of points. The greater the number of points, the greater the memory
used. By setting adaptive=True, interval arithmetic will be used to
plot functions. If the expression cannot be plotted using interval
arithmetic, it defaults to generating a contour using a mesh grid.
With interval arithmetic, the line width can become very small; in those
cases, it is better to use the mesh grid approach.
Parameters:
args
exprExpr, Relational, BooleanFunction
The equation / inequality that is to be plotted.
rangestuples or Symbol
Two tuple denoting the discretization domain, for example:
(x,-10,10),(y,-10,10)
To get a correct plot, at least the horizontal range must be
provided. If no range is given, then the free symbols in the
expression will be assigned in the order they are sorted, which
could ‘invert’ the axis.
Alternatively, a single Symbol corresponding to the horizontal
axis must be provided, which will be internally converted to a
range (sym,-10,10).
labelstr, optional
The label to be shown when multiple expressions are plotted.
If not provided, the string representation of the expression
will be used.
rendering_kwdict, optional
A dictionary of keywords/values which is passed to the backend’s
function to customize the appearance of contours. Refer to the
plotting library (backend) manual for more informations.
adaptivebool, optional
The default value is set to False, meaning that the internal
algorithm uses a mesh grid approach. In such case, Boolean
combinations of expressions cannot be plotted.
If set to True, the internal algorithm uses interval arithmetic.
If the expression cannot be plotted with interval arithmetic, it
switches to the meshgrid approach.
border_colorstr or bool, optional
If given, a limiting border will be added when plotting inequalities
(<, <=, >, >=).
colorstr, optional
Specify the color of lines/regions. Default to None (automatic
coloring by the backend).
aspect(float, float) or str, optional
Set the aspect ratio of the plot. Possible values are "auto" or
"equals". Default to "auto".
depthinteger
The depth of recursion for adaptive grid. Default value is 0.
Takes value in the range (0, 4).
Think of the resulting plot as a picture composed by pixels. By
increasing depth we are increasing the number of pixels, thus
obtaining a more accurate plot.
labelstr or list/tuple, optional
The label 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 expressions.
legendbool, optional
Show/hide the legend. Default to None (the backend determines when
it is appropriate to show it).
rendering_kwdict or list of dicts, optional
A dictionary of keywords/values which is passed to the backend’s
function to customize the appearance. If the adaptive algorithm is
used, then matplotlib’s fill command will be executed.
If adaptive=False, then matplotlib’s contour or contourf
commands will be executed.
If a list of dictionaries is provided, the number of dictionaries must
be equal to the number of expressions.
n1, n2int
Number of discretization points in the horizontal and vertical
directions when adaptive=False. Default to 100.
nint or two-elements tuple (n1, n2), optional
If an integer is provided, the x and y ranges are sampled uniformly
at n of points. If a tuple is provided, it overrides
n1 and n2.
paramsdict
A dictionary mapping symbols to parameters. This keyword argument
enables the interactive-widgets plot. Learn more by reading the
documentation of the interactive sub-module.
showbool
Default value is True. If set to False, the plot will not be shown.
See Plot for further information.
show_in_legendbool
If True, add a legend entry for the expression being plotted.
This option is useful to hide a particular expression when combining
together multiple plots. Default to True.
titlestring
The title for the plot.
use_latexboolean, optional
Turn on/off the rendering of latex labels. If the backend doesn’t
support latex, it will render the string representations instead.
xlabel, ylabelstring
The labels for the x-axis or y-axis, respectively.
Specify both ranges, set the number of discretization points and plot a
region:
>>> plot_implicit(y>x**2,(x,-5,5),(y,-10,10),n=150,grid=False)Plot object containing:[0]: Implicit expression: y > x**2 for x over (-5.0, 5.0) and y over (-10.0, 10.0)
Plot a region using a custom color, highlights the limiting border and
customize its appearance. In this particular case, the content of
rendering_kw will be sent to matplotlib’s contour of contourf
commands.
