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r""" 2D Plotting
Sage provides extensive 2D plotting functionality. The underlying rendering is done using the matplotlib Python library.
The following graphics primitives are supported:
- :func:`~sage.plot.arrow.arrow` - an arrow from a min point to a max point.
- :func:`~sage.plot.circle.circle` - a circle with given radius
- :func:`~sage.plot.ellipse.ellipse` - an ellipse with given radii and angle
- :func:`~sage.plot.arc.arc` - an arc of a circle or an ellipse
- :func:`~sage.plot.disk.disk` - a filled disk (i.e. a sector or wedge of a circle)
- :func:`~sage.plot.line.line` - a line determined by a sequence of points (this need not be straight!)
- :func:`~sage.plot.point.point` - a point
- :func:`~sage.plot.text.text` - some text
- :func:`~sage.plot.polygon.polygon` - a filled polygon
The following plotting functions are supported:
- :func:`plot` - plot of a function or other Sage object (e.g., elliptic curve).
- :func:`parametric_plot`
- :func:`~sage.plot.contour_plot.implicit_plot`
- :func:`polar_plot`
- :func:`~sage.plot.contour_plot.region_plot`
- :func:`list_plot`
- :func:`~sage.plot.scatter_plot.scatter_plot`
- :func:`~sage.plot.bar_chart.bar_chart`
- :func:`~sage.plot.contour_plot.contour_plot`
- :func:`~sage.plot.density_plot.density_plot`
- :func:`~sage.plot.plot_field.plot_vector_field`
- :func:`~sage.plot.plot_field.plot_slope_field`
- :func:`~sage.plot.matrix_plot.matrix_plot`
- :func:`~sage.plot.complex_plot.complex_plot`
- :func:`graphics_array`
- The following log plotting functions:
- :func:`plot_loglog`
- :func:`plot_semilogx` and :func:`plot_semilogy`
- :func:`list_plot_loglog`
- :func:`list_plot_semilogx` and :func:`list_plot_semilogy`
The following miscellaneous Graphics functions are included:
- :func:`Graphics`
- :func:`is_Graphics`
- :func:`~sage.plot.colors.hue`
Type ``?`` after each primitive in Sage for help and examples.
EXAMPLES:
We draw a curve::
sage: plot(x^2, (x,0,5)) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(x**2, (x,0,5)) sphinx_plot(g)
We draw a circle and a curve::
sage: circle((1,1), 1) + plot(x^2, (x,0,5)) Graphics object consisting of 2 graphics primitives
.. PLOT::
g = circle((1,1), 1) + plot(x**2, (x,0,5)) sphinx_plot(g)
Notice that the aspect ratio of the above plot makes the plot very tall because the plot adopts the default aspect ratio of the circle (to make the circle appear like a circle). We can change the aspect ratio to be what we normally expect for a plot by explicitly asking for an 'automatic' aspect ratio::
sage: show(circle((1,1), 1) + plot(x^2, (x,0,5)), aspect_ratio='automatic')
The aspect ratio describes the apparently height/width ratio of a unit square. If you want the vertical units to be twice as big as the horizontal units, specify an aspect ratio of 2::
sage: show(circle((1,1), 1) + plot(x^2, (x,0,5)), aspect_ratio=2)
The ``figsize`` option adjusts the figure size. The default figsize is 4. To make a figure that is roughly twice as big, use ``figsize=8``::
sage: show(circle((1,1), 1) + plot(x^2, (x,0,5)), figsize=8)
You can also give separate horizontal and vertical dimensions. Both will be measured in inches::
sage: show(circle((1,1), 1) + plot(x^2, (x,0,5)), figsize=[4,8])
However, do not make the figsize too big (e.g. one dimension greater than 327 or both in the mid-200s) as this will lead to errors or crashes. See :meth:`~sage.plot.graphics.Graphics.show` for full details.
Note that the axes will not cross if the data is not on both sides of both axes, even if it is quite close::
sage: plot(x^3, (x,1,10)) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(x**3, (x,1,10)) sphinx_plot(g)
When the labels have quite different orders of magnitude or are very large, scientific notation (the `e` notation for powers of ten) is used::
sage: plot(x^2, (x,480,500)) # no scientific notation Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(x**2, (x,480,500)) sphinx_plot(g)
::
sage: plot(x^2, (x,300,500)) # scientific notation on y-axis Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(x**2, (x,300,500)) sphinx_plot(g)
But you can fix your own tick labels, if you know what to expect and have a preference::
sage: plot(x^2, (x,300,500), ticks=[100,50000]) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(x**2, (x,300,500), ticks=[100,50000]) sphinx_plot(g)
To change the ticks on one axis only, use the following notation::
sage: plot(x^2, (x,300,500), ticks=[None,50000]) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(x**2, (x,300,500), ticks=[None,50000]) sphinx_plot(g)
You can even have custom tick labels along with custom positioning. ::
sage: plot(x^2, (x,0,3), ticks=[[1,2.5],pi/2], tick_formatter=[["$x_1$","$x_2$"],pi]) # long time Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(x**2, (x,0,3), ticks=[[1,2.5],pi/2], tick_formatter=[["$x_1$","$x_2$"],pi]) sphinx_plot(g)
We construct a plot involving several graphics objects::
sage: G = plot(cos(x), (x, -5, 5), thickness=5, color='green', title='A plot') sage: P = polygon([[1,2], [5,6], [5,0]], color='red') sage: G + P Graphics object consisting of 2 graphics primitives
.. PLOT::
G = plot(cos(x), (x, -5, 5), thickness=5, color='green', title='A plot') P = polygon([[1,2], [5,6], [5,0]], color='red') sphinx_plot(G + P)
Next we construct the reflection of the above polygon about the `y`-axis by iterating over the list of first-coordinates of the first graphic element of ``P`` (which is the actual Polygon; note that ``P`` is a Graphics object, which consists of a single polygon)::
sage: Q = polygon([(-x,y) for x,y in P[0]], color='blue') sage: Q # show it Graphics object consisting of 1 graphics primitive
.. PLOT::
P = polygon([[1,2], [5,6], [5,0]], color='red') Q = polygon([(-x,y) for x,y in P[0]], color='blue') sphinx_plot(Q)
We combine together different graphics objects using "+"::
sage: H = G + P + Q sage: print(H) Graphics object consisting of 3 graphics primitives sage: type(H) <class 'sage.plot.graphics.Graphics'> sage: H[1] Polygon defined by 3 points sage: list(H[1]) [(1.0, 2.0), (5.0, 6.0), (5.0, 0.0)] sage: H # show it Graphics object consisting of 3 graphics primitives
.. PLOT::
G = plot(cos(x), (x, -5, 5), thickness=5, color='green', title='A plot') P = polygon([[1,2], [5,6], [5,0]], color='red') Q = polygon([(-x,y) for x,y in P[0]], color='blue') H = G + P + Q sphinx_plot(H)
We can put text in a graph::
sage: L = [[cos(pi*i/100)^3,sin(pi*i/100)] for i in range(200)] sage: p = line(L, rgbcolor=(1/4,1/8,3/4)) sage: t = text('A Bulb', (1.5, 0.25)) sage: x = text('x axis', (1.5,-0.2)) sage: y = text('y axis', (0.4,0.9)) sage: g = p+t+x+y sage: g.show(xmin=-1.5, xmax=2, ymin=-1, ymax=1)
.. PLOT::
L = [[cos(pi*i/100)**3,sin(pi*i/100)] for i in range(200)] p = line(L, rgbcolor=(1.0/4.0,1.0/8.0,3.0/4.0)) t = text('A Bulb', (1.5, 0.25)) x = text('x axis', (1.5,-0.2)) y = text('y axis', (0.4,0.9)) g = p+t+x+y g.xmin(-1.5) g.xmax(2) g.ymin(-1) g.ymax(1) sphinx_plot(g)
We can add a title to a graph::
sage: x = var('x') sage: plot(x^2, (x,-2,2), title='A plot of $x^2$') Graphics object consisting of 1 graphics primitive
.. PLOT::
g=plot(x**2, (x,-2,2), title='A plot of $x^2$') sphinx_plot(g)
We can set the position of the title::
sage: plot(x^2, (-2,2), title='Plot of $x^2$', title_pos=(0.5,-0.05)) Graphics object consisting of 1 graphics primitive
.. PLOT::
g=plot(x**2, (-2,2), title='Plot of $x^2$', title_pos=(0.5,-0.05)) sphinx_plot(g)
We plot the Riemann zeta function along the critical line and see the first few zeros::
sage: i = CDF.0 # define i this way for maximum speed. sage: p1 = plot(lambda t: arg(zeta(0.5+t*i)), 1, 27, rgbcolor=(0.8,0,0)) sage: p2 = plot(lambda t: abs(zeta(0.5+t*i)), 1, 27, color=hue(0.7)) sage: print(p1 + p2) Graphics object consisting of 2 graphics primitives sage: p1 + p2 # display it Graphics object consisting of 2 graphics primitives
.. PLOT::
from sage.rings.complex_double import ComplexDoubleElement i = ComplexDoubleElement(0,1) # define i this way for maximum speed. p1 = plot(lambda t: arg(zeta(0.5+t*i)), 1, 27, rgbcolor=(0.8,0,0)) p2 = plot(lambda t: abs(zeta(0.5+t*i)), 1, 27, color=hue(0.7)) g = p1 + p2 sphinx_plot(g)
.. NOTE::
Not all functions in Sage are symbolic. When plotting non-symbolic functions they should be wrapped in ``lambda``::
sage: plot(lambda x:fibonacci(round(x)), (x,1,10)) Graphics object consisting of 1 graphics primitive
.. PLOT::
g=plot(lambda x:fibonacci(round(x)), (x,1,10)) sphinx_plot(g)
Many concentric circles shrinking toward the origin::
sage: show(sum(circle((i,0), i, hue=sin(i/10)) for i in [10,9.9,..,0])) # long time
.. PLOT::
g = sum(circle((i,0), i, hue=sin(i/10)) for i in srange(0,10,0.1)) sphinx_plot(g)
Here is a pretty graph::
sage: g = Graphics() sage: for i in range(60): ....: p = polygon([(i*cos(i),i*sin(i)), (0,i), (i,0)],\ ....: color=hue(i/40+0.4), alpha=0.2) ....: g = g + p sage: g.show(dpi=200, axes=False)
.. PLOT::
