# Plotting¶

## Introduction¶

The plotting module allows you to make 2-dimensional and 3-dimensional plots. Presently the plots are rendered using matplotlib as a backend. It is also possible to plot 2-dimensional plots using a TextBackend if you don’t have matplotlib.

The plotting module has the following functions:

• plot: Plots 2D line plots.

• plot_parametric: Plots 2D parametric plots.

• plot_implicit: Plots 2D implicit and region plots.

• plot3d: Plots 3D plots of functions in two variables.

• plot3d_parametric_line: Plots 3D line plots, defined by a parameter.

• plot3d_parametric_surface: Plots 3D parametric surface plots.

The above functions are only for convenience and ease of use. It is possible to plot any plot by passing the corresponding Series class to Plot as argument.

## Plot Class¶

class sympy.plotting.plot.Plot(*args, title=None, xlabel=None, ylabel=None, zlabel=None, aspect_ratio='auto', xlim=None, ylim=None, axis_center='auto', axis=True, xscale='linear', yscale='linear', legend=False, autoscale=True, margin=0, annotations=None, markers=None, rectangles=None, fill=None, backend='default', size=None, **kwargs)[source]

The central class of the plotting module.

Explanation

For interactive work the function plot is better suited.

This class permits the plotting of sympy expressions using numerous backends (matplotlib, textplot, the old pyglet module for sympy, Google charts api, etc).

The figure can contain an arbitrary number of plots of sympy expressions, lists of coordinates of points, etc. Plot has a private attribute _series that contains all data series to be plotted (expressions for lines or surfaces, lists of points, etc (all subclasses of BaseSeries)). Those data series are instances of classes not imported by from sympy import *.

The customization of the figure is on two levels. Global options that concern the figure as a whole (eg title, xlabel, scale, etc) and per-data series options (eg name) and aesthetics (eg. color, point shape, line type, etc.).

The difference between options and aesthetics is that an aesthetic can be a function of the coordinates (or parameters in a parametric plot). The supported values for an aesthetic are: - None (the backend uses default values) - a constant - a function of one variable (the first coordinate or parameter) - a function of two variables (the first and second coordinate or parameters) - a function of three variables (only in nonparametric 3D plots) Their implementation depends on the backend so they may not work in some backends.

If the plot is parametric and the arity of the aesthetic function permits it the aesthetic is calculated over parameters and not over coordinates. If the arity does not permit calculation over parameters the calculation is done over coordinates.

Only cartesian coordinates are supported for the moment, but you can use the parametric plots to plot in polar, spherical and cylindrical coordinates.

The arguments for the constructor Plot must be subclasses of BaseSeries.

Any global option can be specified as a keyword argument.

The global options for a figure are:

• title : str

• xlabel : str

• ylabel : str

• zlabel : str

• legend : bool

• xscale : {‘linear’, ‘log’}

• yscale : {‘linear’, ‘log’}

• axis : bool

• axis_center : tuple of two floats or {‘center’, ‘auto’}

• xlim : tuple of two floats

• ylim : tuple of two floats

• aspect_ratio : tuple of two floats or {‘auto’}

• autoscale : bool

• margin : float in [0, 1]

• backend : {‘default’, ‘matplotlib’, ‘text’} or a subclass of BaseBackend

• size : optional tuple of two floats, (width, height); default: None

The per data series options and aesthetics are: There are none in the base series. See below for options for subclasses.

Some data series support additional aesthetics or options:

ListSeries, LineOver1DRangeSeries, Parametric2DLineSeries, Parametric3DLineSeries support the following:

Aesthetics:

• line_colorstring, or float, or function, optional

Specifies the color for the plot, which depends on the backend being used.

For example, if MatplotlibBackend is being used, then Matplotlib string colors are acceptable (“red”, “r”, “cyan”, “c”, …). Alternatively, we can use a float number $$0 < color < 1$$ wrapped in a string (for example, $$line_color="0.5"$$) to specify grayscale colors. Alternatively, We can specify a function returning a single float value: this will be used to apply a color-loop (for example, $$line_color=lambda x: math.cos(x)$$).

Note that by setting line_color, it would be applied simultaneously to all the series.

options:

• label : str

• steps : bool

• integers_only : bool

SurfaceOver2DRangeSeries, ParametricSurfaceSeries support the following:

aesthetics:

• surface_color : function which returns a float.

append(arg)[source]

Adds an element from a plot’s series to an existing plot.

Examples

Consider two Plot objects, p1 and p2. To add the second plot’s first series object to the first, use the append method, like so:

>>> from sympy import symbols
>>> from sympy.plotting import plot
>>> x = symbols('x')
>>> p1 = plot(x*x, show=False)
>>> p2 = plot(x, show=False)
>>> p1.append(p2[0])
>>> p1
Plot object containing:
[0]: cartesian line: x**2 for x over (-10.0, 10.0)
[1]: cartesian line: x for x over (-10.0, 10.0)
>>> p1.show()

(png, hires.png, pdf)

extend

extend(arg)[source]

Adds all series from another plot.

