3D general plotting

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, or widgets from the panel module, then users will have to modify the params dictionary in order to make it work with ipywidgets. Refer to Interactive module for more information.

spb.graphics.functions_3d.line_parametric_3d(expr_x, expr_y, expr_z, range_p=None, label=None, rendering_kw=None, colorbar=True, use_cm=True, **kwargs)[source]

Plots a 3D parametric curve.

Parameters:

expr_x

The expression representing the component along the x-axis of the parametric function. It can either be a symbolic expression representing the function of one variable to be plotted, or a numerical function of one variable, supporting vectorization. In the latter case the following keyword arguments are not supported: params, sum_bound.

expr_y

The expression representing the component along the y-axis of the parametric function. It can either be a symbolic expression representing the function of one variable to be plotted, or a numerical function of one variable, supporting vectorization. In the latter case the following keyword arguments are not supported: params, sum_bound.

expr_z

The expression representing the component along the z-axis of the parametric function. It can either be a symbolic expression representing the function of one variable to be plotted, or a numerical function of one variable, supporting vectorization. In the latter case the following keyword arguments are not supported: params, sum_bound.

range_ptuple, Tuple

A 3-tuple (symb, min, max) denoting the range of the parameter. Default values: min=-10 and max=10.

labelstr

Set the label associated to this series, which will be eventually shown on the legend or colorbar.

rendering_kwdict

A dictionary of keyword arguments to be passed to the renderers in order to further customize the appearance of the line. Here are some useful links for the supported plotting libraries:

Matplotlib:
- for solid lines: https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.plot.html
- for colormap-based lines: https://matplotlib.org/stable/api/collections_api.html#matplotlib.collections.LineCollection
- for scatters: https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.scatter.html
Bokeh:
- for solid lines: https://docs.bokeh.org/en/latest/docs/reference/plotting.html#bokeh.plotting.Figure.line
- for scatter: https://docs.bokeh.org/en/latest/docs/reference/plotting/figure.html#bokeh.plotting.Figure.scatter
Plotly: https://plotly.com/python/line-and-scatter/

colorbarbool

Toggle the visibility of the colorbar associated to the current data series. Note that a colorbar is only visible if use_cm=True and color_func is not None. Default value: True.

use_cmbool

Toggle the use of a colormap. By default, some series might use a colormap to display the necessary data. Setting this attribute to False will inform the associated renderer to use solid color. Related parameters: color_func. Default value: False.

color_func

Define a custom 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.
- 3 arguments: f(x, y, z) where x, y, z are the coordinates of the points.
- 4 arguments: f(x, y, z, t).
A symbolic expression having at most as many free symbols as expr_x or expr_y or expr_z.
None: the default value (color mapping according to the parameter).

Alias of this parameters: tp.

colorbar_ticks_formattertick_formatter_multiples_of

An object of type tick_formatter_multiples_of which will be used to place tick values on the colorbar at each multiple of a specified quantity. This only works when use_cm=True.

detect_polesbool, str

Chose whether to detect and correctly plot the roots of the denominator. 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. This algorithm can be used to visualize jump discontinuities as well as essential discontinuities.
a symbolic approach based on the continuous_domain function from the sympy.calculus.util module, which computes the locations of essential discontinuities. If any are found, vertical lines will be shown.

Possible options:

False: No poles detection
True: Poles detection with the numerical algorithm
‘symbolic’: Poles detection with numerical and symbolic algorithms

Default value: False.

epsfloat

An arbitrary small value used by the detect_poles numerical algorithm. Before changing this value, it is recommended to increase the number of discretization points. Related parameters: detect_poles. It must be: 0 ≤ eps < ∞. Default value: 0.01.

excludelist

A list of numerical values along the parameter which are going to be excluded from the evaluation. In practice, it introduces discontinuities in the resulting line.

force_real_evalbool

By default, numerical evaluation is performed over complex numbers, which is slower but produces correct results. However, when the symbolic expression is converted to a numerical function with lambdify, the resulting function may not like to be evaluated over complex numbers. In such cases, forcing the evaluation to be performed over real numbers might be a good choice. The plotting module should be able to detect such occurences and automatically activate this option. If that is not the case, or evaluation performance is of paramount importance, set this parameter to True, but be aware that it might produce wrong results. Default value: False.

is_filledbool

Whether scatter’s markers are filled or void. Default value: True.

is_scatterbool

If True it represent a scatter plot, otherwise a continuous line. Default value: False.

line_color

For back-compatibility with old sympy.plotting. Use rendering_kw in order to fully customize the appearance of the line/scatter.

modules

Specify the evaluation modules to be used by lambdify. If not specified, the evaluation will be done with NumPy/SciPy.

n1int

Number of discretization points along the parameter to be used in the evaluation. An alias of this parameter is n. Related parameters: xscale. It must be: 2 ≤ n1 < ∞. Default value: 1000.

only_integersbool

Discretize the domain using only integer numbers. When this parameter is True, the number of discretization points is choosen by the algorithm. Default value: False.

paramsdict, optional

A dictionary mapping symbols to parameters. If provided, this dictionary enables the interactive-widgets plot.

When calling a plotting function, the parameter can be specified with:

a widget from the ipywidgets module.
a widget from the panel module.
a tuple of the form:
(default, min, max, N, tick_format, label, spacing), which will instantiate a ipywidgets.widgets.widget_float.FloatSlider or a ipywidgets.widgets.widget_float.FloatLogSlider, depending on the spacing strategy. In particular:
- default, min, maxfloat
  Default value, minimum value and maximum value of the slider, respectively. Must be finite numbers. The order of these 3 numbers is not important: the module will figure it out which is what.
- Nint, optional
  Number of steps of the slider.
- tick_formatstr or None, optional
  Provide a formatter for the tick value of the slider. Default to ".2f".
- label: str, optional
  Custom text associated to the slider.
- spacingstr, optional
  Specify the discretization spacing. Default to "linear", can be changed to "log".

Notes:

parameters cannot be linked together (ie, one parameter cannot depend on another one).
If a widget returns multiple numerical values (like panel.widgets.slider.RangeSlider or ipywidgets.widgets.widget_float.FloatRangeSlider), then a corresponding number of symbols must be provided.

Here follows a couple of examples. If imodule="panel":

import panel as pn
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: pn.widgets.FloatSlider(value=1, start=0, end=5), # same slider as above
    (c, d): pn.widgets.RangeSlider(value=(-1, 1), start=-3, end=3, step=0.1)
}

Or with imodule="ipywidgets":

import ipywidgets as w
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: w.FloatSlider(value=1, min=0, max=5), # same slider as above
    (c, d): w.FloatRangeSlider(value=(-1, 1), min=-3, max=3, step=0.1)
}

When instantiating a data series directly, params must be a dictionary mapping symbols to numerical values.

Let series be any data series. Then series.params returns a dictionary mapping symbols to numerical values.

show_in_legendbool

Toggle the visibility of the data series on the legend. Default value: True.

sum_boundint

When plotting sums, the expression will be pre-processed in order to replace lower/upper bounds set to +/- infinity with this +/- numerical value. Note: the higher this number, the slower the evaluation, but the more accurate the plot. It must be: 0 ≤ sum_bound < ∞. Default value: 1000.

txcallable

Numerical transformation function to be applied to the data on the x-axis.

tycallable

Numerical transformation function to be applied to the data on the y-axis.

tzcallable

Numerical transformation function to be applied to the data on the z-axis.

xscalestr

Discretization strategy along the parameter. Possible options: [‘linear’, ‘log’] Default value: ‘linear’.

Returns:

serieslist: A list containing one instance of Parametric3DLineSeries.

See also

spb.graphics.functions_2d.line_parametric_2d, list_3d

Examples

>>> from sympy import symbols, cos, sin, pi, root
>>> from spb import *
>>> t = symbols('t')

(Source code)

Single plot.

>>> graphics(line_parametric_3d(cos(t), sin(t), t, (t, -5, 5)))
Plot object containing:
[0]: 3D parametric cartesian line: (cos(t), sin(t), t) for t over (-5, 5)

(Source code, png)

Customize the appearance by setting a label to the colorbar, changing the colormap, the line width, and the ticks on the colorbar.

>>> graphics(
...     line_parametric_3d(
...         3 * sin(t) + 2 * sin(3 * t), cos(t) - 2 * cos(3 * t), cos(5 * t),
...         (t, 0, 2 * pi), "t [rad]", {"cmap": "hsv", "lw": 1.5},
...         colorbar_ticks_formatter=multiples_of_pi_over_2()
...     )
... )
Plot object containing:
[0]: 3D parametric cartesian line: (3*sin(t) + 2*sin(3*t), cos(t) - 2*cos(3*t), cos(5*t)) for t over (0, 2*pi)

(Source code, png)

Plot multiple parametric 3D lines with different ranges:

>>> a, b, n = 2, 1, 4
>>> p, r, s = symbols("p r s")
>>> xp = a * cos(p) * cos(n * p)
>>> yp = a * sin(p) * cos(n * p)
>>> zp = b * cos(n * p)**2 + pi
>>> xr = root(r, 3) * cos(r)
>>> yr = root(r, 3) * sin(r)
>>> zr = 0
>>> graphics(
...     line_parametric_3d(
...         xp, yp, zp, (p, 0, pi if n % 2 == 1 else 2 * pi), "petals",
...         use_cm=False),
...     line_parametric_3d(xr, yr, zr, (r, 0, 6*pi), "roots",
...         use_cm=False),
...     line_parametric_3d(-sin(s)/3, 0, s, (s, 0, pi), "stem",
...         use_cm=False)
... )
Plot object containing:
[0]: 3D parametric cartesian line: (2*cos(p)*cos(4*p), 2*sin(p)*cos(4*p), cos(4*p)**2 + pi) for p over (0, 2*pi)
[1]: 3D parametric cartesian line: (r**(1/3)*cos(r), r**(1/3)*sin(r), 0) for r over (0, 6*pi)
[2]: 3D parametric cartesian line: (-sin(s)/3, 0, s) for s over (0, pi)

(Source code, png)

Plotting a numerical function instead of a symbolic expression, using Plotly:

from spb import *
import numpy as np
fx = lambda t: (1 + 0.25 * np.cos(75 * t)) * np.cos(t)
fy = lambda t: (1 + 0.25 * np.cos(75 * t)) * np.sin(t)
fz = lambda t: t + 2 * np.sin(75 * t)
graphics(
    line_parametric_3d(fx, fy, fz, ("t", 0, 6 * np.pi),
        rendering_kw={"line": {"colorscale": "bluered"}}, n=1e04),
    title="Helical Toroid", backend=PB)