g=Graphics() for i in range(60): # i/40 doesn't convert to real number p = polygon([(i*cos(i),i*sin(i)), (0,i), (i,0)],\ color=hue(0.025*i+0.4), alpha=0.2) g = g + p g.axes(False) sphinx_plot(g)
Another graph::
sage: x = var('x') sage: P = plot(sin(x)/x, -4, 4, color='blue') + \ ....: plot(x*cos(x), -4, 4, color='red') + \ ....: plot(tan(x), -4, 4, color='green') sage: P.show(ymin=-pi, ymax=pi)
.. PLOT::
g = plot(sin(x)/x, -4, 4, color='blue') + \ plot(x*cos(x), -4, 4, color='red') + \ plot(tan(x), -4, 4, color='green') g.ymin(-pi) g.ymax(pi) sphinx_plot(g)
PYX EXAMPLES: These are some examples of plots similar to some of the plots in the PyX (http://pyx.sourceforge.net) documentation:
Symbolline::
sage: y(x) = x*sin(x^2) sage: v = [(x, y(x)) for x in [-3,-2.95,..,3]] sage: show(points(v, rgbcolor=(0.2,0.6, 0.1), pointsize=30) + plot(spline(v), -3.1, 3))
.. PLOT::
#y(x)=x*sin(x**2) gave SyntaxError: can't assign to function call def y(x): return x*sin(x**2) v=list() for x in srange(-3,3,0.05): v.append((x, y(x))) g = points(v, rgbcolor=(0.2,0.6, 0.1), pointsize=30) + plot(spline(v), -3.1, 3) sphinx_plot(g)
Cycliclink::
sage: x = var('x') sage: g1 = plot(cos(20*x)*exp(-2*x), 0, 1) sage: g2 = plot(2*exp(-30*x) - exp(-3*x), 0, 1) sage: show(graphics_array([g1, g2], 2, 1), xmin=0)
.. PLOT::
g1 = plot(cos(20*x)*exp(-2*x), 0, 1) g2 = plot(2*exp(-30*x) - exp(-3*x), 0, 1) g = graphics_array([g1, g2], 2, 1) sphinx_plot(g)
Pi Axis::
sage: g1 = plot(sin(x), 0, 2*pi) sage: g2 = plot(cos(x), 0, 2*pi, linestyle="--") sage: (g1+g2).show(ticks=pi/6, tick_formatter=pi) # long time # show their sum, nicely formatted
.. PLOT::
g1 = plot(sin(x), 0, 2*pi, ticks=pi/6, tick_formatter=pi) g2 = plot(cos(x), 0, 2*pi, linestyle="--", ticks=pi/6, tick_formatter=pi) sphinx_plot(g1+g2)
An illustration of integration::
sage: f(x) = (x-3)*(x-5)*(x-7)+40 sage: P = line([(2,0),(2,f(2))], color='black') sage: P += line([(8,0),(8,f(8))], color='black') sage: P += polygon([(2,0),(2,f(2))] + [(x, f(x)) for x in [2,2.1,..,8]] + [(8,0),(2,0)], rgbcolor=(0.8,0.8,0.8),aspect_ratio='automatic') sage: P += text("$\\int_{a}^b f(x) dx$", (5, 20), fontsize=16, color='black') sage: P += plot(f, (1, 8.5), thickness=3) sage: P # show the result Graphics object consisting of 5 graphics primitives
.. PLOT::
#inline f substitution to avoid SyntaxError: can't assign to function call in sphinx_plot def f(x): return (x-3)*(x-5)*(x-7)+40 P = line([(2,0),(2,f(2))], color='black') P = P + line([(8,0),(8,f(8))], color='black') L = list(((2,0), (2,f(2)))) for i in srange(2,8.1,0.1): L.append((i,f(i))) L.append((8,0)) L.append((2,0)) P = P + polygon(L, rgbcolor=(0.8,0.8,0.8), aspect_ratio='automatic') P = P + text("$\\int_{a}^b f(x) dx$", (5, 20), fontsize=16, color='black') P = P + plot(f, (1, 8.5), thickness=3) sphinx_plot(P)
NUMERICAL PLOTTING:
Sage includes Matplotlib, which provides 2D plotting with an interface that is a likely very familiar to people doing numerical computation. You can use ``plt.clf()`` to clear the current image frame and ``plt.close()`` to close it. For example,
::
sage: import pylab as plt sage: t = plt.arange(0.0, 2.0, 0.01) sage: s = sin(2*pi*t) sage: P = plt.plot(t, s, linewidth=1.0) sage: xl = plt.xlabel('time (s)') sage: yl = plt.ylabel('voltage (mV)') sage: t = plt.title('About as simple as it gets, folks') sage: plt.grid(True) sage: plt.savefig(os.path.join(SAGE_TMP, 'sage.png')) sage: plt.clf() sage: plt.savefig(os.path.join(SAGE_TMP, 'blank.png')) sage: plt.close() sage: plt.imshow([[1,2],[0,1]]) <matplotlib.image.AxesImage object at ...>
We test that ``imshow`` works as well, verifying that :trac:`2900` is fixed (in Matplotlib).
::
sage: plt.imshow([[(0.0,0.0,0.0)]]) <matplotlib.image.AxesImage object at ...> sage: plt.savefig(os.path.join(SAGE_TMP, 'foo.png'))
Since the above overwrites many Sage plotting functions, we reset the state of Sage, so that the examples below work!
::
sage: reset()
See http://matplotlib.sourceforge.net for complete documentation about how to use Matplotlib.
TESTS:
We test dumping and loading a plot.
::
sage: p = plot(sin(x), (x, 0,2*pi)) sage: Q = loads(dumps(p))
Verify that a clean sage startup does *not* import matplotlib::
sage: os.system("sage -c \"if 'matplotlib' in sys.modules: sys.exit(1)\"") # long time 0
AUTHORS:
- Alex Clemesha and William Stein (2006-04-10): initial version
- David Joyner: examples
- Alex Clemesha (2006-05-04) major update
- William Stein (2006-05-29): fine tuning, bug fixes, better server integration
- William Stein (2006-07-01): misc polish
- Alex Clemesha (2006-09-29): added contour_plot, frame axes, misc polishing
- Robert Miller (2006-10-30): tuning, NetworkX primitive
- Alex Clemesha (2006-11-25): added plot_vector_field, matrix_plot, arrow, bar_chart, Axes class usage (see axes.py)
- Bobby Moretti and William Stein (2008-01): Change plot to specify ranges using the (varname, min, max) notation.
- William Stein (2008-01-19): raised the documentation coverage from a miserable 12 percent to a 'wopping' 35 percent, and fixed and clarified numerous small issues.
- Jason Grout (2009-09-05): shifted axes and grid functionality over to matplotlib; fixed a number of smaller issues.
- Jason Grout (2010-10): rewrote aspect ratio portions of the code
- Jeroen Demeyer (2012-04-19): move parts of this file to graphics.py (:trac:`12857`)
- Aaron Lauve (2016-07-13): reworked handling of 'color' when passed a list of functions; now more in-line with other CAS's. Added list functionality to linestyle and legend_label options as well. (:trac:`12962`) """ #***************************************************************************** # Copyright (C) 2006 Alex Clemesha <clemesha@gmail.com> # Copyright (C) 2006-2008 William Stein <wstein@gmail.com> # Copyright (C) 2010 Jason Grout # # Distributed under the terms of the GNU General Public License (GPL) # as published by the Free Software Foundation; either version 2 of # the License, or (at your option) any later version. # http://www.gnu.org/licenses/ #***************************************************************************** from __future__ import print_function, absolute_import from six.moves import range from six import iteritems
import os from functools import reduce
## IMPORTANT: Do *not* import matplotlib at module scope. It takes a ## surprisingly long time to initialize itself. It's better if it is ## imported in functions, so it only gets started if it is actually ## going to be used.
#DEFAULT_FIGSIZE=(6, 3.70820393249937) EMBEDDED_MODE = False import sage.misc.misc from sage.arith.srange import srange
from sage.misc.randstate import current_randstate #for plot adaptive refinement from math import sin, cos, pi #for polar_plot
from sage.ext.fast_eval import fast_float, fast_float_constant, is_fast_float
from sage.misc.decorators import options, rename_keyword
from .graphics import Graphics, GraphicsArray
#Currently not used - see comment immediately above about #figure.canvas.mpl_connect('draw_event', pad_for_tick_labels) # TODO - figure out how to use this, add documentation #def pad_for_tick_labels(event): # import matplotlib.transforms as mtransforms # figure=event.canvas.figure # bboxes = [] # for ax in figure.axes: # bbox = ax.xaxis.get_label().get_window_extent() # # the figure transform goes from relative coords->pixels and we # # want the inverse of that # bboxi = bbox.inverse_transformed(figure.transFigure) # bboxes.append(bboxi) # # bbox = ax.yaxis.get_label().get_window_extent() # bboxi = bbox.inverse_transformed(figure.transFigure) # bboxes.append(bboxi) # for label in (ax.get_xticklabels()+ax.get_yticklabels() \ # + ax.get_xticklabels(minor=True) \ # +ax.get_yticklabels(minor=True)): # bbox = label.get_window_extent() # bboxi = bbox.inverse_transformed(figure.transFigure) # bboxes.append(bboxi) # # # this is the bbox that bounds all the bboxes, again in relative # # figure coords # bbox = mtransforms.Bbox.union(bboxes) # adjusted=adjust_figure_to_contain_bbox(figure,bbox) # # if adjusted: # figure.canvas.draw() # return False # #Currently not used - see comment above about #figure.canvas.mpl_connect('draw_event', pad_for_tick_labels) # TODO - figure out how to use this, add documentation #def adjust_figure_to_contain_bbox(fig, bbox,pad=1.1): # """ # For each amount we are over (in axes coordinates), we adjust by over*pad # to give ourselves a bit of padding. # """ # left=fig.subplotpars.left # bottom=fig.subplotpars.bottom # right=fig.subplotpars.right # top=fig.subplotpars.top # # adjusted=False # if bbox.xmin<0: # left-=bbox.xmin*pad # adjusted=True # if bbox.ymin<0: # bottom-=bbox.ymin*pad # adjusted=True # if bbox.xmax>1: # right-=(bbox.xmax-1)*pad # adjusted=True # if bbox.ymax>1: # top-=(bbox.ymax-1)*pad # adjusted=True # # if left<right and bottom<top: # fig.subplots_adjust(left=left, bottom=bottom, right=right, top=top) # return adjusted # else: # return False
_SelectiveFormatterClass = None
def SelectiveFormatter(formatter, skip_values): """ This matplotlib formatter selectively omits some tick values and passes the rest on to a specified formatter.