Examples

Consider two Plot objects, p1 and p2. To add the second plot to the first, use the extend method, like so:

>>> from sympy import symbols
>>> from sympy.plotting import plot
>>> x = symbols('x')
>>> p1 = plot(x**2, show=False)
>>> p2 = plot(x, -x, show=False)
>>> p1.extend(p2)
>>> p1
Plot object containing:
[0]: cartesian line: x**2 for x over (-10.0, 10.0)
[1]: cartesian line: x for x over (-10.0, 10.0)
[2]: cartesian line: -x for x over (-10.0, 10.0)
>>> p1.show()

(png, hires.png, pdf)

## Plotting Function Reference¶

sympy.plotting.plot.plot(*args, show=True, **kwargs)[source]

Plots a function of a single variable as a curve.

Parameters

args :

The first argument is the expression representing the function of single variable to be plotted.

The last argument is a 3-tuple denoting the range of the free variable. e.g. (x, 0, 5)

Typical usage examples are in the followings:

• Plotting a single expression with a single range.

plot(expr, range, **kwargs)

• Plotting a single expression with the default range (-10, 10).

plot(expr, **kwargs)

• Plotting multiple expressions with a single range.

plot(expr1, expr2, ..., range, **kwargs)

• Plotting multiple expressions with multiple ranges.

plot((expr1, range1), (expr2, range2), ..., **kwargs)

It is best practice to specify range explicitly because default range may change in the future if a more advanced default range detection algorithm is implemented.

show : bool, 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.

line_color : string, or float, or function, optional

Specifies the color for the plot. See Plot to see how to set color for the plots. Note that by setting line_color, it would be applied simultaneously to all the series.

title : str, optional

Title of the plot. It is set to the latex representation of the expression, if the plot has only one expression.

label : str, optional

The label of the expression in the plot. It will be used when called with legend. Default is the name of the expression. e.g. sin(x)

xlabel : str, optional

Label for the x-axis.

ylabel : str, optional

Label for the y-axis.

xscale : ‘linear’ or ‘log’, optional

Sets the scaling of the x-axis.

yscale : ‘linear’ or ‘log’, optional

Sets the scaling of the y-axis.

axis_center : (float, float), optional

Tuple of two floats denoting the coordinates of the center or {‘center’, ‘auto’}

xlim : (float, float), optional

Denotes the x-axis limits, (min, max).

ylim : (float, float), optional

Denotes the y-axis limits, (min, max).

annotations : list, optional

A list of dictionaries specifying the type of annotation required. The keys in the dictionary should be equivalent to the arguments of the matplotlib’s annotate() function.

markers : list, optional

A list of dictionaries specifying the type the markers required. The keys in the dictionary should be equivalent to the arguments of the matplotlib’s plot() function along with the marker related keyworded arguments.

rectangles : list, optional

A list of dictionaries specifying the dimensions of the rectangles to be plotted. The keys in the dictionary should be equivalent to the arguments of the matplotlib’s patches.Rectangle class.

fill : dict, optional

A dictionary specifying the type of color filling required in the plot. The keys in the dictionary should be equivalent to the arguments of the matplotlib’s fill_between() function.

The default value is set to True. Set adaptive to False and specify nb_of_points if uniform sampling is required.

The plotting uses an adaptive algorithm which samples recursively to accurately plot. The adaptive algorithm uses a random point near the midpoint of two points that has to be further sampled. Hence the same plots can appear slightly different.

depth : int, optional

Recursion depth of the adaptive algorithm. A depth of value n samples a maximum of $$2^{n}$$ points.

If the adaptive flag is set to False, this will be ignored.

nb_of_points : int, optional

Used when the adaptive is set to False. The function is uniformly sampled at nb_of_points number of points.

If the adaptive flag is set to True, this will be ignored.

size : (float, float), optional

A tuple in the form (width, height) in inches to specify the size of the overall figure. The default value is set to None, meaning the size will be set by the default backend.

Examples

>>> from sympy import symbols
>>> from sympy.plotting import plot
>>> x = symbols('x')

Single Plot

>>> plot(x**2, (x, -5, 5))
Plot object containing:
[0]: cartesian line: x**2 for x over (-5.0, 5.0)

(png, hires.png, pdf)

Multiple plots with single range.

>>> plot(x, x**2, x**3, (x, -5, 5))
Plot object containing:
[0]: cartesian line: x for x over (-5.0, 5.0)
[1]: cartesian line: x**2 for x over (-5.0, 5.0)
[2]: cartesian line: x**3 for x over (-5.0, 5.0)

(png, hires.png, pdf)

Multiple plots with different ranges.

>>> plot((x**2, (x, -6, 6)), (x, (x, -5, 5)))
Plot object containing:
[0]: cartesian line: x**2 for x over (-6.0, 6.0)
[1]: cartesian line: x for x over (-5.0, 5.0)

(png, hires.png, pdf)

Plot object containing:
[0]: cartesian line: x**2 for x over (-10.0, 10.0)

(png, hires.png, pdf)

sympy.plotting.plot.plot_parametric(*args, show=True, **kwargs)[source]

Plots a 2D parametric curve.