(Source code, png)

Interactive-widget plot of the parametric line over a tennis ball. Refer to the interactive sub-module documentation to learn more about the params dictionary. This plot illustrates:

combining together different plots.
the use of prange (parametric plotting range).
the use of the params dictionary to specify sliders in their basic form: (default, min, max).

from sympy import *
from spb import *
import k3d
a, b, s, e, t = symbols("a, b, s, e, t")
c = 2 * sqrt(a * b)
r = a + b
params = {
    a: (1.5, 0, 2),
    b: (1, 0, 2),
    s: (0, 0, 2),
    e: (2, 0, 2)
}
graphics(
    surface_revolution(
        (r * cos(t), r * sin(t)), (t, 0, pi),
        params=params, n=50, parallel_axis="x", show_curve=False,
        rendering_kw={"color":0x353535},
        force_real_eval=True
    ),
    line_parametric_3d(
        a * cos(t) + b * cos(3 * t),
        a * sin(t) - b * sin(3 * t),
        c * sin(2 * t), prange(t, s*pi, e*pi),
        rendering_kw={"color_map": k3d.matplotlib_color_maps.Summer},
        params=params
    ),
    backend=KB
)

(Source code, small.png)

spb.graphics.functions_3d.list_3d(list_x, list_y, list_z, label=None, rendering_kw=None, **kwargs)[source]

Plots lists of coordinates in 3D space.

Parameters:

list_x

Coordinates for the x-axis. It can be a list or a numper array.

list_y

Coordinates for the y-axis. It can be a list or a numper array.

list_z

Coordinates for the z-axis. It can be a list or a numper array.

labelstr

Set the label associated to this series, which will be eventually shown on the legend or colorbar.

rendering_kwdict

A dictionary of keyword arguments to be passed to the renderers in order to further customize the appearance of the line. Here are some useful links for the supported plotting libraries:

Matplotlib:
- for solid lines: https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.plot.html
- for colormap-based lines: https://matplotlib.org/stable/api/collections_api.html#matplotlib.collections.LineCollection
- for scatters: https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.scatter.html
Bokeh:
- for solid lines: https://docs.bokeh.org/en/latest/docs/reference/plotting.html#bokeh.plotting.Figure.line
- for scatter: https://docs.bokeh.org/en/latest/docs/reference/plotting/figure.html#bokeh.plotting.Figure.scatter
Plotly: https://plotly.com/python/line-and-scatter/

color_funccallable

A color function to be applied to the numerical data. It can be:

None: no color function.
callable: a function accepting:
- no arguments: this can be used to return an array of precomputed values.
- three arguments, the x, y, z coordinates.

colorbarbool

Toggle the visibility of the colorbar associated to the current data series. Note that a colorbar is only visible if use_cm=True and color_func is not None. Default value: True.

colorbar_ticks_formattertick_formatter_multiples_of

An object of type tick_formatter_multiples_of which will be used to place tick values on the colorbar at each multiple of a specified quantity. This only works when use_cm=True.

is_filledbool

Whether scatter’s markers are filled or void. Default value: True.

is_scatterbool

If True it represent a scatter plot, otherwise a continuous line. Default value: False.

line_color

For back-compatibility with old sympy.plotting. Use rendering_kw in order to fully customize the appearance of the line/scatter.

paramsdict, optional

A dictionary mapping symbols to parameters. If provided, this dictionary enables the interactive-widgets plot.

When calling a plotting function, the parameter can be specified with:

a widget from the ipywidgets module.
a widget from the panel module.
a tuple of the form:
(default, min, max, N, tick_format, label, spacing), which will instantiate a ipywidgets.widgets.widget_float.FloatSlider or a ipywidgets.widgets.widget_float.FloatLogSlider, depending on the spacing strategy. In particular:
- default, min, maxfloat
  Default value, minimum value and maximum value of the slider, respectively. Must be finite numbers. The order of these 3 numbers is not important: the module will figure it out which is what.
- Nint, optional
  Number of steps of the slider.
- tick_formatstr or None, optional
  Provide a formatter for the tick value of the slider. Default to ".2f".
- label: str, optional
  Custom text associated to the slider.
- spacingstr, optional
  Specify the discretization spacing. Default to "linear", can be changed to "log".

Notes:

parameters cannot be linked together (ie, one parameter cannot depend on another one).
If a widget returns multiple numerical values (like panel.widgets.slider.RangeSlider or ipywidgets.widgets.widget_float.FloatRangeSlider), then a corresponding number of symbols must be provided.

Here follows a couple of examples. If imodule="panel":

import panel as pn
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: pn.widgets.FloatSlider(value=1, start=0, end=5), # same slider as above
    (c, d): pn.widgets.RangeSlider(value=(-1, 1), start=-3, end=3, step=0.1)
}

Or with imodule="ipywidgets":

import ipywidgets as w
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: w.FloatSlider(value=1, min=0, max=5), # same slider as above
    (c, d): w.FloatRangeSlider(value=(-1, 1), min=-3, max=3, step=0.1)
}

When instantiating a data series directly, params must be a dictionary mapping symbols to numerical values.

Let series be any data series. Then series.params returns a dictionary mapping symbols to numerical values.

show_in_legendbool

Toggle the visibility of the data series on the legend. Default value: True.

stepsNoneType, bool, str

If set, it connects consecutive points with steps rather than straight segments. Possible options: [‘pre’, ‘post’, ‘mid’, True, False, None] Default value: False.

txcallable

Numerical transformation function to be applied to the data on the x-axis.

tycallable

Numerical transformation function to be applied to the data on the y-axis.

tzcallable

Numerical transformation function to be applied to the data on the z-axis.

unwrapbool, dict

Whether to use numpy.unwrap() on the computed coordinates in order to get rid of discontinuities. It can be:

False: do not use np.unwrap().
True: use np.unwrap() with default keyword arguments.
dictionary of keyword arguments passed to np.unwrap().

use_cmbool

Toggle the use of a colormap. By default, some series might use a colormap to display the necessary data. Setting this attribute to False will inform the associated renderer to use solid color. Related parameters: color_func. Default value: False.

Returns:

serieslist: A list containing one instance of List3DSeries.

See also

spb.graphics.functions_2d.line, line_parametric_3d

Examples

>>> from sympy import *
>>> from spb import *
>>> import numpy as np

(Source code)

Plot the coordinates of a single function:

>>> z = np.linspace(0, 6*np.pi, 100)
>>> x = z * np.cos(z)
>>> y = z * np.sin(z)
>>> graphics(list_3d(x, y, z))
Plot object containing:
[0]: 3D list plot

(Source code, png)

Plotting multiple functions with custom rendering keywords:

>>> graphics(
...     list_3d(x, y, z, "A"),
...     list_3d(x, y, -z, "B", {"linestyle": "--"}))
Plot object containing:
[0]: 3D list plot
[1]: 3D list plot

(Source code, png)

Interactive-widget plot of a dot following a path. Refer to the interactive sub-module documentation to learn more about the params dictionary.

from sympy import *
from spb import *
import numpy as np
t = symbols("t")
z = np.linspace(0, 6*np.pi, 100)
x = z * np.cos(z)
y = z * np.sin(z)
graphics(
    list_3d(x, y, z, is_scatter=False),
    list_3d(t * cos(t), t * sin(t), t,
        params={t: (3*pi, 0, 6*pi)}, is_scatter=True),
    backend=PB
)

(Source code, small.png)

spb.graphics.functions_3d.surface(expr, range_x=None, range_y=None, label=None, rendering_kw=None, colorbar=True, use_cm=False, **kwargs)[source]

Creates the surface of a function of 2 variables.

Parameters:

expr

The expression representing the function of two variables to be plotted. It can be a:

Symbolic expression.
Numerical function of two variable, supporting vectorization. In this case the following keyword arguments are not supported: params.

range_xtuple, Tuple

A 3-tuple (symb, min, max) denoting the range of the x variable. Default values: min=-10 and max=10.

range_ytuple, Tuple

A 3-tuple (symb, min, max) denoting the range of the y variable. Default values: min=-10 and max=10.

labelstr

Set the label associated to this series, which will be eventually shown on the legend or colorbar.

rendering_kwdict

A dictionary of keyword arguments to be passed to the renderers in order to further customize the appearance of the surface. Here are some useful links for the supported plotting libraries:

Matplotlib: https://matplotlib.org/stable/api/_as_gen/mpl_toolkits.mplot3d.axes3d.Axes3D.html#mpl_toolkits.mplot3d.axes3d.Axes3D.plot_surface
Plotly: https://plotly.com/python/3d-surface-plots/
K3D-Jupyter: look at the documentation of k3d.mesh.

colorbarbool

Toggle the visibility of the colorbar associated to the current data series. Note that a colorbar is only visible if use_cm=True and color_func is not None. Default value: True.

use_cmbool

Toggle the use of a colormap. By default, some series might use a colormap to display the necessary data. Setting this attribute to False will inform the associated renderer to use solid color. Related parameters: color_func. Default value: False.

color_func

Define a custom color mapping. It can either be:

A numerical function supporting vectorization. The arity can be:
- 2 arguments: f(x, y) where x, y are the coordinates of the points.
- 3 arguments: f(x, y, z) where x, y, z are the coordinates of the points.
A symbolic expression having at most as many free symbols as expr.
None: the default value (color mapping according to the z coordinate).

colorbar_ticks_formattertick_formatter_multiples_of

An object of type tick_formatter_multiples_of which will be used to place tick values on the colorbar at each multiple of a specified quantity. This only works when use_cm=True.

force_real_evalbool

By default, numerical evaluation is performed over complex numbers, which is slower but produces correct results. However, when the symbolic expression is converted to a numerical function with lambdify, the resulting function may not like to be evaluated over complex numbers. In such cases, forcing the evaluation to be performed over real numbers might be a good choice. The plotting module should be able to detect such occurences and automatically activate this option. If that is not the case, or evaluation performance is of paramount importance, set this parameter to True, but be aware that it might produce wrong results. Default value: False.

is_polarbool

If True, requests a polar discretization. In this case, range_x represents the radius, while range_y represents the angle. Default value: False.

modules

Specify the evaluation modules to be used by lambdify. If not specified, the evaluation will be done with NumPy/SciPy.

n1int

Number of discretization points along the x-axis to be used in the evaluation. Related parameters: xscale. It must be: 2 ≤ n1 < ∞. Default value: 100.

n2int

Number of discretization points along the y-axis to be used in the evaluation. Related parameters: yscale. It must be: 2 ≤ n2 < ∞. Default value: 100.

only_integersbool

Discretize the domain using only integer numbers. When this parameter is True, the number of discretization points is choosen by the algorithm. Default value: False.

paramsdict, optional

A dictionary mapping symbols to parameters. If provided, this dictionary enables the interactive-widgets plot.