EXAMPLES:
This example is almost straight from a matplotlib example.
::
sage: from sage.plot.plot import SelectiveFormatter sage: import matplotlib.pyplot as plt sage: import numpy sage: fig=plt.figure() sage: ax=fig.add_subplot(111) sage: t = numpy.arange(0.0, 2.0, 0.01) sage: s = numpy.sin(2*numpy.pi*t) sage: p = ax.plot(t, s) sage: formatter=SelectiveFormatter(ax.xaxis.get_major_formatter(),skip_values=[0,1]) sage: ax.xaxis.set_major_formatter(formatter) sage: fig.savefig(os.path.join(SAGE_TMP, 'test.png'))
""" global _SelectiveFormatterClass
""" Initialize a SelectiveFormatter object.
INPUT:
- formatter -- the formatter object to which we should pass labels
- skip_values -- a list of values that we should skip when formatting the tick labels
EXAMPLES::
sage: from sage.plot.plot import SelectiveFormatter sage: import matplotlib.pyplot as plt sage: import numpy sage: fig=plt.figure() sage: ax=fig.add_subplot(111) sage: t = numpy.arange(0.0, 2.0, 0.01) sage: s = numpy.sin(2*numpy.pi*t) sage: line=ax.plot(t, s) sage: formatter=SelectiveFormatter(ax.xaxis.get_major_formatter(),skip_values=[0,1]) sage: ax.xaxis.set_major_formatter(formatter) sage: fig.savefig(os.path.join(SAGE_TMP, 'test.png')) """ """ Set the locations for the ticks that are not skipped.
EXAMPLES::
sage: from sage.plot.plot import SelectiveFormatter sage: import matplotlib.ticker sage: formatter=SelectiveFormatter(matplotlib.ticker.Formatter(),skip_values=[0,200]) sage: formatter.set_locs([i*100 for i in range(10)]) """ """ Return the format for tick val *x* at position *pos*
EXAMPLES::
sage: from sage.plot.plot import SelectiveFormatter sage: import matplotlib.ticker sage: formatter=SelectiveFormatter(matplotlib.ticker.FixedFormatter(['a','b']),skip_values=[0,2]) sage: [formatter(i,1) for i in range(10)] ['', 'b', '', 'b', 'b', 'b', 'b', 'b', 'b', 'b'] """ else:
def xydata_from_point_list(points): r""" Returns two lists (xdata, ydata), each coerced to a list of floats, which correspond to the x-coordinates and the y-coordinates of the points.
The points parameter can be a list of 2-tuples or some object that yields a list of one or two numbers.
This function can potentially be very slow for large point sets.
TESTS::
sage: from sage.plot.plot import xydata_from_point_list sage: xydata_from_point_list([CC(0), CC(1)]) # ticket 8082 ([0.0, 1.0], [0.0, 0.0])
This function should work for anything than can be turned into a list, such as iterators and such (see :trac:`10478`)::
sage: xydata_from_point_list(iter([(0,0), (sqrt(3), 2)])) ([0.0, 1.7320508075688772], [0.0, 2.0]) sage: xydata_from_point_list((x, x^2) for x in range(5)) ([0.0, 1.0, 2.0, 3.0, 4.0], [0.0, 1.0, 4.0, 9.0, 16.0]) sage: xydata_from_point_list(enumerate(prime_range(1, 15))) ([0.0, 1.0, 2.0, 3.0, 4.0, 5.0], [2.0, 3.0, 5.0, 7.0, 11.0, 13.0]) sage: from builtins import zip sage: xydata_from_point_list(list(zip([2,3,5,7], [11, 13, 17, 19]))) ([2.0, 3.0, 5.0, 7.0], [11.0, 13.0, 17.0, 19.0]) """
ComplexNumber)):
@options(alpha=1, thickness=1, fill=False, fillcolor='automatic', fillalpha=0.5, plot_points=200, adaptive_tolerance=0.01, adaptive_recursion=5, detect_poles=False, exclude=None, legend_label=None, __original_opts=True, aspect_ratio='automatic') def plot(funcs, *args, **kwds): r""" Use plot by writing
``plot(X, ...)``
where `X` is a Sage object (or list of Sage objects) that either is callable and returns numbers that can be coerced to floats, or has a plot method that returns a ``GraphicPrimitive`` object.
There are many other specialized 2D plot commands available in Sage, such as ``plot_slope_field``, as well as various graphics primitives like :class:`~sage.plot.arrow.Arrow`; type ``sage.plot.plot?`` for a current list.
Type ``plot.options`` for a dictionary of the default options for plots. You can change this to change the defaults for all future plots. Use ``plot.reset()`` to reset to the default options.
PLOT OPTIONS:
- ``plot_points`` - (default: 200) the minimal number of plot points.
- ``adaptive_recursion`` - (default: 5) how many levels of recursion to go before giving up when doing adaptive refinement. Setting this to 0 disables adaptive refinement.
- ``adaptive_tolerance`` - (default: 0.01) how large a difference should be before the adaptive refinement code considers it significant. See the documentation further below for more information, starting at "the algorithm used to insert".
- ``base`` - (default: 10) the base of the logarithm if a logarithmic scale is set. This must be greater than 1. The base can be also given as a list or tuple ``(basex, basey)``. ``basex`` sets the base of the logarithm along the horizontal axis and ``basey`` sets the base along the vertical axis.
- ``scale`` -- (default: ``"linear"``) string. The scale of the axes. Possible values are ``"linear"``, ``"loglog"``, ``"semilogx"``, ``"semilogy"``.
The scale can be also be given as single argument that is a list or tuple ``(scale, base)`` or ``(scale, basex, basey)``.
The ``"loglog"`` scale sets both the horizontal and vertical axes to logarithmic scale. The ``"semilogx"`` scale sets the horizontal axis to logarithmic scale. The ``"semilogy"`` scale sets the vertical axis to logarithmic scale. The ``"linear"`` scale is the default value when :class:`~sage.plot.graphics.Graphics` is initialized.
- ``xmin`` - starting x value in the rendered figure. This parameter is passed directly to the ``show`` procedure and it could be overwritten.
- ``xmax`` - ending x value in the rendered figure. This parameter is passed directly to the ``show`` procedure and it could be overwritten.
- ``ymin`` - starting y value in the rendered figure. This parameter is passed directly to the ``show`` procedure and it could be overwritten.
- ``ymax`` - ending y value in the rendered figure. This parameter is passed directly to the ``show`` procedure and it could be overwritten.
- ``detect_poles`` - (Default: False) If set to True poles are detected. If set to "show" vertical asymptotes are drawn.
- ``legend_label`` - a (TeX) string serving as the label for `X` in the legend. If `X` is a list, then this option can be a single string, or a list or dictionary with strings as entries/values. If a dictionary, then keys are taken from ``range(len(X))``.
.. note::
- If the ``scale`` is ``"linear"``, then irrespective of what ``base`` is set to, it will default to 10 and will remain unused.
- If you want to limit the plot along the horizontal axis in the final rendered figure, then pass the ``xmin`` and ``xmax`` keywords to the :meth:`~sage.plot.graphics.Graphics.show` method. To limit the plot along the vertical axis, ``ymin`` and ``ymax`` keywords can be provided to either this ``plot`` command or to the ``show`` command.
- This function does NOT simply sample equally spaced points between xmin and xmax. Instead it computes equally spaced points and adds small perturbations to them. This reduces the possibility of, e.g., sampling `\sin` only at multiples of `2\pi`, which would yield a very misleading graph.
- If there is a range of consecutive points where the function has no value, then those points will be excluded from the plot. See the example below on automatic exclusion of points.
- For the other keyword options that the ``plot`` function can take, refer to the method :meth:`~sage.plot.graphics.Graphics.show` and the further options below.
COLOR OPTIONS:
- ``color`` - (Default: 'blue') One of:
- an RGB tuple (r,g,b) with each of r,g,b between 0 and 1.
- a color name as a string (e.g., 'purple').
- an HTML color such as '#aaff0b'.
- a list or dictionary of colors (valid only if `X` is a list): if a dictionary, keys are taken from ``range(len(X))``; the entries/values of the list/dictionary may be any of the options above.
- 'automatic' -- maps to default ('blue') if `X` is a single Sage object; and maps to a fixed sequence of regularly spaced colors if `X` is a list.
- ``legend_color`` - the color of the text for `X` (or each item in `X`) in the legend. Default color is 'black'. Options are as in ``color`` above, except that the choice 'automatic' maps to 'black' if `X` is a single Sage object.