Parameters

args

Common specifications are:

• Plotting a single parametric curve with a range

plot_parametric((expr_x, expr_y), range)

• Plotting multiple parametric curves with the same range

plot_parametric((expr_x, expr_y), ..., range)

• Plotting multiple parametric curves with different ranges

plot_parametric((expr_x, expr_y, range), ...)

expr_x is the expression representing $$x$$ component of the parametric function.

expr_y is the expression representing $$y$$ component of the parametric function.

range is 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.

Default range may change in the future if a more advanced algorithm is implemented.

Specifies whether to use the adaptive sampling or not.

The default value is set to True. Set adaptive to False and specify nb_of_points if uniform sampling is required.

depth : int, optional

The recursion depth of the adaptive algorithm. A depth of value $$n$$ samples a maximum of $$2^n$$ points.

nb_of_points : int, optional

Used when the adaptive flag is set to False.

Specifies the number of the points used for the uniform sampling.

line_color : string, or float, or function, optional

Specifies the color for the plot. See Plot to see how to set color for the plots. Note that by setting line_color, it would be applied simultaneously to all the series.

label : str, optional

The label of the expression in the plot. It will be used when called with legend. Default is the name of the expression. e.g. sin(x)

xlabel : str, optional

Label for the x-axis.

ylabel : str, optional

Label for the y-axis.

xscale : ‘linear’ or ‘log’, optional

Sets the scaling of the x-axis.

yscale : ‘linear’ or ‘log’, optional

Sets the scaling of the y-axis.

axis_center : (float, float), optional

Tuple of two floats denoting the coordinates of the center or {‘center’, ‘auto’}

xlim : (float, float), optional

Denotes the x-axis limits, (min, max).

ylim : (float, float), optional

Denotes the y-axis limits, (min, max).

size : (float, float), optional

A tuple in the form (width, height) in inches to specify the size of the overall figure. The default value is set to None, meaning the size will be set by the default backend.

Examples

>>> from sympy import symbols, cos, sin
>>> from sympy.plotting import plot_parametric
>>> u = symbols('u')

A parametric plot with a single expression:

>>> plot_parametric((cos(u), sin(u)), (u, -5, 5))
Plot object containing:
[0]: parametric cartesian line: (cos(u), sin(u)) for u over (-5.0, 5.0)

(png, hires.png, pdf)

A parametric plot with multiple expressions with the same range:

>>> plot_parametric((cos(u), sin(u)), (u, cos(u)), (u, -10, 10))
Plot object containing:
[0]: parametric cartesian line: (cos(u), sin(u)) for u over (-10.0, 10.0)
[1]: parametric cartesian line: (u, cos(u)) for u over (-10.0, 10.0)

(png, hires.png, pdf)

A parametric plot with multiple expressions with different ranges for each curve:

>>> plot_parametric((cos(u), sin(u), (u, -5, 5)),
...     (cos(u), u, (u, -5, 5)))
Plot object containing:
[0]: parametric cartesian line: (cos(u), sin(u)) for u over (-5.0, 5.0)
[1]: parametric cartesian line: (cos(u), u) for u over (-5.0, 5.0)

(png, hires.png, pdf)

Notes

The plotting uses an adaptive algorithm which samples recursively to accurately plot the curve. The adaptive algorithm uses a random point near the midpoint of two points that has to be further sampled. Hence, repeating the same plot command can give slightly different results because of the random sampling.

If there are multiple plots, then the same optional arguments are applied to all the plots drawn in the same canvas. If you want to set these options separately, you can index the returned Plot object and set it.

For example, when you specify line_color once, it would be applied simultaneously to both series.

>>> from sympy import pi
>>> expr1 = (u, cos(2*pi*u)/2 + 1/2)
>>> expr2 = (u, sin(2*pi*u)/2 + 1/2)
>>> p = plot_parametric(expr1, expr2, (u, 0, 1), line_color='blue')

(png, hires.png, pdf)

If you want to specify the line color for the specific series, you should index each item and apply the property manually.

>>> p[0].line_color = 'red'
>>> p.show()

(png, hires.png, pdf)

sympy.plotting.plot.plot3d(*args, show=True, **kwargs)[source]

Plots a 3D surface plot.

Usage

Single plot

plot3d(expr, range_x, range_y, **kwargs)

If the ranges are not specified, then a default range of (-10, 10) is used.

Multiple plot with the same range.

plot3d(expr1, expr2, range_x, range_y, **kwargs)

If the ranges are not specified, then a default range of (-10, 10) is used.

Multiple plots with different ranges.

plot3d((expr1, range_x, range_y), (expr2, range_x, range_y), ..., **kwargs)

Ranges have to be specified for every expression.

Default range may change in the future if a more advanced default range detection algorithm is implemented.