When calling a plotting function, the parameter can be specified with:

a widget from the ipywidgets module.
a widget from the panel module.
a tuple of the form:
(default, min, max, N, tick_format, label, spacing), which will instantiate a ipywidgets.widgets.widget_float.FloatSlider or a ipywidgets.widgets.widget_float.FloatLogSlider, depending on the spacing strategy. In particular:
- default, min, maxfloat
  Default value, minimum value and maximum value of the slider, respectively. Must be finite numbers. The order of these 3 numbers is not important: the module will figure it out which is what.
- Nint, optional
  Number of steps of the slider.
- tick_formatstr or None, optional
  Provide a formatter for the tick value of the slider. Default to ".2f".
- label: str, optional
  Custom text associated to the slider.
- spacingstr, optional
  Specify the discretization spacing. Default to "linear", can be changed to "log".

Notes:

parameters cannot be linked together (ie, one parameter cannot depend on another one).
If a widget returns multiple numerical values (like panel.widgets.slider.RangeSlider or ipywidgets.widgets.widget_float.FloatRangeSlider), then a corresponding number of symbols must be provided.

Here follows a couple of examples. If imodule="panel":

import panel as pn
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: pn.widgets.FloatSlider(value=1, start=0, end=5), # same slider as above
    (c, d): pn.widgets.RangeSlider(value=(-1, 1), start=-3, end=3, step=0.1)
}

Or with imodule="ipywidgets":

import ipywidgets as w
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: w.FloatSlider(value=1, min=0, max=5), # same slider as above
    (c, d): w.FloatRangeSlider(value=(-1, 1), min=-3, max=3, step=0.1)
}

When instantiating a data series directly, params must be a dictionary mapping symbols to numerical values.

Let series be any data series. Then series.params returns a dictionary mapping symbols to numerical values.

show_in_legendbool

Toggle the visibility of the data series on the legend. Default value: True.

sum_boundint

When plotting sums, the expression will be pre-processed in order to replace lower/upper bounds set to +/- infinity with this +/- numerical value. Note: the higher this number, the slower the evaluation, but the more accurate the plot. It must be: 0 ≤ sum_bound < ∞. Default value: 1000.

surface_color

For back-compatibility with old sympy.plotting. Use rendering_kw in order to fully customize the appearance of the surface.

txcallable

Numerical transformation function to be applied to the data on the x-axis.

tycallable

Numerical transformation function to be applied to the data on the y-axis.

tzcallable

Numerical transformation function to be applied to the data on the z-axis.

wf_n1, wf_n2int, optional

Number of wireframe lines along the two ranges, respectively. Default to 10. Note that increasing this number might considerably slow down the plot’s creation. Related parameter: wireframe.

wf_npointsint 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. Related parameter: wireframe.

wf_rendering_kwdict, optional

A dictionary of keywords/values which is passed to the backend’s function to customize the appearance of wireframe lines. Related parameter: wireframe.

wireframeboolean, optional

Enable or disable a wireframe over the surface. Depending on the number of wireframe lines (see wf_n1 and wf_n2), activating this option might add a considerable overhead during the plot’s creation. Default to False (disabled).

xscalestr

Discretization strategy along the x-direction. Related parameters: n1. Possible options: [‘linear’, ‘log’] Default value: ‘linear’.

yscalestr

Discretization strategy along the y-direction. Related parameters: n2. Possible options: [‘linear’, ‘log’] Default value: ‘linear’.

Returns:

serieslist: A list containing one instance of SurfaceOver2DRangeSeries and possibly multiple instances of Parametric3DLineSeries, if wireframe=True.

See also

spb.graphics.functions_2d.contour, surface_parametric, surface_spherical
surface_revolution, wireframe, plane

Examples

>>> from sympy import symbols, cos, sin, pi, exp
>>> from spb import *
>>> x, y = symbols('x y')

(Source code)

Single plot with Matplotlib, with ticks formatted as multiples of pi/2.

>>> graphics(
...     surface(cos((x**2 + y**2)), (x, -pi, pi), (y, -pi, pi)),
...     x_ticks_formatter=multiples_of_pi_over_2(),
...     y_ticks_formatter=multiples_of_pi_over_2(),
... )
Plot object containing:
[0]: cartesian surface: cos(x**2 + y**2) for x over (-pi, pi) and y over (-pi, pi)

(Source code, png)

Single plot with Plotly, illustrating how to apply:

a color map: by default, it will map colors to the z values.
wireframe lines to better understand the discretization and curvature.
transformation to the discretized ranges in order to convert radians to degrees.
custom aspect ratio with Plotly.

from sympy import symbols, sin, cos, pi
from spb import *
import numpy as np
x, y = symbols("x, y")
expr = (cos(x) + sin(x) * sin(y) - sin(x) * cos(y))**2
graphics(
    surface(expr, (x, 0, pi), (y, 0, 2 * pi), use_cm=True,
        tx=np.rad2deg, ty=np.rad2deg,
        wireframe=True, wf_n1=20, wf_n2=20),
    backend=PB, xlabel="x [deg]", ylabel="y [deg]",
    aspect=dict(x=1.5, y=1.5, z=0.5))

(Source code, png)

Multiple plots with same range using color maps. By default, colors are mapped to the z values:

>>> graphics(
...     surface(x*y, (x, -5, 5), (y, -5, 5), use_cm=True),
...     surface(-x*y, (x, -5, 5), (y, -5, 5), use_cm=True))
Plot object containing:
[0]: cartesian surface: x*y for x over (-5, 5) and y over (-5, 5)
[1]: cartesian surface: -x*y for x over (-5, 5) and y over (-5, 5)

(Source code, png)

Multiple plots with different ranges and solid colors.

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

(Source code, png)

Single plot with a polar discretization, a color function mapping a colormap to the radius. Note that the same result can be achieved with plot3d_revolution.

from sympy import *
from spb import *
import numpy as np
r, theta = symbols("r, theta")
expr = cos(r**2) * exp(-r / 3)
graphics(
    surface(expr, (r, 0, 5), (theta, 1.6 * pi, 2 * pi),
        use_cm=True, color_func=lambda x, y, z: np.sqrt(x**2 + y**2),
        is_polar=True,
        wireframe=True, wf_n1=30, wf_n2=10,
        wf_rendering_kw={"width": 0.005}),
    backend=KB, legend=True, grid=False)

(Source code, small.png)

Plotting a numerical function instead of a symbolic expression:

>>> graphics(
...     surface(lambda x, y: x * np.exp(-x**2 - y**2),
...         ("x", -3, 3), ("y", -3, 3), use_cm=True))  

(Source code, png)

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).
the use of panel.widgets.slider.RangeSlider, which is a 2-values widget.

from sympy import *
from spb import *
import panel as pn
x, y, a, b, c, d, e = symbols("x y a b c d e")
graphics(
    surface(
        cos(x**2 + y**2) * exp(-(x**2 + y**2) * a),
        prange(x, b, c), prange(y, d, e),
        params={
            a: (0.25, 0, 1),
            (b, c): pn.widgets.RangeSlider(
                value=(-2, 2), start=-4, end=4, step=0.1),
            (d, e): pn.widgets.RangeSlider(
                value=(-2, 2), start=-4, end=4, step=0.1),
        },
        use_cm=True, n=100,
        wireframe=True, wf_n1=15, wf_n2=15),
    backend=PB, aspect=dict(x=1.5, y=1.5, z=0.75))

(Source code, small.png)

spb.graphics.functions_3d.surface_parametric(expr_x, expr_y, expr_z, range_u=None, range_v=None, label=None, rendering_kw=None, **kwargs)[source]

Creates a 3D parametric surface.

Parameters:

expr_x

The expression representing the component along the x-axis of the parametric function. It can be a:

Symbolic expression.
Numerical function of two variable, f(u, v), supporting vectorization. In this case the following keyword arguments are not supported: params.

expr_y

The expression representing the component along the y-axis of the parametric function. It can be a:

Symbolic expression.
Numerical function of two variable, f(u, v), supporting vectorization. In this case the following keyword arguments are not supported: params.

expr_z

The expression representing the component along the z-axis of the parametric function. It can be a:

Symbolic expression.
Numerical function of two variable, f(u, v), supporting vectorization. In this case the following keyword arguments are not supported: params.

range_utuple, Tuple

A 3-tuple (symb, min, max) denoting the range of the u parameter. Default values: min=-10 and max=10.

range_vtuple, Tuple

A 3-tuple (symb, min, max) denoting the range of the v parameter. Default values: min=-10 and max=10.

labelstr

Set the label associated to this series, which will be eventually shown on the legend or colorbar.

rendering_kwdict

A dictionary of keyword arguments to be passed to the renderers in order to further customize the appearance of the surface. Here are some useful links for the supported plotting libraries:

Matplotlib: https://matplotlib.org/stable/api/_as_gen/mpl_toolkits.mplot3d.axes3d.Axes3D.html#mpl_toolkits.mplot3d.axes3d.Axes3D.plot_surface
Plotly: https://plotly.com/python/3d-surface-plots/
K3D-Jupyter: look at the documentation of k3d.mesh.

color_func

Define the surface color mapping when use_cm=True. It can either be:

None: the default value (color mapping according to the z coordinate).
A numerical function supporting vectorization. The arity can be:
- 1 argument: f(u), where u is the first parameter.
- 2 arguments: f(u, v) where u, v are the parameters.
- 3 arguments: f(x, y, z) where x, y, z are the coordinates of the points.
- 5 arguments: f(x, y, z, u, v).
A symbolic expression having at most as many free symbols as expr_x or expr_y or expr_z.

colorbarbool

Toggle the visibility of the colorbar associated to the current data series. Note that a colorbar is only visible if use_cm=True and color_func is not None. Default value: True.

colorbar_ticks_formattertick_formatter_multiples_of

An object of type tick_formatter_multiples_of which will be used to place tick values on the colorbar at each multiple of a specified quantity. This only works when use_cm=True.

force_real_evalbool

By default, numerical evaluation is performed over complex numbers, which is slower but produces correct results. However, when the symbolic expression is converted to a numerical function with lambdify, the resulting function may not like to be evaluated over complex numbers. In such cases, forcing the evaluation to be performed over real numbers might be a good choice. The plotting module should be able to detect such occurences and automatically activate this option. If that is not the case, or evaluation performance is of paramount importance, set this parameter to True, but be aware that it might produce wrong results. Default value: False.

modules

Specify the evaluation modules to be used by lambdify. If not specified, the evaluation will be done with NumPy/SciPy.

n1int

Number of discretization points along the first parameter to be used in the evaluation. Related parameters: xscale. It must be: 2 ≤ n1 < ∞. Default value: 100.

n2int

Number of discretization points along the second parameter to be used in the evaluation. Related parameters: yscale. It must be: 2 ≤ n2 < ∞. Default value: 100.

only_integersbool

Discretize the domain using only integer numbers. When this parameter is True, the number of discretization points is choosen by the algorithm. Default value: False.

paramsdict, optional

A dictionary mapping symbols to parameters. If provided, this dictionary enables the interactive-widgets plot.