- ``fillcolor`` - The color of the fill for the plot of `X` (or each item in `X`). Default color is 'gray' if `X` is a single Sage object or if ``color`` is a single color. Otherwise, options are as in ``color`` above.
APPEARANCE OPTIONS:
The following options affect the appearance of the line through the points on the graph of `X` (these are the same as for the line function):
INPUT:
- ``alpha`` - How transparent the line is
- ``thickness`` - How thick the line is
- ``rgbcolor`` - The color as an RGB tuple
- ``hue`` - The color given as a hue
LINE OPTIONS:
Any MATPLOTLIB line option may also be passed in. E.g.,
- ``linestyle`` - (default: "-") The style of the line, which is one of
- ``"-"`` or ``"solid"`` - ``"--"`` or ``"dashed"`` - ``"-."`` or ``"dash dot"`` - ``":"`` or ``"dotted"`` - ``"None"`` or ``" "`` or ``""`` (nothing) - a list or dictionary (see below)
The linestyle can also be prefixed with a drawing style (e.g., ``"steps--"``)
- ``"default"`` (connect the points with straight lines) - ``"steps"`` or ``"steps-pre"`` (step function; horizontal line is to the left of point) - ``"steps-mid"`` (step function; points are in the middle of horizontal lines) - ``"steps-post"`` (step function; horizontal line is to the right of point)
If `X` is a list, then ``linestyle`` may be a list (with entries taken from the strings above) or a dictionary (with keys in ``range(len(X))`` and values taken from the strings above).
- ``marker`` - The style of the markers, which is one of
- ``"None"`` or ``" "`` or ``""`` (nothing) -- default - ``","`` (pixel), ``"."`` (point) - ``"_"`` (horizontal line), ``"|"`` (vertical line) - ``"o"`` (circle), ``"p"`` (pentagon), ``"s"`` (square), ``"x"`` (x), ``"+"`` (plus), ``"*"`` (star) - ``"D"`` (diamond), ``"d"`` (thin diamond) - ``"H"`` (hexagon), ``"h"`` (alternative hexagon) - ``"<"`` (triangle left), ``">"`` (triangle right), ``"^"`` (triangle up), ``"v"`` (triangle down) - ``"1"`` (tri down), ``"2"`` (tri up), ``"3"`` (tri left), ``"4"`` (tri right) - ``0`` (tick left), ``1`` (tick right), ``2`` (tick up), ``3`` (tick down) - ``4`` (caret left), ``5`` (caret right), ``6`` (caret up), ``7`` (caret down), ``8`` (octagon) - ``"$...$"`` (math TeX string) - ``(numsides, style, angle)`` to create a custom, regular symbol
- ``numsides`` -- the number of sides - ``style`` -- ``0`` (regular polygon), ``1`` (star shape), ``2`` (asterisk), ``3`` (circle) - ``angle`` -- the angular rotation in degrees
- ``markersize`` - the size of the marker in points
- ``markeredgecolor`` -- the color of the marker edge
- ``markerfacecolor`` -- the color of the marker face
- ``markeredgewidth`` - the size of the marker edge in points
- ``exclude`` - (Default: None) values which are excluded from the plot range. Either a list of real numbers, or an equation in one variable.
FILLING OPTIONS:
- ``fill`` - (Default: False) One of:
- "axis" or True: Fill the area between the function and the x-axis.
- "min": Fill the area between the function and its minimal value.
- "max": Fill the area between the function and its maximal value.
- a number c: Fill the area between the function and the horizontal line y = c.
- a function g: Fill the area between the function that is plotted and g.
- a dictionary ``d`` (only if a list of functions are plotted): The keys of the dictionary should be integers. The value of ``d[i]`` specifies the fill options for the i-th function in the list. If ``d[i] == [j]``: Fill the area between the i-th and the j-th function in the list. (But if ``d[i] == j``: Fill the area between the i-th function in the list and the horizontal line y = j.)
- ``fillalpha`` - (default: 0.5) How transparent the fill is. A number between 0 and 1.
MATPLOTLIB STYLE SHEET OPTION:
- ``stylesheet`` - (Default: classic) Support for loading a full matplotlib style sheet. Any style sheet listed in ``matplotlib.pyplot.style.available`` is acceptable. If a non-existing style is provided the default classic is applied.
EXAMPLES:
We plot the `\sin` function::
sage: P = plot(sin, (0,10)); print(P) Graphics object consisting of 1 graphics primitive sage: len(P) # number of graphics primitives 1 sage: len(P[0]) # how many points were computed (random) 225 sage: P # render Graphics object consisting of 1 graphics primitive
.. PLOT::
P = plot(sin, (0,10)) sphinx_plot(P)
::
sage: P = plot(sin, (0,10), plot_points=10); print(P) Graphics object consisting of 1 graphics primitive sage: len(P[0]) # random output 32 sage: P # render Graphics object consisting of 1 graphics primitive
.. PLOT::
P = plot(sin, (0,10), plot_points=10) sphinx_plot(P)
We plot with ``randomize=False``, which makes the initial sample points evenly spaced (hence always the same). Adaptive plotting might insert other points, however, unless ``adaptive_recursion=0``.
::
sage: p=plot(1, (x,0,3), plot_points=4, randomize=False, adaptive_recursion=0) sage: list(p[0]) [(0.0, 1.0), (1.0, 1.0), (2.0, 1.0), (3.0, 1.0)]
Some colored functions::
sage: plot(sin, 0, 10, color='purple') Graphics object consisting of 1 graphics primitive
.. PLOT::
P=plot(sin,0,10,color='purple') sphinx_plot(P)
::
sage: plot(sin, 0, 10, color='#ff00ff') Graphics object consisting of 1 graphics primitive
.. PLOT::
P=plot(sin, 0, 10, color='#ff00ff') sphinx_plot(P)
We plot several functions together by passing a list of functions as input::
sage: plot([x*exp(-n*x^2)/.4 for n in [1..5]], (0, 2), aspect_ratio=.8) Graphics object consisting of 5 graphics primitives
.. PLOT::
g = plot([x*exp(-n*x**2)/.4 for n in range(1,6)], (0, 2), aspect_ratio=.8) sphinx_plot(g)
By default, color will change from one primitive to the next. This may be controlled by modifying ``color`` option::
sage: g1 = plot([x*exp(-n*x^2)/.4 for n in [1..3]], (0, 2), color='blue', aspect_ratio=.8); g1 Graphics object consisting of 3 graphics primitives sage: g2 = plot([x*exp(-n*x^2)/.4 for n in [1..3]], (0, 2), color=['red','red','green'], linestyle=['-','--','-.'], aspect_ratio=.8); g2 Graphics object consisting of 3 graphics primitives
.. PLOT::
g1 = plot([x*exp(-n*x**2)/.4 for n in range(1,4)], (0, 2), color='blue', aspect_ratio=.8) g2 = plot([x*exp(-n*x**2)/.4 for n in range(1,4)], (0, 2), color=['red','red','green'], linestyle=['-','--','-.'], aspect_ratio=.8) sphinx_plot(graphics_array([[g1], [g2]]))
We can also build a plot step by step from an empty plot::
sage: a = plot([]); a # passing an empty list returns an empty plot (Graphics() object) Graphics object consisting of 0 graphics primitives sage: a += plot(x**2); a # append another plot Graphics object consisting of 1 graphics primitive
.. PLOT::
a = plot([]) a = a + plot(x**2) sphinx_plot(a)
::
sage: a += plot(x**3); a # append yet another plot Graphics object consisting of 2 graphics primitives
.. PLOT::
a = plot([]) a = a + plot(x**2) a = a + plot(x**3) sphinx_plot(a)
The function `\sin(1/x)` wiggles wildly near `0`. Sage adapts to this and plots extra points near the origin.