>>> expr=4*(cos(x)-sin(y)/5)**2+4*(-cos(x)/5+sin(y))**2>>> plot_implicit(expr<=pi,(x,-pi,pi),(y,-pi,pi),... grid=False,color="gold",border_color="k",... rendering_kw={"linestyles":"-.","linewidths":1})Plot object containing:[0]: Implicit expression: 4*(-sin(y)/5 + cos(x))**2 + 4*(sin(y) - cos(x)/5)**2 <= pi for x over (-3.141592653589793, 3.141592653589793) and y over (-3.141592653589793, 3.141592653589793)[1]: Implicit expression: Eq(-4*(-sin(y)/5 + cos(x))**2 - 4*(sin(y) - cos(x)/5)**2 + pi, 0) for x over (-3.141592653589793, 3.141592653589793) and y over (-3.141592653589793, 3.141592653589793)
Boolean expressions will be plotted with the adaptive algorithm. Note the
thin width of lines:
>>> plot_implicit(... Eq(y,sin(x))&(y>0),... Eq(y,sin(x))&(y<0),... (x,-2*pi,2*pi),(y,-4,4))Plot object containing:[0]: Implicit expression: (y > 0) & Eq(y, sin(x)) for x over (-6.283185307179586, 6.283185307179586) and y over (-4.0, 4.0)[1]: Implicit expression: (y < 0) & Eq(y, sin(x)) for x over (-6.283185307179586, 6.283185307179586) and y over (-4.0, 4.0)
Plotting multiple implicit expressions and setting labels:
>>> V,t,b,L=symbols("V, t, b, L")>>> L_array=[5,10,15,20,25]>>> b_val=0.0032>>> expr=b*V*0.277*t-b*L-log(1+b*V*0.277*t)>>> expr_list=[expr.subs({b:b_val,L:L_val})forL_valinL_array]>>> labels=["L = %s"%L_valforL_valinL_array]>>> plot_implicit(*expr_list,(t,0,3),(V,0,1000),label=labels)Plot object containing:[0]: Implicit expression: Eq(0.0008864*V*t - log(0.0008864*V*t + 1) - 0.016, 0) for t over (0.0, 3.0) and V over (0.0, 1000.0)[1]: Implicit expression: Eq(0.0008864*V*t - log(0.0008864*V*t + 1) - 0.032, 0) for t over (0.0, 3.0) and V over (0.0, 1000.0)[2]: Implicit expression: Eq(0.0008864*V*t - log(0.0008864*V*t + 1) - 0.048, 0) for t over (0.0, 3.0) and V over (0.0, 1000.0)[3]: Implicit expression: Eq(0.0008864*V*t - log(0.0008864*V*t + 1) - 0.064, 0) for t over (0.0, 3.0) and V over (0.0, 1000.0)[4]: Implicit expression: Eq(0.0008864*V*t - log(0.0008864*V*t + 1) - 0.08, 0) for t over (0.0, 3.0) and V over (0.0, 1000.0)
Comparison of similar expressions plotted with different algorithms. Note:
Adaptive algorithm (adaptive=True) can be used with any expression,
but it usually creates lines with variable thickness. The depth
keyword argument can be used to improve the accuracy, but reduces line
thickness even further.
Mesh grid algorithm (adaptive=False) creates lines with constant
thickness.
If the expression is plotted with the adaptive algorithm and it produces
“low-quality” results, maybe it’s possible to rewrite it in order to use
the mesh grid approach (contours). For example:
>>> fromspbimportplotgrid>>> expr=Ne(x*y,1)>>> p1=plot_implicit(... expr,(x,-10,10),(y,-10,10),... grid=False,aspect="equal",show=False,... title="$%s$ : First approach"%latex(expr))>>> # plot the entire visible region>>> p2=plot_implicit(... x<20,(x,-10,10),(y,-10,10),... show=False,grid=False,aspect="equal",... title="$%s$ : Second approach"%latex(expr))>>> # plot the excluded contour>>> p3=plot_implicit(... Eq(*expr.args),(x,-10,10),(y,-10,10),... color="w",show_in_legend=False,show=False)>>> plotgrid(p1,(p2+p3),nc=2)
Interactive-widget implicit 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).
fromsympyimport*fromspbimport*x,y,a,b,c,d,e=symbols("x, y, a, b, c, d, e")expr=Eq(a*x**2-b*x+c,d*y+y**2)plot_implicit(expr,(x,-2,2),prange(y,-e,e),params={a:(10,-15,15),b:(7,-15,15),c:(3,-15,15),d:(2,-15,15),e:(10,1,15),},n=150,use_latex=False,ylim=(-10,10))
The expression representing x component of the parametric
function. It can be a:
Symbolic expression representing the function of one variable
to be plotted.
Numerical function of one variable, supporting vectorization.
In this case the following keyword arguments are not supported:
params.
expr_yExpr
The expression representing y component of the parametric
function. It can be a:
Symbolic expression representing the function of one variable
to be plotted.