When calling a plotting function, the parameter can be specified with:

a widget from the ipywidgets module.
a widget from the panel module.
a tuple of the form:
(default, min, max, N, tick_format, label, spacing), which will instantiate a ipywidgets.widgets.widget_float.FloatSlider or a ipywidgets.widgets.widget_float.FloatLogSlider, depending on the spacing strategy. In particular:
- default, min, maxfloat
  Default value, minimum value and maximum value of the slider, respectively. Must be finite numbers. The order of these 3 numbers is not important: the module will figure it out which is what.
- Nint, optional
  Number of steps of the slider.
- tick_formatstr or None, optional
  Provide a formatter for the tick value of the slider. Default to ".2f".
- label: str, optional
  Custom text associated to the slider.
- spacingstr, optional
  Specify the discretization spacing. Default to "linear", can be changed to "log".

Notes:

parameters cannot be linked together (ie, one parameter cannot depend on another one).
If a widget returns multiple numerical values (like panel.widgets.slider.RangeSlider or ipywidgets.widgets.widget_float.FloatRangeSlider), then a corresponding number of symbols must be provided.

Here follows a couple of examples. If imodule="panel":

import panel as pn
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: pn.widgets.FloatSlider(value=1, start=0, end=5), # same slider as above
    (c, d): pn.widgets.RangeSlider(value=(-1, 1), start=-3, end=3, step=0.1)
}

Or with imodule="ipywidgets":

import ipywidgets as w
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: w.FloatSlider(value=1, min=0, max=5), # same slider as above
    (c, d): w.FloatRangeSlider(value=(-1, 1), min=-3, max=3, step=0.1)
}

When instantiating a data series directly, params must be a dictionary mapping symbols to numerical values.

Let series be any data series. Then series.params returns a dictionary mapping symbols to numerical values.

show_in_legendbool

Toggle the visibility of the data series on the legend. Default value: True.

sum_boundint

When plotting sums, the expression will be pre-processed in order to replace lower/upper bounds set to +/- infinity with this +/- numerical value. Note: the higher this number, the slower the evaluation, but the more accurate the plot. It must be: 0 ≤ sum_bound < ∞. Default value: 1000.

surface_color

For back-compatibility with old sympy.plotting. Use rendering_kw in order to fully customize the appearance of the surface.

txcallable

Numerical transformation function to be applied to the data on the x-axis.

tycallable

Numerical transformation function to be applied to the data on the y-axis.

tzcallable

Numerical transformation function to be applied to the data on the z-axis.

use_cmbool

Toggle the use of a colormap. By default, some series might use a colormap to display the necessary data. Setting this attribute to False will inform the associated renderer to use solid color. Related parameters: color_func. Default value: False.

wf_n1, wf_n2int, optional

Number of wireframe lines along the two ranges, respectively. Default to 10. Note that increasing this number might considerably slow down the plot’s creation. Related parameter: wireframe.

wf_npointsint 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. Related parameter: wireframe.

wf_rendering_kwdict, optional

A dictionary of keywords/values which is passed to the backend’s function to customize the appearance of wireframe lines. Related parameter: wireframe.

wireframeboolean, optional

Enable or disable a wireframe over the surface. Depending on the number of wireframe lines (see wf_n1 and wf_n2), activating this option might add a considerable overhead during the plot’s creation. Default to False (disabled).

xscalestr

Discretization strategy along the first parameter. Related parameters: n1. Possible options: [‘linear’, ‘log’] Default value: ‘linear’.

yscalestr

Discretization strategy along the second parameter. Related parameters: n2. Possible options: [‘linear’, ‘log’] Default value: ‘linear’.

Returns:

serieslist: A list containing one instance of ParametricSurfaceSeries and possibly multiple instances of Parametric3DLineSeries, if wireframe=True.

See also

surface, surface_spherical, surface_revolution, wireframe

Examples

>>> from sympy import symbols, cos, sin, pi, I, sqrt, atan2, re, im
>>> from spb import *
>>> u, v = symbols('u v')

(Source code)

Plot a parametric surface:

>>> graphics(
...     surface_parametric(
...         u * cos(v), u * sin(v), u * cos(4 * v) / 2,
...         (u, 0, pi), (v, 0, 2*pi), use_cm=False),
...     title="Sinusoidal Cone")
Plot object containing:
[0]: parametric cartesian surface: (u*cos(v), u*sin(v), u*cos(4*v)/2) for u over (0, pi) and v over (0, 2*pi)

(Source code, png)

Customize the appearance of the surface by changing the colormap. Apply a color function mapping the v values. Activate the wireframe to better visualize the parameterization.

from sympy import *
from spb import *
var("u, v")

x = (1 + v / 2 * cos(u / 2)) * cos(u)
y = (1 + v / 2 * cos(u / 2)) * sin(u)
z = v / 2 * sin(u / 2)
graphics(
surface_parametric(x, y, z, (u, 0, 2*pi), (v, -1, 1),
    "v", {"colorscale": "mygbm"},
    use_cm=True, color_func=lambda u, v: u,
    wireframe=True, wf_n1=20,
    colorbar_ticks_formatter=multiples_of_pi_over_2(label="π")
),
xlabel="x", ylabel="y", zlabel="z",
backend=PB, title="Möbius strip")

(Source code, png)

Riemann surfaces of the real part of the multivalued function z**n, using Plotly:

from sympy import symbols, sqrt, re, im, pi, atan2, sin, cos, I
from spb import *
r, theta, x, y = symbols("r, theta, x, y", real=True)
mag = lambda z: sqrt(re(z)**2 + im(z)**2)
phase = lambda z, k=0: atan2(im(z), re(z)) + 2 * k * pi
n = 2 # exponent (integer)
z = x + I * y # cartesian
d = {x: r * cos(theta), y: r * sin(theta)} # cartesian to polar
branches = [(mag(z)**(1 / n) * cos(phase(z, i) / n)).subs(d)
    for i in range(n)]
exprs = [(r * cos(theta), r * sin(theta), rb) for rb in branches]
series = [
    surface_parametric(*e, (r, 0, 3), (theta, -pi, pi),
        label="branch %s" % (i + 1), wireframe=True, wf_n2=20)
    for i, e in enumerate(exprs)]
graphics(*series, backend=PB, zlabel="f(z)")

(Source code, png)

Plotting a numerical function instead of a symbolic expression.

from spb import *
import numpy as np
fx = lambda u, v: (4 + np.cos(u)) * np.cos(v)
fy = lambda u, v: (4 + np.cos(u)) * np.sin(v)
fz = lambda u, v: np.sin(u)
graphics(
    surface_parametric(fx, fy, fz, ("u", 0, 2 * np.pi),
        ("v", 0, 2 * np.pi)),
    zlim=(-2.5, 2.5), title="Torus", backend=KB, grid=False)

(Source code, small.png)

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).

from sympy import *
from spb import *
import k3d
alpha, u, v, up, vp = symbols("alpha u v u_p v_p")
graphics(
    surface_parametric(
        exp(u) * cos(v - alpha) / 2 + exp(-u) * cos(v + alpha) / 2,
        exp(u) * sin(v - alpha) / 2 + exp(-u) * sin(v + alpha) / 2,
        cos(alpha) * u + sin(alpha) * v,
        prange(u, -up, up), prange(v, 0, vp * pi),
        n=50, use_cm=True, color_func=lambda u, v: v,
        rendering_kw={"color_map": k3d.colormaps.paraview_color_maps.Hue_L60},
        wireframe=True, wf_n2=15, wf_rendering_kw={"width": 0.005},
        params={
            alpha: (0, 0, pi),
            up: (1, 0, 2),
            vp: (2, 0, 2),
        }),
    backend=KB, grid=False,
    title="Catenoid \, to \, Right \, Helicoid \, Transformation"
)

(Source code, small.png)

Interactive-widget plot. Refer to the interactive sub-module documentation to learn more about the params dictionary. Note that the plot’s creation might be slow due to the wireframe lines.

from sympy import *
from spb import *
import panel as pn
n, u, v = symbols("n, u, v")
x = v * cos(u)
y = v * sin(u)
z = sin(n * u)
graphics(
    surface_parametric(x, y, z, (u, 0, 2*pi), (v, -1, 0),
        params = {
            n: pn.widgets.IntInput(value=3, name="n")
        },
        use_cm=True, wireframe=True, wf_n1=75, wf_n2=6),
    backend=PB,
    title="Plücker's conoid",
    imodule="panel"
)

(Source code, small.png)

spb.graphics.functions_3d.surface_revolution(curve, range_t, range_phi=None, axis=(0, 0), parallel_axis='z', show_curve=False, curve_kw={}, **kwargs)[source]

Creates a surface of revolution by rotating a curve around an axis of rotation.