::
sage: plot(sin(1/x), (x, -1, 1)) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(sin(1/x), (x, -1, 1)) sphinx_plot(g)
Via the matplotlib library, Sage makes it easy to tell whether a graph is on both sides of both axes, as the axes only cross if the origin is actually part of the viewing area::
sage: plot(x^3,(x,0,2)) # this one has the origin Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(x**3,(x,0,2)) sphinx_plot(g)
::
sage: plot(x^3,(x,1,2)) # this one does not Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(x**3,(x,1,2)) sphinx_plot(g)
Another thing to be aware of with axis labeling is that when the labels have quite different orders of magnitude or are very large, scientific notation (the `e` notation for powers of ten) is used::
sage: plot(x^2,(x,480,500)) # this one has no scientific notation Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(x**2,(x,480,500)) sphinx_plot(g)
::
sage: plot(x^2,(x,300,500)) # this one has scientific notation on y-axis Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(x**2,(x,300,500)) sphinx_plot(g)
You can put a legend with ``legend_label`` (the legend is only put once in the case of multiple functions)::
sage: plot(exp(x), 0, 2, legend_label='$e^x$') Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(exp(x), 0, 2, legend_label='$e^x$') sphinx_plot(g)
Sage understands TeX, so these all are slightly different, and you can choose one based on your needs::
sage: plot(sin, legend_label='sin') Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(sin,legend_label='sin') sphinx_plot(g)
::
sage: plot(sin, legend_label='$sin$') Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(sin,legend_label='$sin$') sphinx_plot(g)
::
sage: plot(sin, legend_label='$\sin$') Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(sin,legend_label='$\sin$') sphinx_plot(g)
It is possible to use a different color for the text of each label::
sage: p1 = plot(sin, legend_label='sin', legend_color='red') sage: p2 = plot(cos, legend_label='cos', legend_color='green') sage: p1 + p2 Graphics object consisting of 2 graphics primitives
.. PLOT::
p1 = plot(sin, legend_label='sin', legend_color='red') p2 = plot(cos, legend_label='cos', legend_color='green') g = p1 + p2 sphinx_plot(g)
Prior to :trac:`19485`, legends by default had a shadowless gray background. This behavior can be recovered by setting the legend options on your plot object::
sage: p = plot(sin(x), legend_label='$\sin(x)$') sage: p.set_legend_options(back_color=(0.9,0.9,0.9), shadow=False)
.. PLOT::
g = plot(sin(x), legend_label='$\sin(x)$') g.set_legend_options(back_color=(0.9,0.9,0.9), shadow=False) sphinx_plot(g)
If `X` is a list of Sage objects and ``legend_label`` is 'automatic', then Sage will create labels for each function according to their internal representation::
sage: plot([sin(x), tan(x), 1-x^2], legend_label='automatic') Graphics object consisting of 3 graphics primitives
.. PLOT::
g = plot([sin(x), tan(x), 1-x**2], legend_label='automatic') sphinx_plot(g)
If ``legend_label`` is any single string other than 'automatic', then it is repeated for all members of `X`::
sage: plot([sin(x), tan(x)], color='blue', legend_label='trig') Graphics object consisting of 2 graphics primitives
.. PLOT::
g = plot([sin(x), tan(x)], color='blue', legend_label='trig') sphinx_plot(g)
Note that the independent variable may be omitted if there is no ambiguity::
sage: plot(sin(1.0/x), (-1, 1)) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(sin(1.0/x), (-1, 1)) sphinx_plot(g)
Plotting in logarithmic scale is possible for 2D plots. There are two different syntaxes supported::
sage: plot(exp, (1, 10), scale='semilogy') # log axis on vertical Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(exp, (1, 10), scale='semilogy') sphinx_plot(g)
::
sage: plot_semilogy(exp, (1, 10)) # same thing Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot_semilogy(exp, (1, 10)) sphinx_plot(g)
::
sage: plot_loglog(exp, (1, 10)) # both axes are log Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot_loglog(exp, (1, 10)) sphinx_plot(g)
::
sage: plot(exp, (1, 10), scale='loglog', base=2) # long time # base of log is 2 Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(exp, (1, 10), scale='loglog', base=2) sphinx_plot(g)
We can also change the scale of the axes in the graphics just before displaying::
sage: G = plot(exp, 1, 10) # long time sage: G.show(scale=('semilogy', 2)) # long time
The algorithm used to insert extra points is actually pretty simple. On the picture drawn by the lines below::
sage: p = plot(x^2, (-0.5, 1.4)) + line([(0,0), (1,1)], color='green') sage: p += line([(0.5, 0.5), (0.5, 0.5^2)], color='purple') sage: p += point(((0, 0), (0.5, 0.5), (0.5, 0.5^2), (1, 1)), color='red', pointsize=20) sage: p += text('A', (-0.05, 0.1), color='red') sage: p += text('B', (1.01, 1.1), color='red') sage: p += text('C', (0.48, 0.57), color='red') sage: p += text('D', (0.53, 0.18), color='red') sage: p.show(axes=False, xmin=-0.5, xmax=1.4, ymin=0, ymax=2)
.. PLOT::
g = plot(x**2, (-0.5, 1.4)) + line([(0,0), (1,1)], color='green') g = g + line([(0.5, 0.5), (0.5, 0.5**2)], color='purple') g = g + point(((0, 0), (0.5, 0.5), (0.5, 0.5**2), (1, 1)), color='red', pointsize=20) g = g + text('A', (-0.05, 0.1), color='red') g = g + text('B', (1.01, 1.1), color='red') g = g + text('C', (0.48, 0.57), color='red') g = g + text('D', (0.53, 0.18), color='red') g.axes(False) g.xmin(-0.5) g.xmax(1.4) g.ymin(0) g.ymax(2) sphinx_plot(g)
You have the function (in blue) and its approximation (in green) passing through the points A and B. The algorithm finds the midpoint C of AB and computes the distance between C and D. If that distance exceeds the ``adaptive_tolerance`` threshold (*relative* to the size of the initial plot subintervals), the point D is added to the curve. If D is added to the curve, then the algorithm is applied recursively to the points A and D, and D and B. It is repeated ``adaptive_recursion`` times (5, by default).
The actual sample points are slightly randomized, so the above plots may look slightly different each time you draw them.
We draw the graph of an elliptic curve as the union of graphs of 2 functions.
::
sage: def h1(x): return abs(sqrt(x^3 - 1)) sage: def h2(x): return -abs(sqrt(x^3 - 1)) sage: P = plot([h1, h2], 1,4) sage: P # show the result Graphics object consisting of 2 graphics primitives
.. PLOT::
def h1(x): return abs(sqrt(x**3 - 1)) def h2(x): return -abs(sqrt(x**3 - 1)) P = plot([h1, h2], 1,4) sphinx_plot(P)
It is important to mention that when we draw several graphs at the same time, parameters ``xmin``, ``xmax``, ``ymin`` and ``ymax`` are just passed directly to the ``show`` procedure. In fact, these parameters would be overwritten::
sage: p=plot(x^3, x, xmin=-1, xmax=1,ymin=-1, ymax=1) sage: q=plot(exp(x), x, xmin=-2, xmax=2, ymin=0, ymax=4) sage: (p+q).show()
As a workaround, we can perform the trick::
sage: p1 = line([(a,b) for a,b in zip(p[0].xdata,p[0].ydata) if (b>=-1 and b<=1)]) sage: q1 = line([(a,b) for a,b in zip(q[0].xdata,q[0].ydata) if (b>=0 and b<=4)]) sage: (p1+q1).show()
We can also directly plot the elliptic curve::
sage: E = EllipticCurve([0,-1]) sage: plot(E, (1, 4), color=hue(0.6)) Graphics object consisting of 1 graphics primitive
.. PLOT::
E = EllipticCurve([0,-1]) g = plot(E, (1, 4), color=hue(0.6)) sphinx_plot(g)
We can change the line style as well::
sage: plot(sin(x), (x, 0, 10), linestyle='-.') Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(sin(x), (x, 0, 10), linestyle='-.') sphinx_plot(g)
If we have an empty linestyle and specify a marker, we can see the points that are actually being plotted::
sage: plot(sin(x), (x,0,10), plot_points=20, linestyle='', marker='.') Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(sin(x), (x,0,10), plot_points=20, linestyle='', marker='.') sphinx_plot(g)
The marker can be a TeX symbol as well::
sage: plot(sin(x), (x,0,10), plot_points=20, linestyle='', marker=r'$\checkmark$') Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(sin(x), (x,0,10), plot_points=20, linestyle='', marker=r'$\checkmark$') sphinx_plot(g)
Sage currently ignores points that cannot be evaluated
::
sage: set_verbose(-1) sage: plot(-x*log(x), (x,0,1)) # this works fine since the failed endpoint is just skipped. Graphics object consisting of 1 graphics primitive sage: set_verbose(0)
This prints out a warning and plots where it can (we turn off the warning by setting the verbose mode temporarily to -1.)
::
sage: set_verbose(-1) sage: plot(x^(1/3), (x,-1,1)) Graphics object consisting of 1 graphics primitive sage: set_verbose(0)
.. PLOT::
set_verbose(-1) g = plot(x**(1.0/3.0), (x,-1,1)) sphinx_plot(g) set_verbose(0)
Plotting the real cube root function for negative input requires avoiding the complex numbers one would usually get. The easiest way is to use absolute value::
sage: plot(sign(x)*abs(x)^(1/3), (x,-1,1)) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(sign(x)*abs(x)**(1.0/3.0), (x,-1,1)) sphinx_plot(g)
We can also use the following::
sage: plot(sign(x)*(x*sign(x))^(1/3), (x,-4,4)) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(sign(x)*(x*sign(x))**(1.0/3.0), (x,-4,4)) sphinx_plot(g)
A way that points to how to plot other functions without symbolic variants is using lambda functions::
sage: plot(lambda x : RR(x).nth_root(3), (x,-1, 1)) Graphics object consisting of 1 graphics primitive
.. PLOT::
sphinx_plot(plot(lambda x : RR(x).nth_root(3), (x,-1, 1)))
We can detect the poles of a function::
sage: plot(gamma, (-3, 4), detect_poles=True).show(ymin=-5, ymax=5)
.. PLOT::
g = plot(gamma, (-3, 4), detect_poles=True) g.ymin(-5) g.ymax(5) sphinx_plot(g)
We draw the Gamma-Function with its poles highlighted::
sage: plot(gamma, (-3, 4), detect_poles='show').show(ymin=-5, ymax=5)
.. PLOT::
g = plot(gamma, (-3, 4), detect_poles='show') g.ymin(-5) g.ymax(5) sphinx_plot(g)
The basic options for filling a plot::
sage: p1 = plot(sin(x), -pi, pi, fill='axis') sage: p2 = plot(sin(x), -pi, pi, fill='min', fillalpha=1) sage: p3 = plot(sin(x), -pi, pi, fill='max') sage: p4 = plot(sin(x), -pi, pi, fill=(1-x)/3, fillcolor='blue', fillalpha=.2) sage: graphics_array([[p1, p2], [p3, p4]]).show(frame=True, axes=False) # long time
.. PLOT::
p1 = plot(sin(x), -pi, pi, fill='axis'); print p1 p2 = plot(sin(x), -pi, pi, fill='min', fillalpha=1) p3 = plot(sin(x), -pi, pi, fill='max') p4 = plot(sin(x), -pi, pi, fill=(1-x)/3, fillcolor='blue', fillalpha=.2) g = graphics_array([[p1, p2], [p3, p4]]) sphinx_plot(g) # Needs to accept options 'frame', 'axes', ...