Numerical function of one variable, supporting vectorization.
In this case the following keyword arguments are not supported:
params.
range(symbol, min, max)
A 3-tuple denoting the parameter symbol, start and stop. For
example, (u,0,5). If the range is not specified, then a
default range of (-10, 10) is used.
However, if the arguments are specified as
(expr_x,expr_y,range),..., you must specify the ranges
for each expressions manually.
labelstr, optional
The label to be shown in the legend. If not provided, the string
representation of expr_x and expr_y will be used.
rendering_kwdict, 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.
adaptivebool, optional
Setting adaptive=True activates the adaptive algorithm
implemented in [2] to create smooth plots. Use adaptive_goal
and loss_fn to further customize the output.
The default value is False, which uses an uniform sampling
strategy, where the number of discretization points is specified by
the n keyword argument.
adaptive_goalcallable, int, float or None
Controls the “smoothness” of the evaluation. Possible values:
None (default): it will use the following goal:
lambdal: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:
lambdal:l.loss()<number
callable: a function requiring one input element, the learner. It
must return a float number. Refer to [2] 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.
axis_center(float, float), optional
Tuple of two floats denoting the coordinates of the center or
{‘center’, ‘auto’}. Only available with MatplotlibBackend.
backendPlot, optional
A subclass of Plot, which will perform the rendering.
Default to MatplotlibBackend.
colorbarboolean, optional
Show/hide the colorbar. Default to True (colorbar is visible).
Only works when use_cm=True.
color_funccallable, optional
Define the line color mapping when use_cm=True. It can either be:
A numerical function supporting vectorization. The arity can be:
1 argument: f(t), where t is the parameter.
2 arguments: f(x,y) where x,y are the coordinates of
the points.
3 arguments: f(x,y,t).
A symbolic expression having at most as many free symbols as
expr_x or expr_y.
None: the default value (color mapping applied to the parameter).
excludelist, optional
A list of numerical values along the parameter which are going to
be excluded from the plot. In practice, it introduces discontinuities
in the resulting line.
force_real_evalboolean, optional
Default to False, with which the numerical evaluation is attempted
over a complex domain, which is slower but produces correct results.
Set this to True if performance is of paramount importance, but be
aware that it might produce wrong results. It only works with
adaptive=False.
labelstr or list/tuple, optional
The label to be shown in the legend or in the colorbar. If not
provided, the string representation of expr will be used. The number
of labels must be equal to the number of expressions.
legendbool, optional
Show/hide the legend. Default to None (the backend determines when
it is appropriate to show it). Only works when use_cm=False.
loss_fncallable 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 [2] for more information. Specifically,
look at adaptive.learner.learner1D to find more loss functions.
nint, optional
Used when the adaptive=False. The function is uniformly sampled
at n number of points. Default value to 1000.
If the adaptive=True, this parameter will be ignored.
paramsdict
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_kwdict 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 expressions.
showbool, optional
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.
titlestr, optional
Title of the plot. It is set to the latex representation of
the expression, if the plot has only one expression.
tx, ty, tpcallable, optional
Apply a numerical function to the x-direction, y-direction and
parameter, respectively.
use_cmboolean, optional
If True, apply a color map to the parametric lines.
If False, solid colors will be used instead. Default to True.
use_latexboolean, optional
Turn on/off the rendering of latex labels. If the backend doesn’t
support latex, it will render the string representations instead.
xlabel, ylabelstr, 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), visible in the chart.
A parametric plot with multiple expressions with the same range with solid
line colors:
>>> plot_parametric((2*cos(t),sin(t)),(cos(t),2*sin(t)),... (t,0,2*pi),use_cm=False)Plot object containing:[0]: parametric cartesian line: (2*cos(t), sin(t)) for t over (0.0, 6.283185307179586)[1]: parametric cartesian line: (cos(t), 2*sin(t)) for t over (0.0, 6.283185307179586)
A parametric plot with multiple expressions with different ranges,
custom labels, custom rendering options and a transformation function
applied to the discretized parameter to convert radians to degrees:
>>> importnumpyasnp>>> plot_parametric(... (3*cos(u),3*sin(u),(u,0,2*pi),"u [deg]",{"lw":3}),... (3*cos(2*v),5*sin(4*v),(v,0,pi),"v [deg]"),... aspect="equal",tp=np.rad2deg)Plot object containing:[0]: parametric cartesian line: (3*cos(u), 3*sin(u)) for u over (0.0, 6.283185307179586)[1]: parametric cartesian line: (3*cos(2*v), 5*sin(4*v)) for v over (0.0, 3.141592653589793)
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).
fromsympyimport*fromspbimport*x,a,s,e=symbols("x a s, e")plot_parametric(cos(a*x),sin(x),prange(x,s*pi,e*pi),params={a:(0.5,0,2),s:(0,0,2),e:(2,0,2),},aspect="equal",use_latex=False,xlim=(-1.25,1.25),ylim=(-1.25,1.25))
NOTE: this is an experimental plotting function as it only draws lines
without fills. The resulting visualization might change when new features
will be implemented.