Parameters:

curveExpr, list ortuple of 2 or 3 elements

The curve to be revolved, which can be either:

a symbolic expression
a 2-tuple representing a parametric curve in 2D space
a 3-tuple representing a parametric curve in 3D space

range_ttuple

A 3-tuple (symbol, min, max) denoting the range of the parameter of the curve.

range_phituple

A 3-tuple (symbol, min, max) denoting the range of the azimuthal angle where the curve will be revolved. Default to (phi, 0, 2*pi).

axistuple

A 2-tuple (coord1, coord2) that specifies the position of the rotation axis. Depending on the value of parallel_axis:

"x": the rotation axis intersects the YZ plane at (coord1, coord2).
"y": the rotation axis intersects the XZ plane at (coord1, coord2).
"z": the rotation axis intersects the XY plane at (coord1, coord2).

Default to (0, 0).

parallel_axisstr

Specify the axis parallel to the axis of rotation. Must be one of the following options: “x”, “y” or “z”. Default to “z”.

show_curvebool

Add the initial curve to the plot. Default to False.

curve_kwdict

A dictionary of options that will be passed to plot3d_parametric_line if show_curve=True in order to customize the appearance of the initial curve. Refer to its documentation for more information.

color_func

Define the surface color mapping when use_cm=True. It can either be:

None: the default value (color mapping according to the z coordinate).
A numerical function supporting vectorization. The arity can be:
- 1 argument: f(u), where u is the first parameter.
- 2 arguments: f(u, v) where u, v are the parameters.
- 3 arguments: f(x, y, z) where x, y, z are the coordinates of the points.
- 5 arguments: f(x, y, z, u, v).
A symbolic expression having at most as many free symbols as expr_x or expr_y or expr_z.

colorbarbool

Toggle the visibility of the colorbar associated to the current data series. Note that a colorbar is only visible if use_cm=True and color_func is not None. Default value: True.

colorbar_ticks_formattertick_formatter_multiples_of

An object of type tick_formatter_multiples_of which will be used to place tick values on the colorbar at each multiple of a specified quantity. This only works when use_cm=True.

force_real_evalbool

By default, numerical evaluation is performed over complex numbers, which is slower but produces correct results. However, when the symbolic expression is converted to a numerical function with lambdify, the resulting function may not like to be evaluated over complex numbers. In such cases, forcing the evaluation to be performed over real numbers might be a good choice. The plotting module should be able to detect such occurences and automatically activate this option. If that is not the case, or evaluation performance is of paramount importance, set this parameter to True, but be aware that it might produce wrong results. Default value: False.

labelstr

Set the label associated to this series, which will be eventually shown on the legend or colorbar.

modules

Specify the evaluation modules to be used by lambdify. If not specified, the evaluation will be done with NumPy/SciPy.

n1int

Number of discretization points along the first parameter to be used in the evaluation. Related parameters: xscale. It must be: 2 ≤ n1 < ∞. Default value: 100.

n2int

Number of discretization points along the second parameter to be used in the evaluation. Related parameters: yscale. It must be: 2 ≤ n2 < ∞. Default value: 100.

only_integersbool

Discretize the domain using only integer numbers. When this parameter is True, the number of discretization points is choosen by the algorithm. Default value: False.

paramsdict, optional

A dictionary mapping symbols to parameters. If provided, this dictionary enables the interactive-widgets plot.

When calling a plotting function, the parameter can be specified with:

a widget from the ipywidgets module.
a widget from the panel module.
a tuple of the form:
(default, min, max, N, tick_format, label, spacing), which will instantiate a ipywidgets.widgets.widget_float.FloatSlider or a ipywidgets.widgets.widget_float.FloatLogSlider, depending on the spacing strategy. In particular:
- default, min, maxfloat
  Default value, minimum value and maximum value of the slider, respectively. Must be finite numbers. The order of these 3 numbers is not important: the module will figure it out which is what.
- Nint, optional
  Number of steps of the slider.
- tick_formatstr or None, optional
  Provide a formatter for the tick value of the slider. Default to ".2f".
- label: str, optional
  Custom text associated to the slider.
- spacingstr, optional
  Specify the discretization spacing. Default to "linear", can be changed to "log".

Notes:

parameters cannot be linked together (ie, one parameter cannot depend on another one).
If a widget returns multiple numerical values (like panel.widgets.slider.RangeSlider or ipywidgets.widgets.widget_float.FloatRangeSlider), then a corresponding number of symbols must be provided.

Here follows a couple of examples. If imodule="panel":

import panel as pn
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: pn.widgets.FloatSlider(value=1, start=0, end=5), # same slider as above
    (c, d): pn.widgets.RangeSlider(value=(-1, 1), start=-3, end=3, step=0.1)
}

Or with imodule="ipywidgets":

import ipywidgets as w
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: w.FloatSlider(value=1, min=0, max=5), # same slider as above
    (c, d): w.FloatRangeSlider(value=(-1, 1), min=-3, max=3, step=0.1)
}

When instantiating a data series directly, params must be a dictionary mapping symbols to numerical values.

Let series be any data series. Then series.params returns a dictionary mapping symbols to numerical values.

rendering_kwdict

A dictionary of keyword arguments to be passed to the renderers in order to further customize the appearance of the surface. Here are some useful links for the supported plotting libraries:

Matplotlib: https://matplotlib.org/stable/api/_as_gen/mpl_toolkits.mplot3d.axes3d.Axes3D.html#mpl_toolkits.mplot3d.axes3d.Axes3D.plot_surface
Plotly: https://plotly.com/python/3d-surface-plots/
K3D-Jupyter: look at the documentation of k3d.mesh.

show_in_legendbool

Toggle the visibility of the data series on the legend. Default value: True.

sum_boundint

When plotting sums, the expression will be pre-processed in order to replace lower/upper bounds set to +/- infinity with this +/- numerical value. Note: the higher this number, the slower the evaluation, but the more accurate the plot. It must be: 0 ≤ sum_bound < ∞. Default value: 1000.

surface_color

For back-compatibility with old sympy.plotting. Use rendering_kw in order to fully customize the appearance of the surface.

txcallable

Numerical transformation function to be applied to the data on the x-axis.

tycallable

Numerical transformation function to be applied to the data on the y-axis.

tzcallable

Numerical transformation function to be applied to the data on the z-axis.

use_cmbool

Toggle the use of a colormap. By default, some series might use a colormap to display the necessary data. Setting this attribute to False will inform the associated renderer to use solid color. Related parameters: color_func. Default value: False.

xscalestr

Discretization strategy along the first parameter. Related parameters: n1. Possible options: [‘linear’, ‘log’] Default value: ‘linear’.

yscalestr

Discretization strategy along the second parameter. Related parameters: n2. Possible options: [‘linear’, ‘log’] Default value: ‘linear’.

Returns:

serieslist: A list containing one instance of ParametricSurfaceSeries, possibly multiple instances of Parametric3DLineSeries representing wireframe lines (if wireframe=True) and possible another instance of Parametric3DLineSeries representing the curve (if show_curve=True).

See also

surface, surface_spherical, surface_parametric, wireframe

Examples

>>> from sympy import symbols, cos, sin, pi
>>> from spb import *
>>> t, phi = symbols('t phi')

(Source code)

Revolve a function around the z axis:

graphics(
    surface_revolution(
        cos(t), (t, 0, pi),
        use_cm=True, color_func=lambda t, phi: phi,
        rendering_kw={"alpha": 0.6, "cmap": "twilight"},
        # indicates the azimuthal angle on the colorbar label
        label="$\phi$ [rad]",
        show_curve=True,
        # this dictionary will be passes to plot3d_parametric_line in
        # order to draw the initial curve
        curve_kw=dict(rendering_kw={"color": "r", "label": "cos(t)"}),
        # activate the wireframe to visualize the parameterization
        wireframe=True, wf_n1=15, wf_n2=15,
        wf_rendering_kw={"lw": 0.5, "alpha": 0.75}
    )
)

(Source code, png)

Revolve the same function around an axis parallel to the x axis, using Plotly:

from sympy import symbols, cos, sin, pi
from spb import *
t, phi = symbols('t phi')
graphics(
    surface_revolution(
        cos(t), (t, 0, pi), parallel_axis="x", axis=(1, 0),
        label="phi [rad]", rendering_kw={"colorscale": "twilight"},
        use_cm=True, color_func=lambda t, phi: phi,
        show_curve=True,
        curve_kw=dict(rendering_kw={"line": {"color": "red", "width": 8},
            "name": "cos(t)"}),
        wireframe=True, wf_n1=15, wf_n2=15,
        wf_rendering_kw={"line_width": 1}
    ),
    backend=PB
)

(Source code, png)

Revolve a 2D parametric circle around the z axis:

from sympy import *
from spb import *
t = symbols("t")
circle = (3 + cos(t), sin(t))
graphics(
    surface_revolution(circle, (t, 0, 2 * pi),
        show_curve=True, rendering_kw={"opacity": 0.65},
        curve_kw={"rendering_kw": {"width": 0.05}}),
    backend=KB)

(Source code, small.png)

Revolve a 3D parametric curve around the z axis for a given azimuthal angle, using Plotly:

from sympy import *
from spb import *
t = symbols("t")
graphics(
    surface_revolution(
        (cos(t), sin(t), t), (t, 0, 2*pi), (phi, 0, pi),
        use_cm=True, color_func=lambda t, phi: t, label="t [rad]",
        show_curve=True,
        wireframe=True, wf_n1=2, wf_n2=5),
    backend=PB, aspect="cube")

(Source code, png)

Interactive-widget plot of a goblet. 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).

from sympy import *
from spb import *
t, phi, u, v, w = symbols("t phi u v w")
graphics(
    surface_revolution(
        (t, cos(u * t), t**2), prange(t, 0, v), prange(phi, 0, w*pi),
        axis=(1, 0.2), n=50,
        wireframe=True, wf_n1=15, wf_n2=15,
        wf_rendering_kw={"width": 0.004},
        show_curve=True, curve_kw={"rendering_kw": {"width": 0.025}},
        params={
            u: (2.5, 0, 6),
            v: (2, 0, 3),
            w: (2, 0, 2)
        }),
    backend=KB, force_real_eval=True)

(Source code, small.png)

spb.graphics.functions_3d.surface_spherical(r, range_theta=None, range_phi=None, label=None, rendering_kw=None, **kwargs)[source]

Plots a radius as a function of the spherical coordinates theta and phi.