The basic options for filling a list of plots::
sage: (f1, f2) = x*exp(-1*x^2)/.35, x*exp(-2*x^2)/.35 sage: p1 = plot([f1, f2], -pi, pi, fill={1: [0]}, fillcolor='blue', fillalpha=.25, color='blue') sage: p2 = plot([f1, f2], -pi, pi, fill={0: x/3, 1:[0]}, color=['blue']) sage: p3 = plot([f1, f2], -pi, pi, fill=[0, [0]], fillcolor=['orange','red'], fillalpha=1, color={1: 'blue'}) sage: p4 = plot([f1, f2], (x,-pi, pi), fill=[x/3, 0], fillcolor=['grey'], color=['red', 'blue']) sage: graphics_array([[p1, p2], [p3, p4]]).show(frame=True, axes=False) # long time
.. PLOT::
(f1, f2) = x*exp(-1*x**2)/.35, x*exp(-2*x**2)/.35 p1 = plot([f1, f2], -pi, pi, fill={1: [0]}, fillcolor='blue', fillalpha=.25, color='blue') p2 = plot([f1, f2], -pi, pi, fill={0: x/3, 1:[0]}, color=['blue']) p3 = plot([f1, f2], -pi, pi, fill=[0, [0]], fillcolor=['orange','red'], fillalpha=1, color={1: 'blue'}) p4 = plot([f1, f2], (x,-pi, pi), fill=[x/3, 0], fillcolor=['grey'], color=['red', 'blue']) g = graphics_array([[p1, p2], [p3, p4]]) sphinx_plot(g) # Needs to accept options 'frame', 'axes', ...
A example about the growth of prime numbers::
sage: plot(1.13*log(x), 1, 100, fill=lambda x: nth_prime(x)/floor(x), fillcolor='red') Graphics object consisting of 2 graphics primitives
.. PLOT::
sphinx_plot(plot(1.13*log(x), 1, 100, fill=lambda x: nth_prime(x)/floor(x), fillcolor='red'))
Fill the area between a function and its asymptote::
sage: f = (2*x^3+2*x-1)/((x-2)*(x+1)) sage: plot([f, 2*x+2], -7,7, fill={0: [1]}, fillcolor='#ccc').show(ymin=-20, ymax=20)
.. PLOT::
f = (2*x**3+2*x-1)/((x-2)*(x+1)) g = plot([f, 2*x+2], -7,7, fill={0: [1]}, fillcolor='#ccc') g.ymin(-20) g.ymax(20) sphinx_plot(g)
Fill the area between a list of functions and the x-axis::
sage: def b(n): return lambda x: bessel_J(n, x) sage: plot([b(n) for n in [1..5]], 0, 20, fill='axis') Graphics object consisting of 10 graphics primitives
.. PLOT::
def b(n): return lambda x: bessel_J(n, x) g = plot([b(n) for n in range(1,6)], 0, 20, fill='axis') sphinx_plot(g)
Note that to fill between the ith and jth functions, you must use the dictionary key-value syntax ``i:[j]``; using key-value pairs like ``i:j`` will fill between the ith function and the line y=j::
sage: def b(n): return lambda x: bessel_J(n, x) + 0.5*(n-1) sage: plot([b(c) for c in [1..5]], 0, 20, fill={i:[i-1] for i in [1..4]}, color={i:'blue' for i in [1..5]}, aspect_ratio=3, ymax=3); Graphics object consisting of 9 graphics primitives sage: plot([b(c) for c in [1..5]], 0, 20, fill={i:i-1 for i in [1..4]}, color='blue', aspect_ratio=3) # long time Graphics object consisting of 9 graphics primitives
.. PLOT::
def b(n): return lambda x: bessel_J(n, x) + 0.5*(n-1) g1 = plot([b(n) for n in range(1,6)], 0, 20, fill={i:[i-1] for i in range(1,5)}, color={i:'blue' for i in range(1,6)}, aspect_ratio=3) g2 = plot([b(n) for n in range(1,6)], 0, 20, fill={i:i-1 for i in range(1,5)}, color='blue', aspect_ratio=3) # long time g1.ymax(3) g = graphics_array([[g1], [g2]]) sphinx_plot(g)
Extra options will get passed on to :meth:`~sage.plot.graphics.Graphics.show`, as long as they are valid::
sage: plot(sin(x^2), (x, -3, 3), title='Plot of $\sin(x^2)$', axes_labels=['$x$','$y$']) # These labels will be nicely typeset Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(sin(x**2), (x, -3, 3), title='Plot of $\sin(x^2)$', axes_labels=['$x$','$y$']) # These labels will be nicely typeset sphinx_plot(g)
::
sage: plot(sin(x^2), (x, -3, 3), title='Plot of sin(x^2)', axes_labels=['x','y']) # These will not Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(sin(x**2), (x, -3, 3), title='Plot of sin(x^2)', axes_labels=['x','y']) # These will not sphinx_plot(g)
::
sage: plot(sin(x^2), (x, -3, 3), axes_labels=['x','y'], axes_labels_size=2.5) # Large axes labels (w.r.t. the tick marks) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(sin(x**2), (x, -3, 3), axes_labels=['x','y'], axes_labels_size=2.5) # Large axes labels (w.r.t. the tick marks) sphinx_plot(g)
::
sage: plot(sin(x^2), (x, -3, 3), figsize=[8,2]) Graphics object consisting of 1 graphics primitive sage: plot(sin(x^2), (x, -3, 3)).show(figsize=[8,2]) # These are equivalent
.. PLOT::
g = plot(sin(x**2), (x, -3, 3), figsize=[8,2]) sphinx_plot(g)
This includes options for custom ticks and formatting. See documentation for :meth:`show` for more details.
::
sage: plot(sin(pi*x), (x, -8, 8), ticks=[[-7,-3,0,3,7],[-1/2,0,1/2]]) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(sin(pi*x), (x, -8, 8), ticks=[[-7,-3,0,3,7],[-1/2,0,1/2]]) sphinx_plot(g)
::
sage: plot(2*x+1,(x,0,5),ticks=[[0,1,e,pi,sqrt(20)],2],tick_formatter="latex") Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(2*x+1,(x,0,5),ticks=[[0,1,e,pi,sqrt(20)],2],tick_formatter="latex") sphinx_plot(g)
This is particularly useful when setting custom ticks in multiples of `pi`.
::
sage: plot(sin(x),(x,0,2*pi),ticks=pi/3,tick_formatter=pi) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(sin(x),(x,0,2*pi),ticks=pi/3,tick_formatter=pi) sphinx_plot(g)
You can even have custom tick labels along with custom positioning. ::
sage: plot(x**2, (x,0,3), ticks=[[1,2.5],[0.5,1,2]], tick_formatter=[["$x_1$","$x_2$"],["$y_1$","$y_2$","$y_3$"]]) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot(x**2, (x,0,3), ticks=[[1,2.5],[0.5,1,2]], tick_formatter=[["$x_1$","$x_2$"],["$y_1$","$y_2$","$y_3$"]]) sphinx_plot(g)
You can force Type 1 fonts in your figures by providing the relevant option as shown below. This also requires that LaTeX, dvipng and Ghostscript be installed::
sage: plot(x, typeset='type1') # optional - latex Graphics object consisting of 1 graphics primitive
A example with excluded values::
sage: plot(floor(x), (x, 1, 10), exclude=[1..10]) Graphics object consisting of 11 graphics primitives
.. PLOT::
g = plot(floor(x), (x, 1, 10), exclude=list(range(1,11))) sphinx_plot(g)
We exclude all points where :class:`~sage.functions.prime_pi.PrimePi` makes a jump::
sage: jumps = [n for n in [1..100] if prime_pi(n) != prime_pi(n-1)] sage: plot(lambda x: prime_pi(x), (x, 1, 100), exclude=jumps) Graphics object consisting of 26 graphics primitives
.. PLOT::
#jumps = [n for n in [1..100] if prime_pi(n) != prime_pi(n-1)] #syntaxError: invalid syntax, so we need more code jumps=list() for n in range(1,101): if prime_pi(n) != prime_pi(n-1): jumps.append(n) g = plot(lambda x: prime_pi(x), (x, 1, 100), exclude=jumps) sphinx_plot(g)
Excluded points can also be given by an equation::
sage: g(x) = x^2-2*x-2 sage: plot(1/g(x), (x, -3, 4), exclude=g(x)==0, ymin=-5, ymax=5) # long time Graphics object consisting of 3 graphics primitives
.. PLOT::
def g(x): return x**2-2*x-2 G = plot(1/g(x), (x, -3, 4), exclude=g(x)==0, ymin=-5, ymax=5) sphinx_plot(G)
``exclude`` and ``detect_poles`` can be used together::
sage: f(x) = (floor(x)+0.5) / (1-(x-0.5)^2) sage: plot(f, (x, -3.5, 3.5), detect_poles='show', exclude=[-3..3], ymin=-5, ymax=5) Graphics object consisting of 12 graphics primitives
.. PLOT::
def f(x): return (floor(x)+0.5) / (1-(x-0.5)**2) g = plot(f, (x, -3.5, 3.5), detect_poles='show', exclude=list(range(-3,4)), ymin=-5, ymax=5) sphinx_plot(g)
Regions in which the plot has no values are automatically excluded. The regions thus excluded are in addition to the exclusion points present in the ``exclude`` keyword argument.::
sage: set_verbose(-1) sage: plot(arcsec, (x, -2, 2)) # [-1, 1] is excluded automatically Graphics object consisting of 2 graphics primitives
.. PLOT::
set_verbose(-1) g = plot(arcsec, (x, -2, 2)) # [-1, 1] is excluded automatically sphinx_plot(g)
::
sage: plot(arcsec, (x, -2, 2), exclude=[1.5]) # x=1.5 is also excluded Graphics object consisting of 3 graphics primitives
.. PLOT::
set_verbose(-1) g = plot(arcsec, (x, -2, 2), exclude=[1.5]) # x=1.5 is also excluded sphinx_plot(g)
::
sage: plot(arcsec(x/2), -2, 2) # plot should be empty; no valid points Graphics object consisting of 0 graphics primitives sage: plot(sqrt(x^2-1), -2, 2) # [-1, 1] is excluded automatically Graphics object consisting of 2 graphics primitives
.. PLOT::
set_verbose(-1) g = plot(sqrt(x**2-1), -2, 2) # [-1, 1] is excluded automatically sphinx_plot(g)
::
sage: plot(arccsc, -2, 2) # [-1, 1] is excluded automatically Graphics object consisting of 2 graphics primitives sage: set_verbose(0)
.. PLOT::
set_verbose(-1) g = plot(arccsc, -2, 2) # [-1, 1] is excluded automatically sphinx_plot(g)
TESTS:
We do not randomize the endpoints::
sage: p = plot(x, (x,-1,1)) sage: p[0].xdata[0] == -1 True sage: p[0].xdata[-1] == 1 True
We check to make sure that the x/y min/max data get set correctly when there are multiple functions.