Arguments

expr : Expression representing the function along x.

range_x: (x, 0, 5), A 3-tuple denoting the range of the x variable.

range_y: (y, 0, 5), A 3-tuple denoting the range of the y

variable.

Keyword Arguments

Arguments for SurfaceOver2DRangeSeries class:

nb_of_points_x: int. The x range is sampled uniformly at nb_of_points_x of points.

nb_of_points_y: int. The y range is sampled uniformly at nb_of_points_y of points.

Aesthetics:

surface_color: Function which returns a float. Specifies the color for the surface of the plot. See sympy.plotting.Plot for more details.

If there are multiple plots, then the same series arguments are applied to all the plots. If you want to set these options separately, you can index the returned Plot object and set it.

Arguments for Plot class:

title : str. Title of the plot. size : (float, float), optional A tuple in the form (width, height) in inches to specify the size of the overall figure. The default value is set to None, meaning the size will be set by the default backend.

Examples

>>> from sympy import symbols
>>> from sympy.plotting import plot3d
>>> x, y = symbols('x y')

Single plot

>>> plot3d(x*y, (x, -5, 5), (y, -5, 5))
Plot object containing:
[0]: cartesian surface: x*y for x over (-5.0, 5.0) and y over (-5.0, 5.0)

(png, hires.png, pdf)

Multiple plots with same range

>>> plot3d(x*y, -x*y, (x, -5, 5), (y, -5, 5))
Plot object containing:
[0]: cartesian surface: x*y for x over (-5.0, 5.0) and y over (-5.0, 5.0)
[1]: cartesian surface: -x*y for x over (-5.0, 5.0) and y over (-5.0, 5.0)

(png, hires.png, pdf)

Multiple plots with different ranges.

>>> plot3d((x**2 + y**2, (x, -5, 5), (y, -5, 5)),
...     (x*y, (x, -3, 3), (y, -3, 3)))
Plot object containing:
[0]: cartesian surface: x**2 + y**2 for x over (-5.0, 5.0) and y over (-5.0, 5.0)
[1]: cartesian surface: x*y for x over (-3.0, 3.0) and y over (-3.0, 3.0)

(png, hires.png, pdf)

sympy.plotting.plot.plot3d_parametric_line(*args, show=True, **kwargs)[source]

Plots a 3D parametric line plot.

Usage

Single plot:

plot3d_parametric_line(expr_x, expr_y, expr_z, range, **kwargs)

If the range is not specified, then a default range of (-10, 10) is used.

Multiple plots.

plot3d_parametric_line((expr_x, expr_y, expr_z, range), ..., **kwargs)

Ranges have to be specified for every expression.

Default range may change in the future if a more advanced default range detection algorithm is implemented.

Arguments

expr_x : Expression representing the function along x.

expr_y : Expression representing the function along y.

expr_z : Expression representing the function along z.

range: (u, 0, 5), A 3-tuple denoting the range of the parameter variable.

Keyword Arguments

Arguments for Parametric3DLineSeries class.

nb_of_points: The range is uniformly sampled at nb_of_points number of points.

Aesthetics:

line_color: string, or float, or function, optional

Specifies the color for the plot. See Plot to see how to set color for the plots. Note that by setting line_color, it would be applied simultaneously to all the series.

label: str

The label to the plot. It will be used when called with legend=True to denote the function with the given label in the plot.

If there are multiple plots, then the same series arguments are applied to all the plots. If you want to set these options separately, you can index the returned Plot object and set it.

Arguments for Plot class.

title : str. Title of the plot.

size(float, float), optional

A tuple in the form (width, height) in inches to specify the size of the overall figure. The default value is set to None, meaning the size will be set by the default backend.

Examples

>>> from sympy import symbols, cos, sin
>>> from sympy.plotting import plot3d_parametric_line
>>> u = symbols('u')

Single plot.

>>> plot3d_parametric_line(cos(u), sin(u), u, (u, -5, 5))
Plot object containing:
[0]: 3D parametric cartesian line: (cos(u), sin(u), u) for u over (-5.0, 5.0)

(png, hires.png, pdf)

Multiple plots.

>>> plot3d_parametric_line((cos(u), sin(u), u, (u, -5, 5)),
...     (sin(u), u**2, u, (u, -5, 5)))
Plot object containing:
[0]: 3D parametric cartesian line: (cos(u), sin(u), u) for u over (-5.0, 5.0)
[1]: 3D parametric cartesian line: (sin(u), u**2, u) for u over (-5.0, 5.0)

(png, hires.png, pdf)

sympy.plotting.plot.plot3d_parametric_surface(*args, show=True, **kwargs)[source]

Plots a 3D parametric surface plot.

Explanation

Single plot.

plot3d_parametric_surface(expr_x, expr_y, expr_z, range_u, range_v, **kwargs)

If the ranges is not specified, then a default range of (-10, 10) is used.