Parameters:

rExpr or callable

Expression representing the radius. It can be a:

Symbolic expression.
Numerical function of two variable, f(theta, phi), supporting vectorization. In this case the following keyword arguments are not supported: params.

range_thetatuple

A 3-tuple (symbol, min, max) denoting the range of the polar angle, which is limited in [0, pi]. Consider a sphere:

theta=0 indicates the north pole.
theta=pi/2 indicates the equator.
theta=pi indicates the south pole.

range_phituple

A 3-tuple (symbol, min, max) denoting the range of the azimuthal angle, which is limited in [0, 2*pi].

labelstr

Set the label associated to this series, which will be eventually shown on the legend or colorbar.

rendering_kwdict

A dictionary of keyword arguments to be passed to the renderers in order to further customize the appearance of the surface. Here are some useful links for the supported plotting libraries:

Matplotlib: https://matplotlib.org/stable/api/_as_gen/mpl_toolkits.mplot3d.axes3d.Axes3D.html#mpl_toolkits.mplot3d.axes3d.Axes3D.plot_surface
Plotly: https://plotly.com/python/3d-surface-plots/
K3D-Jupyter: look at the documentation of k3d.mesh.

color_func

Define the surface color mapping when use_cm=True. It can either be:

None: the default value (color mapping according to the z coordinate).
A numerical function supporting vectorization. The arity can be:
- 1 argument: f(u), where u is the first parameter.
- 2 arguments: f(u, v) where u, v are the parameters.
- 3 arguments: f(x, y, z) where x, y, z are the coordinates of the points.
- 5 arguments: f(x, y, z, u, v).
A symbolic expression having at most as many free symbols as expr_x or expr_y or expr_z.

colorbarbool

Toggle the visibility of the colorbar associated to the current data series. Note that a colorbar is only visible if use_cm=True and color_func is not None. Default value: True.

colorbar_ticks_formattertick_formatter_multiples_of

An object of type tick_formatter_multiples_of which will be used to place tick values on the colorbar at each multiple of a specified quantity. This only works when use_cm=True.

force_real_evalbool

By default, numerical evaluation is performed over complex numbers, which is slower but produces correct results. However, when the symbolic expression is converted to a numerical function with lambdify, the resulting function may not like to be evaluated over complex numbers. In such cases, forcing the evaluation to be performed over real numbers might be a good choice. The plotting module should be able to detect such occurences and automatically activate this option. If that is not the case, or evaluation performance is of paramount importance, set this parameter to True, but be aware that it might produce wrong results. Default value: False.

modules

Specify the evaluation modules to be used by lambdify. If not specified, the evaluation will be done with NumPy/SciPy.

n1int

Number of discretization points along the first parameter to be used in the evaluation. Related parameters: xscale. It must be: 2 ≤ n1 < ∞. Default value: 100.

n2int

Number of discretization points along the second parameter to be used in the evaluation. Related parameters: yscale. It must be: 2 ≤ n2 < ∞. Default value: 100.

only_integersbool

Discretize the domain using only integer numbers. When this parameter is True, the number of discretization points is choosen by the algorithm. Default value: False.

paramsdict, optional

A dictionary mapping symbols to parameters. If provided, this dictionary enables the interactive-widgets plot.

When calling a plotting function, the parameter can be specified with:

a widget from the ipywidgets module.
a widget from the panel module.
a tuple of the form:
(default, min, max, N, tick_format, label, spacing), which will instantiate a ipywidgets.widgets.widget_float.FloatSlider or a ipywidgets.widgets.widget_float.FloatLogSlider, depending on the spacing strategy. In particular:
- default, min, maxfloat
  Default value, minimum value and maximum value of the slider, respectively. Must be finite numbers. The order of these 3 numbers is not important: the module will figure it out which is what.
- Nint, optional
  Number of steps of the slider.
- tick_formatstr or None, optional
  Provide a formatter for the tick value of the slider. Default to ".2f".
- label: str, optional
  Custom text associated to the slider.
- spacingstr, optional
  Specify the discretization spacing. Default to "linear", can be changed to "log".

Notes:

parameters cannot be linked together (ie, one parameter cannot depend on another one).
If a widget returns multiple numerical values (like panel.widgets.slider.RangeSlider or ipywidgets.widgets.widget_float.FloatRangeSlider), then a corresponding number of symbols must be provided.

Here follows a couple of examples. If imodule="panel":

import panel as pn
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: pn.widgets.FloatSlider(value=1, start=0, end=5), # same slider as above
    (c, d): pn.widgets.RangeSlider(value=(-1, 1), start=-3, end=3, step=0.1)
}

Or with imodule="ipywidgets":

import ipywidgets as w
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: w.FloatSlider(value=1, min=0, max=5), # same slider as above
    (c, d): w.FloatRangeSlider(value=(-1, 1), min=-3, max=3, step=0.1)
}

When instantiating a data series directly, params must be a dictionary mapping symbols to numerical values.

Let series be any data series. Then series.params returns a dictionary mapping symbols to numerical values.

show_in_legendbool

Toggle the visibility of the data series on the legend. Default value: True.

sum_boundint

When plotting sums, the expression will be pre-processed in order to replace lower/upper bounds set to +/- infinity with this +/- numerical value. Note: the higher this number, the slower the evaluation, but the more accurate the plot. It must be: 0 ≤ sum_bound < ∞. Default value: 1000.

surface_color

For back-compatibility with old sympy.plotting. Use rendering_kw in order to fully customize the appearance of the surface.

txcallable

Numerical transformation function to be applied to the data on the x-axis.

tycallable

Numerical transformation function to be applied to the data on the y-axis.

tzcallable

Numerical transformation function to be applied to the data on the z-axis.

use_cmbool

Toggle the use of a colormap. By default, some series might use a colormap to display the necessary data. Setting this attribute to False will inform the associated renderer to use solid color. Related parameters: color_func. Default value: False.

xscalestr

Discretization strategy along the first parameter. Related parameters: n1. Possible options: [‘linear’, ‘log’] Default value: ‘linear’.

yscalestr

Discretization strategy along the second parameter. Related parameters: n2. Possible options: [‘linear’, ‘log’] Default value: ‘linear’.

Returns:

serieslist: A list containing one instance of ParametricSurfaceSeries and possibly multiple instances of Parametric3DLineSeries, if wireframe=True.

See also

surface, surface_parametric, surface_revolution, wireframe

Examples

>>> from sympy import symbols, cos, sin, pi, Ynm, re, lambdify
>>> from spb import *
>>> theta, phi = symbols('theta phi')

(Source code)

Sphere cap:

>>> graphics(
...     surface_spherical(1, (theta, 0, 0.7 * pi), (phi, 0, 1.8 * pi)))
Plot object containing:
[0]: parametric cartesian surface: (sin(theta)*cos(phi), sin(phi)*sin(theta), cos(theta)) for theta over (0, 0.7*pi) and phi over (0, 1.8*pi)

(Source code, png)

Plot real spherical harmonics, highlighting the regions in which the real part is positive and negative, using Plotly:

from sympy import symbols, sin, pi, Ynm, re, lambdify
from spb import *
theta, phi = symbols('theta phi')
r = re(Ynm(3, 3, theta, phi).expand(func=True).rewrite(sin).expand())
graphics(
    surface_spherical(
        abs(r), (theta, 0, pi), (phi, 0, 2 * pi), "real",
        use_cm=True, n2=200,
        color_func=lambdify([theta, phi], r)),
    backend=PB)

(Source code, png)

Multiple surfaces with wireframe lines, using Plotly. Note that activating the wireframe option might add a considerable overhead during the plot’s creation.

from sympy import symbols, sin, pi
from spb import *
theta, phi = symbols('theta phi')
r1 = 1
r2 = 1.5 + sin(5 * phi) * sin(10 * theta) / 10
graphics(
    surface_spherical(r1, (theta, 0, pi / 2), (phi, 0.35 * pi, 2 * pi),
        label="r1", wireframe=True, wf_n2=25),
    surface_spherical(r2, (theta, 0, pi / 2), (phi, 0.35 * pi, 2 * pi),
        label="r2", wireframe=True, wf_n2=25),
    backend=PB)

(Source code, png)

Interactive-widget plot of real spherical harmonics, highlighting the regions in which the real part is positive and negative. Note that the plot’s creation and update might be slow and that it must be m < n at all times. Refer to the interactive sub-module documentation to learn more about the params dictionary.

from sympy import *
from spb import *
import panel as pn
n, m = symbols("n, m")
phi, theta = symbols("phi, theta", real=True)
r = re(Ynm(n, m, theta, phi).expand(func=True).rewrite(sin).expand())
graphics(
    surface_spherical(abs(r), (theta, 0, pi), (phi, 0, 2*pi),
        label="real",
        params = {
            n: pn.widgets.IntInput(value=2, name="n"),
            m: pn.widgets.IntInput(value=0, name="m"),
        },
        use_cm=True, color_func=r, force_real_eval=True),
    backend=PB, imodule="panel")

(Source code, small.png)

spb.graphics.functions_3d.plane(p, range_x=None, range_y=None, range_z=None, label=None, rendering_kw=None, **kwargs)[source]

Plot a plane in a 3D space.

Parameters:

pPlane

The plane to be plotted.

range_xtuple, Tuple

A 3-tuple (symb, min, max) denoting the range of the x variable. Default values: min=-10 and max=10.

range_ytuple, Tuple

A 3-tuple (symb, min, max) denoting the range of the y variable. Default values: min=-10 and max=10.

range_ztuple, Tuple

A 3-tuple (symb, min, max) denoting the range of the z variable. Default values: min=-10 and max=10.

labelstr

Set the label associated to this series, which will be eventually shown on the legend or colorbar.

rendering_kwdict

A dictionary of keyword arguments to be passed to the renderers in order to further customize the appearance of the surface. Here are some useful links for the supported plotting libraries:

Matplotlib: https://matplotlib.org/stable/api/_as_gen/mpl_toolkits.mplot3d.axes3d.Axes3D.html#mpl_toolkits.mplot3d.axes3d.Axes3D.plot_surface
Plotly: https://plotly.com/python/3d-surface-plots/
K3D-Jupyter: look at the documentation of k3d.mesh.

color_func

A color function to be applied to the numerical data. It can be:

None: no color function.
callable: a function of 3 arguments (the coordinates x, y, z) returning numerical data.

colorbarbool

Toggle the visibility of the colorbar associated to the current data series. Note that a colorbar is only visible if use_cm=True and color_func is not None. Default value: True.

colorbar_ticks_formattertick_formatter_multiples_of

An object of type tick_formatter_multiples_of which will be used to place tick values on the colorbar at each multiple of a specified quantity. This only works when use_cm=True.

force_real_evalbool

By default, numerical evaluation is performed over complex numbers, which is slower but produces correct results. However, when the symbolic expression is converted to a numerical function with lambdify, the resulting function may not like to be evaluated over complex numbers. In such cases, forcing the evaluation to be performed over real numbers might be a good choice. The plotting module should be able to detect such occurences and automatically activate this option. If that is not the case, or evaluation performance is of paramount importance, set this parameter to True, but be aware that it might produce wrong results. Default value: False.

modules

Specify the evaluation modules to be used by lambdify. If not specified, the evaluation will be done with NumPy/SciPy.

n1int

Number of discretization points along the x-axis to be used in the evaluation. Related parameters: xscale. It must be: 2 ≤ n1 < ∞. Default value: 20.

n2int

Number of discretization points along the y-axis to be used in the evaluation. Related parameters: yscale. It must be: 2 ≤ n2 < ∞. Default value: 20.

n3int

Number of discretization points along the z-axis to be used in the evaluation. Related parameters: zscale. It must be: 2 ≤ n3 < ∞. Default value: 20.

only_integersbool

Discretize the domain using only integer numbers. When this parameter is True, the number of discretization points is choosen by the algorithm. Default value: False.

paramsdict, optional

A dictionary mapping symbols to parameters. If provided, this dictionary enables the interactive-widgets plot.