::
sage: d = plot([sin(x), cos(x)], 100, 120).get_minmax_data() sage: d['xmin'] 100.0 sage: d['xmax'] 120.0
We check various combinations of tuples and functions, ending with tests that lambda functions work properly with explicit variable declaration, without a tuple.
::
sage: p = plot(lambda x: x,(x,-1,1)) sage: p = plot(lambda x: x,-1,1) sage: p = plot(x,x,-1,1) sage: p = plot(x,-1,1) sage: p = plot(x^2,x,-1,1) sage: p = plot(x^2,xmin=-1,xmax=2) sage: p = plot(lambda x: x,x,-1,1) sage: p = plot(lambda x: x^2,x,-1,1) sage: p = plot(lambda x: 1/x,x,-1,1) sage: f(x) = sin(x+3)-.1*x^3 sage: p = plot(lambda x: f(x),x,-1,1)
We check to handle cases where the function gets evaluated at a point which causes an 'inf' or '-inf' result to be produced.
::
sage: p = plot(1/x, 0, 1) sage: p = plot(-1/x, 0, 1)
Bad options now give better errors::
sage: P = plot(sin(1/x), (x,-1,3), foo=10) Traceback (most recent call last): ... RuntimeError: Error in line(): option 'foo' not valid. sage: P = plot(x, (x,1,1)) # trac ticket #11753 Traceback (most recent call last): ... ValueError: plot start point and end point must be different
We test that we can plot `f(x)=x` (see :trac:`10246`)::
sage: f(x)=x; f x |--> x sage: plot(f,(x,-1,1)) Graphics object consisting of 1 graphics primitive
Check that :trac:`15030` is fixed::
sage: plot(abs(log(x)), x) Graphics object consisting of 1 graphics primitive
Check that if excluded points are less than xmin then the exclusion still works for polar and parametric plots. The following should show two excluded points::
sage: set_verbose(-1) sage: polar_plot(sin(sqrt(x^2-1)), (x,0,2*pi), exclude=[1/2,2,3]) Graphics object consisting of 3 graphics primitives
sage: parametric_plot((sqrt(x^2-1),sqrt(x^2-1/2)), (x,0,5), exclude=[1,2,3]) Graphics object consisting of 3 graphics primitives
sage: set_verbose(0)
Legends can contain variables with long names, :trac:`13543`::
sage: hello = var('hello') sage: label = '$' + latex(hello) + '$' sage: plot(x, x, 0, 1, legend_label=label) Graphics object consisting of 1 graphics primitive
Extra keywords should be saved if object has a plot method, :trac:`20924`::
sage: G = graphs.PetersenGraph() sage: p = G.plot() sage: p.aspect_ratio() 1.0 sage: pp = plot(G) sage: pp.aspect_ratio() 1.0 """ raise ValueError('only one of color or rgbcolor should be specified')
# If we have extra keywords already set, then update them
# if we are using the generic plotting method else: # if there are no extra args, try to get xmin,xmax from # keyword arguments or pick some silly default
# if there is one extra arg, then it had better be a tuple # if there are two extra args, then pull them out and pass them as a tuple # if there are three extra args, then pull them out and pass them as a tuple elif ('xmin' in kwds) or ('xmax' in kwds): xmin = kwds.pop('xmin', -1) xmax = kwds.pop('xmax', 1) G = _plot(funcs, (xmin, xmax), *args, **kwds) pass else: sage.misc.misc.verbose("there were %s extra arguments (besides %s)" % (n, funcs), level=0)
G.show()
def _plot(funcs, xrange, parametric=False, polar=False, fill=False, label='', randomize=True, **options): """ Internal function which does the actual plotting.
INPUT:
- ``funcs`` - function or list of functions to be plotted - ``xrange`` - two or three tuple of [input variable], min and max - ``parametric`` - (default: False) a boolean for whether this is a parametric plot - ``polar`` - (default: False) a boolean for whether this is a polar plot - ``fill`` - (default: False) an input for whether this plot is filled - ``randomize`` - (default: True) a boolean for whether to use random plot points
The following option is deprecated in favor of ``legend_label``:
- ``label`` - (default: '') a string for the label
All other usual plot options are also accepted, and a number are required (see the example below) which are normally passed through the options decorator to :func:`plot`.
OUTPUT:
- A ``Graphics`` object
EXAMPLES::
See :func:`plot` for many, many implicit examples. Here is an explicit one::
sage: from sage.plot.plot import _plot sage: P = _plot(e^(-x^2),(-3,3),fill=True,color='red',plot_points=50,adaptive_tolerance=2,adaptive_recursion=True,exclude=None) sage: P.show(aspect_ratio='automatic')
TESTS:
Make sure that we get the right number of legend entries as the number of functions varies (:trac:`10514`)::
sage: p1 = plot(1*x, legend_label='1x') sage: p2 = plot(2*x, legend_label='2x', color='green') sage: p1+p2 Graphics object consisting of 2 graphics primitives
::
sage: len(p1.matplotlib().axes[0].legend().texts) 1 sage: len((p1+p2).matplotlib().axes[0].legend().texts) 2 sage: q1 = plot([sin(x), tan(x)], color='blue', legend_label='trig') sage: len(q1.matplotlib().axes[0].legend().texts) 2 sage: q1 Graphics object consisting of 2 graphics primitives sage: q2 = plot([sin(x), tan(x)], legend_label={1:'tan'}) sage: len(q2.matplotlib().axes[0].legend().texts) 2 sage: q2 Graphics object consisting of 2 graphics primitives
::
Make sure that we don't get multiple legend labels for plot segments (:trac:`11998`)::
sage: p1 = plot(1/(x^2-1),(x,-2,2),legend_label="foo",detect_poles=True) sage: len(p1.matplotlib().axes[0].legend().texts) 1 sage: p1.show(ymin=-10,ymax=10) # should be one legend
Parametric plots that get evaluated at invalid points should still plot properly (:trac:`13246`)::
sage: parametric_plot((x, arcsec(x)), (x, -2, 2)) Graphics object consisting of 1 graphics primitive
""" # parametric_plot will be a list or tuple of two functions (f,g) # and will plotted as (f(x), g(x)) for all x in the given range # or we have only a single function to be plotted: else:
# Keep ``rgbcolor`` option 'automatic' only for lists of functions. # Otherwise, let the plot color be 'blue'. options['rgbcolor'] = (0, 0, 1) # default color for a single curve.
# Check to see if funcs is a list of functions that will be all plotted together. # note: sage's "blue" has hue-saturation-lightness values (2/3, 1, 1/2).
# passed more than one color directive? else: else: else: # assume a single color was assigned for all plots
# passed more than one fill directive? else: fill_entry = False else: else: else: fill_entry = False else: else: fill_entry = False else:
# passed more than one fillcolor directive? if i in fillcolor_temp: fillcolor_entry = fillcolor_temp[i] else: fillcolor_entry = golden_rainbow(i,0.85) else: # check that we haven't overwritten automatic multi-colors in color_temp
# passed more than one linestyle directive? if i in linestyle_temp: linestyle_entry = linestyle_temp[i] else: linestyle_entry = default_line_styles[i] else: linestyle_entry = default_line_styles[i] linestyle_entry = default_line_styles[i] else: linestyle_entry = linestyle_temp
# passed more than one legend_label directive? if i < len(legend_label_temp): legend_label_entry = legend_label_temp[i] # assume it is None or str.
# passed more than one legend_color directive? if i in legend_color_temp: legend_color_entry = legend_color_temp[i] if i < len(legend_color_temp): legend_color_entry = legend_color_temp[i] if len(funcs)>1: legend_color_entry = golden_rainbow(i) legend_color_entry = legend_color_temp
rgbcolor=color_entry, linestyle=linestyle_entry, \ legend_label=legend_label_entry, legend_color=legend_color_entry, **options_temp)
if len(exclude.variables()) > 1: raise ValueError('exclude has to be an equation of only one variable') v = exclude.variables()[0] points = [e.right() for e in exclude.solve(v) if e.left() == v and (v not in e.right().variables())] # We are only interested in real solutions for x in points: try: excluded_points.append(float(x)) except TypeError: pass excluded_points.sort()
# We should either have a list in excluded points or exclude # itself must be a list else: raise ValueError('exclude needs to be a list of numbers or an equation')
# We make sure that points plot points close to the excluded points are computed [[x - epsilon, x + epsilon] for x in excluded_points], []) adaptive_tolerance, adaptive_recursion, randomize, initial_points) else: adaptive_tolerance, adaptive_recursion, randomize)
# If the difference between consecutive x-values is more than # 2 times the difference between two consecutive plot points, then # add an exclusion point.
# We need the original x-values to be able to exclude points in parametric plots except (ValueError, TypeError): newdata.append((fdata, 0)) # append a dummy value 0 excluded_points.append(x)
# TODO: Use matplotlib's fill and fill_between commands. else: if fill == max: fstr = 'max' else: fstr = 'min' msg = "WARNING: You use the built-in function %s for filling. You probably wanted the string '%s'." % (fstr, fstr) sage.misc.misc.verbose(msg, level=0) else:
adaptive_recursion, randomize) else: except TypeError: base_level = 0
adaptive_recursion, randomize)
# We need the original data to be able to exclude points in polar plots
# setup for pole detection
# setup for exclusion points
# detect poles and ((y1 > 0 and y0 < 0) or (y1 < 0 and y0 > 0)): # calculate the slope of the line segment # draw a vertical asymptote
# exclude points # all excluded points were considered
# all excluded points were considered
else:
########## misc functions ###################
@options(aspect_ratio=1.0) def parametric_plot(funcs, *args, **kwargs): r""" Plot a parametric curve or surface in 2d or 3d.