Multiple plots.

plot3d_parametric_surface((expr_x, expr_y, expr_z, range_u, range_v), ..., **kwargs)

Ranges have to be specified for every expression.

Default range may change in the future if a more advanced default range detection algorithm is implemented.

Arguments

expr_x: Expression representing the function along x.

expr_y: Expression representing the function along y.

expr_z: Expression representing the function along z.

range_u: (u, 0, 5), A 3-tuple denoting the range of the u variable.

range_v: (v, 0, 5), A 3-tuple denoting the range of the v variable.

Keyword Arguments

Arguments for ParametricSurfaceSeries class:

nb_of_points_u: int. The u range is sampled uniformly at nb_of_points_v of points

nb_of_points_y: int. The v range is sampled uniformly at nb_of_points_y of points

Aesthetics:

surface_color: Function which returns a float. Specifies the color for the surface of the plot. See sympy.plotting.Plot for more details.

If there are multiple plots, then the same series arguments are applied for all the plots. If you want to set these options separately, you can index the returned Plot object and set it.

Arguments for Plot class:

title : str. Title of the plot. size : (float, float), optional A tuple in the form (width, height) in inches to specify the size of the overall figure. The default value is set to None, meaning the size will be set by the default backend.

Examples

>>> from sympy import symbols, cos, sin
>>> from sympy.plotting import plot3d_parametric_surface
>>> u, v = symbols('u v')

Single plot.

>>> plot3d_parametric_surface(cos(u + v), sin(u - v), u - v,
...     (u, -5, 5), (v, -5, 5))
Plot object containing:
[0]: parametric cartesian surface: (cos(u + v), sin(u - v), u - v) for u over (-5.0, 5.0) and v over (-5.0, 5.0)

(png, hires.png, pdf)

sympy.plotting.plot_implicit.plot_implicit(expr, x_var=None, y_var=None, adaptive=True, depth=0, points=300, line_color='blue', show=True, **kwargs)[source]

A plot function to plot implicit equations / inequalities.

Arguments

• expr : The equation / inequality that is to be plotted.

• x_var (optional) : symbol to plot on x-axis or tuple giving symbol and range as (symbol, xmin, xmax)

• y_var (optional) : symbol to plot on y-axis or tuple giving symbol and range as (symbol, ymin, ymax)

If neither x_var nor y_var are given then the free symbols in the expression will be assigned in the order they are sorted.

The following keyword arguments can also be used:

• adaptive Boolean. The default value is set to True. It has to be

set to False if you want to use a mesh grid.

• depth integer. The depth of recursion for adaptive mesh grid.

Default value is 0. Takes value in the range (0, 4).

• points integer. The number of points if adaptive mesh grid is not

used. Default value is 300.

• show Boolean. Default value is True. If set to False, the plot will

not be shown. See Plot for further information.

• title string. The title for the plot.

• xlabel string. The label for the x-axis

• ylabel string. The label for the y-axis

Aesthetics options:

• line_color: float or string. Specifies the color for the plot.

See Plot to see how to set color for the plots. Default value is “Blue”

plot_implicit, by default, uses interval arithmetic to plot functions. If the expression cannot be plotted using interval arithmetic, it defaults to a generating a contour using a mesh grid of fixed number of points. By setting adaptive to False, you can force plot_implicit to use the mesh grid. The mesh grid method can be effective when adaptive plotting using interval arithmetic, fails to plot with small line width.

Examples

Plot expressions:

>>> from sympy import plot_implicit, symbols, Eq, And
>>> x, y = symbols('x y')

Without any ranges for the symbols in the expression:

>>> p1 = plot_implicit(Eq(x**2 + y**2, 5))

(png, hires.png, pdf)

With the range for the symbols:

>>> p2 = plot_implicit(
...     Eq(x**2 + y**2, 3), (x, -3, 3), (y, -3, 3))

(png, hires.png, pdf)

With depth of recursion as argument:

>>> p3 = plot_implicit(
...     Eq(x**2 + y**2, 5), (x, -4, 4), (y, -4, 4), depth = 2)

(png, hires.png, pdf)

Using mesh grid and not using adaptive meshing:

>>> p4 = plot_implicit(
...     Eq(x**2 + y**2, 5), (x, -5, 5), (y, -2, 2),

(png, hires.png, pdf)

Using mesh grid without using adaptive meshing with number of points specified:

>>> p5 = plot_implicit(
...     Eq(x**2 + y**2, 5), (x, -5, 5), (y, -2, 2),

(png, hires.png, pdf)

Plotting regions:

>>> p6 = plot_implicit(y > x**2)

(png, hires.png, pdf)

Plotting Using boolean conjunctions:

>>> p7 = plot_implicit(And(y > x, y > -x))

(png, hires.png, pdf)

When plotting an expression with a single variable (y - 1, for example), specify the x or the y variable explicitly:

>>> p8 = plot_implicit(y - 1, y_var=y)
>>> p9 = plot_implicit(x - 1, x_var=x)

## PlotGrid Class¶

class sympy.plotting.plot.PlotGrid(nrows, ncolumns, *args, show=True, size=None, **kwargs)[source]

This class helps to plot subplots from already created sympy plots in a single figure.