When calling a plotting function, the parameter can be specified with:

a widget from the ipywidgets module.
a widget from the panel module.
a tuple of the form:
(default, min, max, N, tick_format, label, spacing), which will instantiate a ipywidgets.widgets.widget_float.FloatSlider or a ipywidgets.widgets.widget_float.FloatLogSlider, depending on the spacing strategy. In particular:
- default, min, maxfloat
  Default value, minimum value and maximum value of the slider, respectively. Must be finite numbers. The order of these 3 numbers is not important: the module will figure it out which is what.
- Nint, optional
  Number of steps of the slider.
- tick_formatstr or None, optional
  Provide a formatter for the tick value of the slider. Default to ".2f".
- label: str, optional
  Custom text associated to the slider.
- spacingstr, optional
  Specify the discretization spacing. Default to "linear", can be changed to "log".

Notes:

parameters cannot be linked together (ie, one parameter cannot depend on another one).
If a widget returns multiple numerical values (like panel.widgets.slider.RangeSlider or ipywidgets.widgets.widget_float.FloatRangeSlider), then a corresponding number of symbols must be provided.

Here follows a couple of examples. If imodule="panel":

import panel as pn
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: pn.widgets.FloatSlider(value=1, start=0, end=5), # same slider as above
    (c, d): pn.widgets.RangeSlider(value=(-1, 1), start=-3, end=3, step=0.1)
}

Or with imodule="ipywidgets":

import ipywidgets as w
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: w.FloatSlider(value=1, min=0, max=5), # same slider as above
    (c, d): w.FloatRangeSlider(value=(-1, 1), min=-3, max=3, step=0.1)
}

When instantiating a data series directly, params must be a dictionary mapping symbols to numerical values.

Let series be any data series. Then series.params returns a dictionary mapping symbols to numerical values.

show_in_legendbool

Toggle the visibility of the data series on the legend. Default value: True.

sum_boundint

When plotting sums, the expression will be pre-processed in order to replace lower/upper bounds set to +/- infinity with this +/- numerical value. Note: the higher this number, the slower the evaluation, but the more accurate the plot. It must be: 0 ≤ sum_bound < ∞. Default value: 1000.

surface_color

For back-compatibility with old sympy.plotting. Use rendering_kw in order to fully customize the appearance of the surface.

txcallable

Numerical transformation function to be applied to the data on the x-axis.

tycallable

Numerical transformation function to be applied to the data on the y-axis.

tzcallable

Numerical transformation function to be applied to the data on the z-axis.

use_cmbool

Toggle the use of a colormap. By default, some series might use a colormap to display the necessary data. Setting this attribute to False will inform the associated renderer to use solid color. Related parameters: color_func. Default value: False.

xscalestr

Discretization strategy along the x-direction. Related parameters: n1. Possible options: [‘linear’, ‘log’] Default value: ‘linear’.

yscalestr

Discretization strategy along the y-direction. Related parameters: n2. Possible options: [‘linear’, ‘log’] Default value: ‘linear’.

zscalestr

Discretization strategy along the z-direction. Related parameters: n3. Possible options: [‘linear’, ‘log’] Default value: ‘linear’.

Returns:

serieslist: A list containing an instance of PlaneSeries.

See also

surface

Examples

from sympy import *
from spb import *
from sympy.abc import x, y, z
ranges = [(x, -5, 5), (y, -5, 5), (z, -5, 5)]
graphics(
    plane(Plane((0, 0, 0), (1, 0, 0)), *ranges, label="yz"),
    plane(Plane((0, 0, 0), (0, 1, 0)), *ranges, label="xz"),
    plane(Plane((0, 0, 0), (0, 0, 1)), *ranges, label="yz"),
    plane(Plane((0, 0, 0), (1, 1, 1)), *ranges, label="inclined", n=150),
    backend=PB, xlabel="x", ylabel="y", zlabel="z"
)

(Source code, png)

spb.graphics.functions_3d.implicit_3d(expr, range_x=None, range_y=None, range_z=None, label=None, rendering_kw=None, **kwargs)[source]

Plots an isosurface of a function.

Parameters:

expr

Implicit expression. It can be a:

Symbolic expression.
Numerical function of three variable, f(x, y, z), supporting
vectorization.

range_xtuple, Tuple

A 3-tuple (symb, min, max) denoting the range of the x variable. Default values: min=-10 and max=10.

range_ytuple, Tuple

A 3-tuple (symb, min, max) denoting the range of the y variable. Default values: min=-10 and max=10.

range_ztuple, Tuple

A 3-tuple (symb, min, max) denoting the range of the z variable. Default values: min=-10 and max=10.

labelstr

Set the label associated to this series, which will be eventually shown on the legend or colorbar.

rendering_kwdict

A dictionary of keyword arguments to be passed to the renderers in order to further customize the appearance of the surface. Here are some useful links for the supported plotting libraries:

Plotly: https://plotly.com/python/3d-isosurface-plots/
K3D-Jupyter: look at the documentation of k3d.marching_cubes.

colorbarbool

Toggle the visibility of the colorbar associated to the current data series. Note that a colorbar is only visible if use_cm=True and color_func is not None. Default value: True.

colorbar_ticks_formattertick_formatter_multiples_of

An object of type tick_formatter_multiples_of which will be used to place tick values on the colorbar at each multiple of a specified quantity. This only works when use_cm=True.

force_real_evalbool

By default, numerical evaluation is performed over complex numbers, which is slower but produces correct results. However, when the symbolic expression is converted to a numerical function with lambdify, the resulting function may not like to be evaluated over complex numbers. In such cases, forcing the evaluation to be performed over real numbers might be a good choice. The plotting module should be able to detect such occurences and automatically activate this option. If that is not the case, or evaluation performance is of paramount importance, set this parameter to True, but be aware that it might produce wrong results. Default value: False.

modules

Specify the evaluation modules to be used by lambdify. If not specified, the evaluation will be done with NumPy/SciPy.

n1int

Number of discretization points along the x-axis to be used in the evaluation. Related parameters: xscale. It must be: 2 ≤ n1 < ∞. Default value: 60.

n2int

Number of discretization points along the y-axis to be used in the evaluation. Related parameters: yscale. It must be: 2 ≤ n2 < ∞. Default value: 60.

n3int

Number of discretization points along the z-axis to be used in the evaluation. Related parameters: zscale. It must be: 2 ≤ n3 < ∞. Default value: 60.

only_integersbool

Discretize the domain using only integer numbers. When this parameter is True, the number of discretization points is choosen by the algorithm. Default value: False.

paramsdict, optional

A dictionary mapping symbols to parameters. If provided, this dictionary enables the interactive-widgets plot.

When calling a plotting function, the parameter can be specified with:

a widget from the ipywidgets module.
a widget from the panel module.
a tuple of the form:
(default, min, max, N, tick_format, label, spacing), which will instantiate a ipywidgets.widgets.widget_float.FloatSlider or a ipywidgets.widgets.widget_float.FloatLogSlider, depending on the spacing strategy. In particular:
- default, min, maxfloat
  Default value, minimum value and maximum value of the slider, respectively. Must be finite numbers. The order of these 3 numbers is not important: the module will figure it out which is what.
- Nint, optional
  Number of steps of the slider.
- tick_formatstr or None, optional
  Provide a formatter for the tick value of the slider. Default to ".2f".
- label: str, optional
  Custom text associated to the slider.
- spacingstr, optional
  Specify the discretization spacing. Default to "linear", can be changed to "log".

Notes:

parameters cannot be linked together (ie, one parameter cannot depend on another one).
If a widget returns multiple numerical values (like panel.widgets.slider.RangeSlider or ipywidgets.widgets.widget_float.FloatRangeSlider), then a corresponding number of symbols must be provided.

Here follows a couple of examples. If imodule="panel":

import panel as pn
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: pn.widgets.FloatSlider(value=1, start=0, end=5), # same slider as above
    (c, d): pn.widgets.RangeSlider(value=(-1, 1), start=-3, end=3, step=0.1)
}

Or with imodule="ipywidgets":

import ipywidgets as w
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: w.FloatSlider(value=1, min=0, max=5), # same slider as above
    (c, d): w.FloatRangeSlider(value=(-1, 1), min=-3, max=3, step=0.1)
}

When instantiating a data series directly, params must be a dictionary mapping symbols to numerical values.

Let series be any data series. Then series.params returns a dictionary mapping symbols to numerical values.

show_in_legendbool

Toggle the visibility of the data series on the legend. Default value: True.

sum_boundint

When plotting sums, the expression will be pre-processed in order to replace lower/upper bounds set to +/- infinity with this +/- numerical value. Note: the higher this number, the slower the evaluation, but the more accurate the plot. It must be: 0 ≤ sum_bound < ∞. Default value: 1000.

surface_color

For back-compatibility with old sympy.plotting. Use rendering_kw in order to fully customize the appearance of the surface.

txcallable

Numerical transformation function to be applied to the data on the x-axis.

tycallable

Numerical transformation function to be applied to the data on the y-axis.

tzcallable

Numerical transformation function to be applied to the data on the z-axis.

use_cmbool

Toggle the use of a colormap. By default, some series might use a colormap to display the necessary data. Setting this attribute to False will inform the associated renderer to use solid color. Related parameters: color_func. Default value: False.

xscalestr

Discretization strategy along the x-direction. Related parameters: n1. Possible options: [‘linear’, ‘log’] Default value: ‘linear’.

yscalestr

Discretization strategy along the y-direction. Related parameters: n2. Possible options: [‘linear’, ‘log’] Default value: ‘linear’.

zscalestr

Discretization strategy along the z-direction. Related parameters: n3. Possible options: [‘linear’, ‘log’] Default value: ‘linear’.