Typical usage examples are in the followings:
Plotting a single parametric curve with a range
plot_parametric(expr_x, expr_y, range_u, range_v)
Plotting multiple parametric curves with the same range
The expression representing x and y component, respectively, of
the parametric function. It can be a:
Symbolic expression representing the function of one variable
to be plotted.
Numerical function of one variable, supporting vectorization.
In this case the following keyword arguments are not supported:
params.
range_u, range_v(symbol, min, max)
A 3-tuple denoting the parameter symbols, start and stop. For
example, (u, 0, 5), (v, 0, 5). If the ranges are not specified,
then they default to (-10, 10).
However, if the arguments are specified as
(expr_x, expr_y, range_u, range_v), …, you must specify the
ranges for each expressions manually.
rendering_kwdict, 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.
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.
backendPlot, optional
A subclass of Plot, which will perform the rendering.
Default to MatplotlibBackend.
nint, optional
The functions are uniformly sampled at n number of points.
Default value to 1000.
n1, n2int, optional
Number of lines to create along each direction. Default to 10.
Note: the higher the number, the slower the rendering.
rkw_u, rkw_vdict
A dictionary of keywords/values which is passed to the backend’s
function to customize the appearance of lines along the u and v
directions, respectively. These overrides rendering_kw if provided.
Refer to the plotting library (backend) manual for more informations.
showbool, optional
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.
titlestr, optional
Title of the plot. It is set to the latex representation of
the expression, if the plot has only one expression.
use_latexboolean, optional
Turn on/off the rendering of latex labels. If the backend doesn’t
support latex, it will render the string representations instead.
xlabel, ylabelstr, optional
Label 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 or y-axis limits, (min,max), visible in the
chart.
>>> plot_contour(cos((x**2+y**2))*exp(-(x**2+y**2)/10),... (x,-5,5),(y,-5,5))Plot object containing:[0]: contour: exp(-x**2/10 - y**2/10)*cos(x**2 + y**2) for x over (-5.0, 5.0) and y over (-5.0, 5.0)
>>> expr=5*(cos(x)-0.2*sin(y))**2+5*(-0.2*cos(x)+sin(y))**2>>> plot_contour(expr,(x,0,2*pi),(y,0,2*pi),is_filled=False)Plot object containing:[0]: contour: 5*(-0.2*sin(y) + cos(x))**2 + 5*(sin(y) - 0.2*cos(x))**2 for x over (0.0, 6.283185307179586) and y over (0.0, 6.283185307179586)
Visually inspect the solutions of a system of 2 non-linear equations.
The intersections between the contour lines represent the solutions.
>>> eq1=Eq((cos(x)-sin(y)/2)**2+3*(-sin(x)+cos(y)/2)**2,2)>>> eq2=Eq((cos(x)-2*sin(y))**2-(sin(x)+2*cos(y))**2,3)>>> plot_contour(eq1.rewrite(Add),eq2.rewrite(Add),{"levels":[0]},... (x,0,2*pi),(y,0,2*pi),is_filled=False,clabels=False)Plot object containing:[0]: contour: 3*(-sin(x) + cos(y)/2)**2 + (-sin(y)/2 + cos(x))**2 - 2 for x over (0.0, 6.283185307179586) and y over (0.0, 6.283185307179586)[1]: contour: -(sin(x) + 2*cos(y))**2 + (-2*sin(y) + cos(x))**2 - 3 for x over (0.0, 6.283185307179586) and y over (0.0, 6.283185307179586)
>>> r,theta=symbols("r, theta")>>> plot_contour(sin(2*r)*cos(theta),(theta,0,2*pi),(r,0,7),... {"levels":100},polar_axis=True,aspect="equal")Plot object containing:[0]: contour: sin(2*r)*cos(theta) for theta over (0.0, 6.283185307179586) and r over (0.0, 7.0)
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).
fromsympyimport*fromspbimport*x,y,a,b,xp,yp=symbols("x y a b x_p y_p")expr=(cos(x)+a*sin(x)*sin(y)-b*sin(x)*cos(y))**2plot_contour(expr,prange(x,0,xp*pi),prange(y,0,yp*pi),params={a:(1,0,2),b:(1,0,2),xp:(1,0,2),yp:(2,0,2)},grid=False,use_latex=False)
Interactive-widget plot of Guilloché Pattern. 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 the widgets to be
created by Holoviz’s Panel.