Examples

>>> from sympy import symbols
>>> from sympy.plotting import plot, plot3d, PlotGrid
>>> x, y = symbols('x, y')
>>> p1 = plot(x, x**2, x**3, (x, -5, 5))
>>> p2 = plot((x**2, (x, -6, 6)), (x, (x, -5, 5)))
>>> p3 = plot(x**3, (x, -5, 5))
>>> p4 = plot3d(x*y, (x, -5, 5), (y, -5, 5))

Plotting vertically in a single line:

>>> PlotGrid(2, 1 , p1, p2)
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: x for x over (-5.0, 5.0)
[1]: cartesian line: x**2 for x over (-5.0, 5.0)
[2]: cartesian line: x**3 for x over (-5.0, 5.0)
Plot[1]:Plot object containing:
[0]: cartesian line: x**2 for x over (-6.0, 6.0)
[1]: cartesian line: x for x over (-5.0, 5.0)

(png, hires.png, pdf)

Plotting horizontally in a single line:

>>> PlotGrid(1, 3 , p2, p3, p4)
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: x**2 for x over (-6.0, 6.0)
[1]: cartesian line: x for x over (-5.0, 5.0)
Plot[1]:Plot object containing:
[0]: cartesian line: x**3 for x over (-5.0, 5.0)
Plot[2]:Plot object containing:
[0]: cartesian surface: x*y for x over (-5.0, 5.0) and y over (-5.0, 5.0)

(png, hires.png, pdf)

Plotting in a grid form:

>>> PlotGrid(2, 2, p1, p2 ,p3, p4)
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: x for x over (-5.0, 5.0)
[1]: cartesian line: x**2 for x over (-5.0, 5.0)
[2]: cartesian line: x**3 for x over (-5.0, 5.0)
Plot[1]:Plot object containing:
[0]: cartesian line: x**2 for x over (-6.0, 6.0)
[1]: cartesian line: x for x over (-5.0, 5.0)
Plot[2]:Plot object containing:
[0]: cartesian line: x**3 for x over (-5.0, 5.0)
Plot[3]:Plot object containing:
[0]: cartesian surface: x*y for x over (-5.0, 5.0) and y over (-5.0, 5.0)

(png, hires.png, pdf)

## Series Classes¶

class sympy.plotting.plot.BaseSeries[source]

Base class for the data objects containing stuff to be plotted.

Explanation

The backend should check if it supports the data series that it’s given. (eg TextBackend supports only LineOver1DRange). It’s the backend responsibility to know how to use the class of data series that it’s given.

Some data series classes are grouped (using a class attribute like is_2Dline) according to the api they present (based only on convention). The backend is not obliged to use that api (eg. The LineOver1DRange belongs to the is_2Dline group and presents the get_points method, but the TextBackend does not use the get_points method).

class sympy.plotting.plot.Line2DBaseSeries[source]

A base class for 2D lines.

• adding the label, steps and only_integers options

• making is_2Dline true

• defining get_segments and get_color_array

get_data()[source]

Return lists of coordinates for plotting the line.

Returns

x: list

List of x-coordinates

y: list

List of y-coordinates

y: list

List of z-coordinates in case of Parametric3DLineSeries

class sympy.plotting.plot.LineOver1DRangeSeries(expr, var_start_end, **kwargs)[source]

Representation for a line consisting of a SymPy expression over a range.

get_points()[source]

Return lists of coordinates for plotting. Depending on the $$adaptive$$ option, this function will either use an adaptive algorithm or it will uniformly sample the expression over the provided range.

Returns

x: list

List of x-coordinates

y: list

List of y-coordinates

Explanation

The adaptive sampling is done by recursively checking if three points are almost collinear. If they are not collinear, then more points are added between those points.

References

R637

Adaptive polygonal approximation of parametric curves, Luiz Henrique de Figueiredo.

class sympy.plotting.plot.Parametric2DLineSeries(expr_x, expr_y, var_start_end, **kwargs)[source]

Representation for a line consisting of two parametric sympy expressions over a range.

get_points()[source]

Return lists of coordinates for plotting. Depending on the $$adaptive$$ option, this function will either use an adaptive algorithm or it will uniformly sample the expression over the provided range.

Returns

x: list

List of x-coordinates

y: list

List of y-coordinates

Explanation

The adaptive sampling is done by recursively checking if three points are almost collinear. If they are not collinear, then more points are added between those points.

References

R638

Adaptive polygonal approximation of parametric curves, Luiz Henrique de Figueiredo.

class sympy.plotting.plot.Line3DBaseSeries[source]

A base class for 3D lines.