Returns:

serieslist: A list containing one instance of Implicit3DSeries.

See also

surface, spb.graphics.functions_2d.implicit_2d

Notes

the number of discretization points is crucial as the algorithm will discretize a volume. A high number of discretization points creates a smoother mesh, at the cost of a much higher memory consumption and slower computation.
Only PlotlyBackend and K3DBackend support 3D implicit plotting.
To plot f(x, y, z) = c either write expr = f(x, y, z) - c or pass the appropriate keyword to rendering_kw. Read the backends documentation to find out the available options.

Examples

from sympy import symbols
from spb import *
x, y, z = symbols('x, y, z')
graphics(
    implicit_3d(x**2 + y**3 - z**2, (x, -2, 2), (y, -2, 2), (z, -2, 2)),
    backend=PB)

(Source code, png)

graphics(
    implicit_3d(x**4 + y**4 + z**4 - (x**2 + y**2 + z**2 - 0.3),
        (x, -2, 2), (y, -2, 2), (z, -2, 2)),
    backend=PB)

(Source code, png)

Visualize the isocontours from isomin=0 to isomax=2 by providing a rendering_kw dictionary:

graphics(
    implicit_3d(1/x**2 - 1/y**2 + 1/z**2,
        (x, -2, 2), (y, -2, 2), (z, -2, 2),
        rendering_kw={
            "isomin": 0, "isomax": 2,
            "colorscale":"aggrnyl", "showscale":True
        }),
    backend=PB
)

(Source code, png)

spb.graphics.functions_3d.wireframe(surface_series, n1=10, n2=10, n=None, rendering_kw=None, **kwargs)[source]

Creates a wireframe of a 3D surface.

Parameters:

surface_seriesBaseSeries

A data series representing a surface.

n1int, optional

Number of wireframe lines along the first direction (or range).

n2int, optional

Number of wireframe lines along the second direction (or range).

nint, optional

Number of evaluation points for each wireframe line. If not provided, the algorithm will chose the number of discretization points from the surface_series. The higher this number, the slower the creation of the plot.

rendering_kwdict

A dictionary of keyword arguments to be passed to the renderers in order to further customize the appearance of the line. Here are some useful links for the supported plotting libraries:

Matplotlib:
- for solid lines: https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.plot.html
- for colormap-based lines: https://matplotlib.org/stable/api/collections_api.html#matplotlib.collections.LineCollection
- for scatters: https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.scatter.html
Bokeh:
- for solid lines: https://docs.bokeh.org/en/latest/docs/reference/plotting.html#bokeh.plotting.Figure.line
- for scatter: https://docs.bokeh.org/en/latest/docs/reference/plotting/figure.html#bokeh.plotting.Figure.scatter
Plotly: https://plotly.com/python/line-and-scatter/

color_func

Define a custom 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.
- 3 arguments: f(x, y, z) where x, y, z are the coordinates of the points.
- 4 arguments: f(x, y, z, t).
A symbolic expression having at most as many free symbols as expr_x or expr_y or expr_z.
None: the default value (color mapping according to the parameter).

Alias of this parameters: tp.

colorbarbool

Toggle the visibility of the colorbar associated to the current data series. Note that a colorbar is only visible if use_cm=True and color_func is not None. Default value: True.

colorbar_ticks_formattertick_formatter_multiples_of

An object of type tick_formatter_multiples_of which will be used to place tick values on the colorbar at each multiple of a specified quantity. This only works when use_cm=True.

detect_polesbool, str

Chose whether to detect and correctly plot the roots of the denominator. 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. This algorithm can be used to visualize jump discontinuities as well as essential discontinuities.
a symbolic approach based on the continuous_domain function from the sympy.calculus.util module, which computes the locations of essential discontinuities. If any are found, vertical lines will be shown.

Possible options:

False: No poles detection
True: Poles detection with the numerical algorithm
‘symbolic’: Poles detection with numerical and symbolic algorithms

Default value: False.

epsfloat

An arbitrary small value used by the detect_poles numerical algorithm. Before changing this value, it is recommended to increase the number of discretization points. Related parameters: detect_poles. It must be: 0 ≤ eps < ∞. Default value: 0.01.

excludelist

A list of numerical values along the parameter which are going to be excluded from the evaluation. In practice, it introduces discontinuities in the resulting line.

force_real_evalbool

By default, numerical evaluation is performed over complex numbers, which is slower but produces correct results. However, when the symbolic expression is converted to a numerical function with lambdify, the resulting function may not like to be evaluated over complex numbers. In such cases, forcing the evaluation to be performed over real numbers might be a good choice. The plotting module should be able to detect such occurences and automatically activate this option. If that is not the case, or evaluation performance is of paramount importance, set this parameter to True, but be aware that it might produce wrong results. Default value: False.

is_filledbool

Whether scatter’s markers are filled or void. Default value: True.

is_scatterbool

If True it represent a scatter plot, otherwise a continuous line. Default value: False.

labelstr

Set the label associated to this series, which will be eventually shown on the legend or colorbar.

line_color

For back-compatibility with old sympy.plotting. Use rendering_kw in order to fully customize the appearance of the line/scatter.

modules

Specify the evaluation modules to be used by lambdify. If not specified, the evaluation will be done with NumPy/SciPy.

only_integersbool

Discretize the domain using only integer numbers. When this parameter is True, the number of discretization points is choosen by the algorithm. Default value: False.

paramsdict, optional

A dictionary mapping symbols to parameters. If provided, this dictionary enables the interactive-widgets plot.

When calling a plotting function, the parameter can be specified with:

a widget from the ipywidgets module.
a widget from the panel module.
a tuple of the form:
(default, min, max, N, tick_format, label, spacing), which will instantiate a ipywidgets.widgets.widget_float.FloatSlider or a ipywidgets.widgets.widget_float.FloatLogSlider, depending on the spacing strategy. In particular:
- default, min, maxfloat
  Default value, minimum value and maximum value of the slider, respectively. Must be finite numbers. The order of these 3 numbers is not important: the module will figure it out which is what.
- Nint, optional
  Number of steps of the slider.
- tick_formatstr or None, optional
  Provide a formatter for the tick value of the slider. Default to ".2f".
- label: str, optional
  Custom text associated to the slider.
- spacingstr, optional
  Specify the discretization spacing. Default to "linear", can be changed to "log".

Notes:

parameters cannot be linked together (ie, one parameter cannot depend on another one).
If a widget returns multiple numerical values (like panel.widgets.slider.RangeSlider or ipywidgets.widgets.widget_float.FloatRangeSlider), then a corresponding number of symbols must be provided.

Here follows a couple of examples. If imodule="panel":

import panel as pn
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: pn.widgets.FloatSlider(value=1, start=0, end=5), # same slider as above
    (c, d): pn.widgets.RangeSlider(value=(-1, 1), start=-3, end=3, step=0.1)
}

Or with imodule="ipywidgets":

import ipywidgets as w
params = {
    a: (1, 0, 5), # slider from 0 to 5, with default value of 1
    b: w.FloatSlider(value=1, min=0, max=5), # same slider as above
    (c, d): w.FloatRangeSlider(value=(-1, 1), min=-3, max=3, step=0.1)
}

When instantiating a data series directly, params must be a dictionary mapping symbols to numerical values.

Let series be any data series. Then series.params returns a dictionary mapping symbols to numerical values.

show_in_legendbool

Toggle the visibility of the data series on the legend. Default value: True.

sum_boundint

When plotting sums, the expression will be pre-processed in order to replace lower/upper bounds set to +/- infinity with this +/- numerical value. Note: the higher this number, the slower the evaluation, but the more accurate the plot. It must be: 0 ≤ sum_bound < ∞. Default value: 1000.

txcallable

Numerical transformation function to be applied to the data on the x-axis.

tycallable

Numerical transformation function to be applied to the data on the y-axis.

tzcallable

Numerical transformation function to be applied to the data on the z-axis.

use_cmbool

Toggle the use of a colormap. By default, some series might use a colormap to display the necessary data. Setting this attribute to False will inform the associated renderer to use solid color. Related parameters: color_func. Default value: False.

xscalestr

Discretization strategy along the parameter. Possible options: [‘linear’, ‘log’] Default value: ‘linear’.

Returns:

serieslist: A list containing instances of Parametric3DLineSeries.

See also

surface, surface_parametric, surface_spherical, surface_revolution

Examples

Provide better code separation between the surface and wireframe:

from sympy import *
from spb import *
import numpy as np
x, y = symbols("x y")
surf_series = surface(
    cos(x*y) * exp(-sqrt(x**2 + y**2) / 4), (x, -pi, pi), (y, -pi, pi),
    use_cm=True, color_func=lambda x, y, z: np.sqrt(x**2 + y**2)
)
graphics(
    surf_series,
    wireframe(surf_series[0], n1=11, n2=11),
    backend=PB, aspect=dict(x=1.5, y=1.5, z=1)
)

(Source code, png)

Show wireframe without showing the actual surface. Here, dotted wireframe represent the half-unit sphere.

from sympy import *
from spb import *
var("t u v theta phi")
r_sphere = 1
sphere = surface_spherical(1, (theta, 0, pi), (phi, pi, 2*pi))[0]
t = pi / 3 # half-cone angle
r_cone = r_sphere * sin(t)
graphics(
    wireframe(sphere, n1=13, rendering_kw={"line_dash": "dot"}),
    surface_spherical(1, (theta, pi - t, pi), (phi, pi, 2*pi),
        label="sphere cap", wireframe=True, wf_n1=5),
    surface_parametric(
        u * cos(v), u * sin(v), -u / tan(t), (u, 0, r_cone), (v, pi , 2*pi),
        label="cone", wireframe=True, wf_n1=7),
    backend=PB, grid=False
)

(Source code, png)