The name of the geometry entity to be eventually shown on the
legend. If not provided, the string representation of geom
will be used.
rendering_kwdict, optional
A dictionary of keywords/values which is passed to the backend’s
function to customize the appearance of lines or fills. Refer to
the plotting library (backend) manual for more informations.
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.
backendPlot, optional
A subclass of Plot, which will perform the rendering.
Default to MatplotlibBackend.
is_filledboolean
Default to True. Fill the polygon/circle/ellipse.
labelstr or list/tuple, optional
The label to be shown in the legend. If not provided, the string
representation of geom will be used. The number of labels must be
equal to the number of geometric entities.
legendbool, optional
Show/hide the legend. Default to None (the backend determines when
it is appropriate to show it).
paramsdict
A dictionary in which the keys are symbols, enabling two different
modes of operation:
If the values are numbers, the dictionary acts like a substitution
dictionary for the provided geometric entities.
If the values are tuples representing parameters, the dictionary
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.
axis_center(float, float), optional
Tuple of two floats denoting the coordinates of the center or
{‘center’, ‘auto’}. Only available with MatplotlibBackend.
rendering_kwdict or list of dicts, optional
A dictionary of keywords/values which is passed to the backend’s
functions to customize the appearance of lines and/or fills. 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 expressions.
showbool, optional
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.
titlestr, optional
Title of the plot. It is set to the latex representation of
the expression, if the plot has only one expression.
use_latexboolean, 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, zlabelstr, optional
Labels for the x-axis, y-axis or z-axis, respectively.
xlim, ylim, zlim(float, float), optional
Denotes the x-axis limits, y-axis limits or z-axis limits,
respectively, (min,max), visible in the chart.
Plot several numeric geometric entitiesy. By default, circles, ellipses and
polygons are going to be filled. Plotting Curve objects is the same as
plot_parametric.
Plot several symbolic geometric entities. We need to pass in the params
dictionary, which will be used to substitute symbols before numerical
evaluation. Note: here we also set custom labels:
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.
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.
axis_center(float, float), optional
Tuple of two floats denoting the coordinates of the center or
{‘center’, ‘auto’}. Only available with MatplotlibBackend.
backendPlot, optional
A subclass of Plot, which will perform the rendering.
Default to MatplotlibBackend.
is_pointboolean, optional
Default to False, which will render a line connecting all the points.
If True, a scatter plot will be generated.
is_filledboolean, optional
Default to False, which will render empty circular markers. It only
works if is_point=True.
If True, filled circular markers will be rendered.
labelstr or list/tuple, optional
The label to be shown in the legend. The number of labels must be
equal to the number of expressions.
legendbool, optional
Show/hide the legend. Default to None (the backend determines when
it is appropriate to show it).
paramsdict
A dictionary mapping symbols to parameters. This keyword argument
enables the interactive-widgets plot. Learn more by reading the
documentation of the interactive sub-module.
rendering_kwdict 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 expressions.
showbool, optional
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.
titlestr, optional
Title of the plot. It is set to the latex representation of
the expression, if the plot has only one expression.
use_latexboolean, optional
Turn on/off the rendering of latex labels. If the backend doesn’t
support latex, it will render the string representations instead.
xlabel, ylabelstr, 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), visible in the chart.
>>> plot_list(([0],[0],"A"),([1],[1],"B"),([2],[0],"C"),... is_point=True,is_filled=True)Plot object containing:[0]: 2D list plot[1]: 2D list plot[2]: 2D list plot
Expression representing the function of one variable to be
plotted.
range: (symbol, min, max)
A 3-tuple denoting the range of the x variable. Default values:
min=-10 and max=10.
labelstr, optional
The label to be shown in the legend. If not provided, the string
representation of expr will be used.
rendering_kwdict, 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.
adaptivebool, optional
Setting adaptive=True activates the adaptive algorithm
implemented in [3] to create smooth plots. Use adaptive_goal
and loss_fn to further customize the output.