Most of the stuff is derived from Line2DBaseSeries.

class sympy.plotting.plot.Parametric3DLineSeries(expr_x, expr_y, expr_z, var_start_end, **kwargs)[source]

Representation for a 3D line consisting of three parametric sympy expressions and a range.

class sympy.plotting.plot.SurfaceBaseSeries[source]

A base class for 3D surfaces.

class sympy.plotting.plot.SurfaceOver2DRangeSeries(expr, var_start_end_x, var_start_end_y, **kwargs)[source]

Representation for a 3D surface consisting of a sympy expression and 2D range.

class sympy.plotting.plot.ParametricSurfaceSeries(expr_x, expr_y, expr_z, var_start_end_u, var_start_end_v, **kwargs)[source]

Representation for a 3D surface consisting of three parametric sympy expressions and a range.

class sympy.plotting.plot_implicit.ImplicitSeries(expr, var_start_end_x, var_start_end_y, has_equality, use_interval_math, depth, nb_of_points, line_color)[source]

Representation for Implicit plot

## Backends¶

class sympy.plotting.plot.BaseBackend(parent)[source]

Base class for all backends. A backend represents the plotting library, which implements the necessary functionalities in order to use SymPy plotting functions.

How the plotting module works:

1. Whenever a plotting function is called, the provided expressions are

processed and a list of instances of the $$BaseSeries$$ class is created, containing the necessary information to plot the expressions (eg the expression, ranges, series name, …). Eventually, these objects will generate the numerical data to be plotted.

2. A Plot object is instantiated, which stores the list of series and the

main attributes of the plot (eg axis labels, title, …).

3. When the “show” command is executed, a new backend is instantiated,

which loops through each series object to generate and plot the numerical data. The backend is also going to set the axis labels, title, …, according to the values stored in the Plot instance.

The backend should check if it supports the data series that it’s given (eg TextBackend supports only LineOver1DRange).

It’s the backend responsibility to know how to use the class of data series that it’s given. Note that the current implementation of the $$*Series$$ classes is “matplotlib-centric”: the numerical data returned by the $$get_points$$ and $$get_meshes$$ methods is meant to be used directly by Matplotlib. Therefore, the new backend will have to pre-process the numerical data to make it compatible with the chosen plotting library. Keep in mind that future SymPy versions may improve the $$*Series$$ classes in order to return numerical data “non-matplotlib-centric”, hence if you code a new backend you have the responsibility to check if its working on each SymPy release.

Please, explore the $$MatplotlibBackend$$ source code to understand how a backend should be coded.

class sympy.plotting.plot.MatplotlibBackend(parent)[source]

This class implements the functionalities to use Matplotlib with SymPy plotting functions.

static get_segments(x, y, z=None)[source]

Convert two list of coordinates to a list of segments to be used with Matplotlib’s LineCollection.

Parameters

x: list

List of x-coordinates

y: list

List of y-coordinates

z: list

List of z-coordinates for a 3D line.

process_series()[source]

Iterates over every Plot object and further calls _process_series()

class sympy.plotting.plot.TextBackend(parent)[source]

## Pyglet Plotting¶

This is the documentation for the old plotting module that uses pyglet. This module has some limitations and is not actively developed anymore. For an alternative you can look at the new plotting module.

The pyglet plotting module can do nice 2D and 3D plots that can be controlled by console commands as well as keyboard and mouse, with the only dependency being pyglet.

Here is the simplest usage:

>>> from sympy import var
>>> from sympy.plotting.pygletplot import PygletPlot as Plot
>>> var('x y z')
>>> Plot(x*y**3-y*x**3)

To see lots of plotting examples, see examples/pyglet_plotting.py and try running it in interactive mode (python -i plotting.py):

\$ python -i examples/pyglet_plotting.py

And type for instance example(7) or example(11).

## Plot Window Controls¶

Camera

Keys

Sensitivity Modifier

SHIFT

Zoom

R and F, Page Up and Down, Numpad + and -

Rotate View X,Y axis

Rotate View Z axis

Q and E, Numpad 7 and 9

Rotate Ordinate Z axis

Z and C, Numpad 1 and 3

View XY

F1

View XZ

F2

View YZ

F3

View Perspective

F4

Reset

Axes

Keys

Toggle Visible

F5

Toggle Colors

F6

Window

Keys

Close

ESCAPE

Screenshot

F8

The mouse can be used to rotate, zoom, and translate by dragging the left, middle, and right mouse buttons respectively.

## Coordinate Modes¶

Plot supports several curvilinear coordinate modes, and they are independent for each plotted function. You can specify a coordinate mode explicitly with the ‘mode’ named argument, but it can be automatically determined for cartesian or parametric plots, and therefore must only be specified for polar, cylindrical, and spherical modes.

Specifically, Plot(function arguments) and Plot.__setitem__(i, function arguments) (accessed using array-index syntax on the Plot instance) will interpret your arguments as a cartesian plot if you provide one function and a parametric plot if you provide two or three functions. Similarly, the arguments will be interpreted as a curve is one variable is used, and a surface if two are used.