The default value is False, which uses an uniform sampling
strategy, where the number of discretization points is specified by
the n keyword argument.
adaptive_goalcallable, int, float or None
Controls the “smoothness” of the evaluation. Possible values:
None (default): it will use the following goal:
lambdal: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:
lambdal:l.loss()<number
callable: a function requiring one input element, the learner. It
must return a float number. Refer to [3] 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.
axis_center(float, float), optional
Tuple of two floats denoting the coordinates of the center or
{‘center’, ‘auto’}. Only available with MatplotlibBackend.
backendPlot, optional
A subclass of Plot, which will perform the rendering.
Default to MatplotlibBackend.
detect_polesboolean or str, optional
Chose whether to detect and correctly plot poles. There are two
algorithms at work:
based on the gradient of the numerical data, it introduces NaN
values at locations where the steepness is greater than some
threshold. This splits the line into multiple segments. To improve
detection, increase the number of discretization points n
and/or change the value of eps.
a symbolic approach based on the continuous_domain function
from the sympy.calculus.util module, which computes the
locations of discontinuities. If any is found, vertical lines
will be shown.
Possible options:
True: activate poles detection computed with the numerical
gradient.
False: no poles detection.
"symbolic": use both numerical and symbolic algorithms.
Default to False.
dotsboolean
Wheter to show circular markers at the endpoints. Default to True.
epsfloat, optional
An arbitrary small value used by the detect_poles algorithm.
Default value to 0.1. Before changing this value, it is recommended to
increase the number of discretization points.
force_real_evalboolean, optional
Default to False, with which the numerical evaluation is attempted
over a complex domain, which is slower but produces correct results.
Set this to True if performance is of paramount importance, but be
aware that it might produce wrong results. It only works with
adaptive=False.
labelstr or list/tuple, optional
The label to be shown in the legend. If not provided, the string
representation of expr will be used. If a list/tuple is provided, the
number of labels must be equal to the number of expressions.
loss_fncallable 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 [3] for more information. Specifically,
look at adaptive.learner.learner1D to find more loss functions.
nint, optional
Used when the adaptive=False. The function is uniformly
sampled at n number of points. Default value to 1000.
If the adaptive=True, this parameter will be ignored.
showbool, optional
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.
titlestr, optional
Title of the plot. It is set to the latex representation of
the expression, if the plot has only one expression.
tx, tycallable, optional
Apply a numerical function to the discretized domain in the
x and y directions, respectively.
use_latexboolean, optional
Turn on/off the rendering of latex labels. If the backend doesn’t
support latex, it will render the string representations instead.
xlabel, ylabelstr, 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(float, float), optional
Denotes the x-axis limits, (min,max), visible in the chart.
Note that the function is still being evaluated over the specified
range.
ylim(float, float), optional
Denotes the y-axis limits, (min,max), visible in the chart.
>>> f=Piecewise((x**2,x<2),(5,Eq(x,2)),(10-x,True))>>> plot_piecewise(f,(x,-2,5))Plot object containing:[0]: cartesian line: x**2 for x over (-2.0, 1.999999)[1]: 2D list plot[2]: cartesian line: 10 - x for x over (2.000001, 5.0)[3]: 2D list plot[4]: 2D list plot
>>> plot_piecewise(Heaviside(x,0).rewrite(Piecewise),... (x,-10,10),dots=False)Plot object containing:[0]: cartesian line: 0 for x over (-10.0, 0.0)[1]: cartesian line: 1 for x over (1e-06, 10.0)
Plot multiple expressions in which the second piecewise expression has
a dotted line style.
>>> plot_piecewise(... (Heaviside(x,0).rewrite(Piecewise),(x,-10,10)),... (Piecewise(... (sin(x),x<-5),... (cos(x),x>5),... (1/x,True)),(x,-8,8),{"linestyle":":"}),... ylim=(-2,2),detect_poles=True)Plot object containing:[0]: cartesian line: 0 for x over (-10.0, 0.0)[1]: cartesian line: 1 for x over (1e-06, 10.0)[2]: 2D list plot[3]: 2D list plot[4]: cartesian line: sin(x) for x over (-8.0, -5.000001)[5]: 2D list plot[6]: cartesian line: cos(x) for x over (5.000001, 8.0)[7]: 2D list plot[8]: cartesian line: 1/x for x over (-5.0, 5.0)[9]: 2D list plot[10]: 2D list plot