Supported mode names by number of variables:

• 1 (curves): parametric, cartesian, polar

• 2 (surfaces): parametric, cartesian, cylindrical, spherical

Note that function parameters are given as option strings of the form “key1=value1; key2 = value2” (spaces are truncated). Keyword arguments given directly to plot apply to the plot itself.

## Specifying Intervals for Variables¶

The basic format for variable intervals is [var, min, max, steps]. However, the syntax is quite flexible, and arguments not specified are taken from the defaults for the current coordinate mode:

>>> Plot(x**2) # implies [x,-5,5,100]
>>> Plot(x**2, [], []) # [x,-1,1,40], [y,-1,1,40]
>>> Plot(x**2-y**2, [100], [100]) # [x,-1,1,100], [y,-1,1,100]
>>> Plot(x**2, [x,-13,13,100])
>>> Plot(x**2, [-13,13]) # [x,-13,13,100]
>>> Plot(x**2, [x,-13,13]) # [x,-13,13,100]
>>> Plot(1*x, [], [x], 'mode=cylindrical') # [unbound_theta,0,2*Pi,40], [x,-1,1,20]

## Using the Interactive Interface¶

>>> p = Plot(visible=False)
>>> f = x**2
>>> p[1] = f
>>> p[2] = f.diff(x)
>>> p[3] = f.diff(x).diff(x)
>>> p
[1]: x**2, 'mode=cartesian'
[2]: 2*x, 'mode=cartesian'
[3]: 2, 'mode=cartesian'
>>> p.show()
>>> p.clear()
>>> p
<blank plot>
>>> p[1] =  x**2+y**2
>>> p[1].style = 'solid'
>>> p[2] = -x**2-y**2
>>> p[2].style = 'wireframe'
>>> p[1].color = z, (0.4,0.4,0.9), (0.9,0.4,0.4)
>>> p[1].style = 'both'
>>> p[2].style = 'both'
>>> p.close()

## Using Custom Color Functions¶

The following code plots a saddle and color it by the magnitude of its gradient:

>>> fz = x**2-y**2
>>> Fx, Fy, Fz = fz.diff(x), fz.diff(y), 0
>>> p[1] = fz, 'style=solid'
>>> p[1].color = (Fx**2 + Fy**2 + Fz**2)**(0.5)

The coloring algorithm works like this:

1. Evaluate the color function(s) across the curve or surface.

2. Find the minimum and maximum value of each component.

3. Scale each component to the color gradient.

When not specified explicitly, the default color gradient is f(0.0)=(0.4,0.4,0.4) -> f(1.0)=(0.9,0.9,0.9). In our case, everything is gray-scale because we have applied the default color gradient uniformly for each color component. When defining a color scheme in this way, you might want to supply a color gradient as well:

>>> p[1].color = (Fx**2 + Fy**2 + Fz**2)**(0.5), (0.1,0.1,0.9), (0.9,0.1,0.1)

Here’s a color gradient with four steps:

>>> gradient = [ 0.0, (0.1,0.1,0.9), 0.3, (0.1,0.9,0.1),
...              0.7, (0.9,0.9,0.1), 1.0, (1.0,0.0,0.0) ]
>>> p[1].color = (Fx**2 + Fy**2 + Fz**2)**(0.5), gradient

The other way to specify a color scheme is to give a separate function for each component r, g, b. With this syntax, the default color scheme is defined:

>>> p[1].color = z,y,x, (0.4,0.4,0.4), (0.9,0.9,0.9)

This maps z->red, y->green, and x->blue. In some cases, you might prefer to use the following alternative syntax:

>>> p[1].color = z,(0.4,0.9), y,(0.4,0.9), x,(0.4,0.9)

You can still use multi-step gradients with three-function color schemes.

## Plotting Geometric Entities¶

The plotting module is capable of plotting some 2D geometric entities like line, circle and ellipse. The following example plots a circle centred at origin and of radius 2 units.

>>> from sympy import *
>>> x,y = symbols('x y')
>>> plot_implicit(Eq(x**2+y**2, 4))

Similarly, plot_implicit() may be used to plot any 2-D geometric structure from its implicit equation.

Plotting polygons (Polygon, RegularPolygon, Triangle) are not supported directly.

## Plotting with ASCII art¶

sympy.plotting.textplot.textplot(expr, a, b, W=55, H=21)[source]

Print a crude ASCII art plot of the SymPy expression ‘expr’ (which should contain a single symbol, e.g. x or something else) over the interval [a, b].

Examples

>>> from sympy import Symbol, sin
>>> from sympy.plotting import textplot
>>> t = Symbol('t')
>>> textplot(sin(t)*t, 0, 15)
14 |                                                  ...
|                                                     .
|                                                 .
|                                                      .
|                                                .
|                            ...
|                           /   .               .
|                          /
|                         /      .
|                        .        .            .
1.5 |----.......--------------------------------------------
|....       \           .          .
|            \         /                      .
|             ..      /             .
|               \    /                       .
|                ....
|                                    .
|                                     .     .
|
|                                      .   .
-11 |_______________________________________________________
0                          7.5                        15