:func:`parametric_plot` takes two or three functions as a list or a tuple and makes a plot with the first function giving the `x` coordinates, the second function giving the `y` coordinates, and the third function (if present) giving the `z` coordinates.
In the 2d case, :func:`parametric_plot` is equivalent to the :func:`plot` command with the option ``parametric=True``. In the 3d case, :func:`parametric_plot` is equivalent to :func:`~sage.plot.plot3d.parametric_plot3d.parametric_plot3d`. See each of these functions for more help and examples.
INPUT:
- ``funcs`` - 2 or 3-tuple of functions, or a vector of dimension 2 or 3.
- ``other options`` - passed to :func:`plot` or :func:`~sage.plot.plot3d.parametric_plot3d.parametric_plot3d`
EXAMPLES: We draw some 2d parametric plots. Note that the default aspect ratio is 1, so that circles look like circles. ::
sage: t = var('t') sage: parametric_plot( (cos(t), sin(t)), (t, 0, 2*pi)) Graphics object consisting of 1 graphics primitive
.. PLOT::
t = var('t') g = parametric_plot( (cos(t), sin(t)), (t, 0, 2*pi)) sphinx_plot(g)
::
sage: parametric_plot( (sin(t), sin(2*t)), (t, 0, 2*pi), color=hue(0.6) ) Graphics object consisting of 1 graphics primitive
.. PLOT::
t = var('t') g = parametric_plot( (sin(t), sin(2*t)), (t, 0, 2*pi), color=hue(0.6)) sphinx_plot(g)
::
sage: parametric_plot((1, t), (t, 0, 4)) Graphics object consisting of 1 graphics primitive
.. PLOT::
t =var('t') g = parametric_plot((1, t), (t, 0, 4)) sphinx_plot(g)
Note that in parametric_plot, there is only fill or no fill.
::
sage: parametric_plot((t, t^2), (t, -4, 4), fill=True) Graphics object consisting of 2 graphics primitives
.. PLOT::
t =var('t') g = parametric_plot((t, t**2), (t, -4, 4), fill=True) sphinx_plot(g)
A filled Hypotrochoid::
sage: parametric_plot([cos(x) + 2 * cos(x/4), sin(x) - 2 * sin(x/4)], (x,0, 8*pi), fill=True) Graphics object consisting of 2 graphics primitives
.. PLOT::
g = parametric_plot([cos(x) + 2 * cos(x/4), sin(x) - 2 * sin(x/4)], (x,0, 8*pi), fill=True) sphinx_plot(g)
::
sage: parametric_plot( (5*cos(x), 5*sin(x), x), (x,-12, 12), plot_points=150, color="red") # long time Graphics3d Object
.. PLOT::
#AttributeError: 'Line' object has no attribute 'plot' #g = parametric_plot( (5*cos(x), 5*sin(x), x), (x,-12, 12), plot_points=150, color="red") # long time #sphinx_plot(g)
::
sage: y=var('y') sage: parametric_plot( (5*cos(x), x*y, cos(x*y)), (x, -4,4), (y,-4,4)) # long time` Graphics3d Object
.. PLOT::
#AttributeError: 'sage.plot.plot3d.parametric_surface.ParametricSurf' object has no attribute 'plot' #y = var('y') #g = parametric_plot( (5*cos(x), x*y, cos(x*y)), (x, -4,4), (y,-4,4)) # long time` #sphinx_plot(g)
::
sage: t=var('t') sage: parametric_plot( vector((sin(t), sin(2*t))), (t, 0, 2*pi), color='green') # long time Graphics object consisting of 1 graphics primitive
.. PLOT::
t = var('t') g = parametric_plot( vector((sin(t), sin(2*t))), (t, 0, 2*pi), color='green') # long time sphinx_plot(g)
::
sage: t = var('t') sage: parametric_plot( vector([t, t+1, t^2]), (t, 0, 1)) # long time Graphics3d Object
.. PLOT::
#t = var('t') #g = parametric_plot( vector([t, t+1, t**2]), (t, 0, 1)) # long time #sphinx_plot(g)
Plotting in logarithmic scale is possible with 2D plots. The keyword ``aspect_ratio`` will be ignored if the scale is not ``'loglog'`` or ``'linear'``.::
sage: parametric_plot((x, x**2), (x, 1, 10), scale='loglog') Graphics object consisting of 1 graphics primitive
.. PLOT::
g = parametric_plot((x, x**2), (x, 1, 10), scale='loglog') sphinx_plot(g)
We can also change the scale of the axes in the graphics just before displaying. In this case, the ``aspect_ratio`` must be specified as ``'automatic'`` if the ``scale`` is set to ``'semilogx'`` or ``'semilogy'``. For other values of the ``scale`` parameter, any ``aspect_ratio`` can be used, or the keyword need not be provided.::
sage: p = parametric_plot((x, x**2), (x, 1, 10)) sage: p.show(scale='semilogy', aspect_ratio='automatic')
TESTS::
sage: parametric_plot((x, t^2), (x, -4, 4)) Traceback (most recent call last): ... ValueError: there are more variables than variable ranges
sage: parametric_plot((1, x+t), (x, -4, 4)) Traceback (most recent call last): ... ValueError: there are more variables than variable ranges
sage: parametric_plot((-t, x+t), (x, -4, 4)) Traceback (most recent call last): ... ValueError: there are more variables than variable ranges
sage: parametric_plot((1, x+t, y), (x, -4, 4), (t, -4, 4)) Traceback (most recent call last): ... ValueError: there are more variables than variable ranges
sage: parametric_plot((1, x, y), 0, 4) Traceback (most recent call last): ... ValueError: there are more variables than variable ranges """ else:
# Reset aspect_ratio to 'automatic' in case scale is 'semilog[xy]'. # Otherwise matplotlib complains. scale = scale[0] kwargs['aspect_ratio'] = 'automatic'
elif (num_funcs == 3 and num_ranges <= 2): return sage.plot.plot3d.parametric_plot3d.parametric_plot3d(funcs, *args, **kwargs) else: raise ValueError("the number of functions and the number of variable ranges is not a supported combination for a 2d or 3d parametric plots")
@options(aspect_ratio=1.0) def polar_plot(funcs, *args, **kwds): r""" ``polar_plot`` takes a single function or a list or tuple of functions and plots them with polar coordinates in the given domain.
This function is equivalent to the :func:`plot` command with the options ``polar=True`` and ``aspect_ratio=1``. For more help on options, see the documentation for :func:`plot`.
INPUT:
- ``funcs`` - a function - other options are passed to plot
EXAMPLES:
Here is a blue 8-leaved petal::
sage: polar_plot(sin(5*x)^2, (x, 0, 2*pi), color='blue') Graphics object consisting of 1 graphics primitive
.. PLOT::
g = polar_plot(sin(5*x)**2, (x, 0, 2*pi), color='blue') sphinx_plot(g)
A red figure-8::
sage: polar_plot(abs(sqrt(1 - sin(x)^2)), (x, 0, 2*pi), color='red') Graphics object consisting of 1 graphics primitive
.. PLOT::
g = polar_plot(abs(sqrt(1 - sin(x)**2)), (x, 0, 2*pi), color='red') sphinx_plot(g)
A green limacon of Pascal::
sage: polar_plot(2 + 2*cos(x), (x, 0, 2*pi), color=hue(0.3)) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = polar_plot(2 + 2*cos(x), (x, 0, 2*pi), color=hue(0.3)) sphinx_plot(g)
Several polar plots::
sage: polar_plot([2*sin(x), 2*cos(x)], (x, 0, 2*pi)) Graphics object consisting of 2 graphics primitives
.. PLOT::
g = polar_plot([2*sin(x), 2*cos(x)], (x, 0, 2*pi)) sphinx_plot(g)
A filled spiral::
sage: polar_plot(sqrt, 0, 2 * pi, fill=True) Graphics object consisting of 2 graphics primitives
.. PLOT::
g = polar_plot(sqrt, 0, 2 * pi, fill=True) sphinx_plot(g)
Fill the area between two functions::
sage: polar_plot(cos(4*x) + 1.5, 0, 2*pi, fill=0.5 * cos(4*x) + 2.5, fillcolor='orange') Graphics object consisting of 2 graphics primitives
.. PLOT::
g = polar_plot(cos(4*x) + 1.5, 0, 2*pi, fill=0.5 * cos(4*x) + 2.5, fillcolor='orange') sphinx_plot(g)
Fill the area between several spirals::
sage: polar_plot([(1.2+k*0.2)*log(x) for k in range(6)], 1, 3 * pi, fill={0: [1], 2: [3], 4: [5]}) Graphics object consisting of 9 graphics primitives
.. PLOT::
g = polar_plot([(1.2+k*0.2)*log(x) for k in range(6)], 1, 3 * pi, fill={0: [1], 2: [3], 4: [5]}) sphinx_plot(g)
Exclude points at discontinuities::
sage: polar_plot(log(floor(x)), (x, 1, 4*pi), exclude=[1..12]) Graphics object consisting of 12 graphics primitives
.. PLOT::
g = polar_plot(log(floor(x)), (x, 1, 4*pi), exclude=list(range(1,13))) sphinx_plot(g)
"""
@options(aspect_ratio='automatic') def list_plot(data, plotjoined=False, **kwargs): r""" ``list_plot`` takes either a list of numbers, a list of tuples, a numpy array, or a dictionary and plots the corresponding points.
If given a list of numbers (that is, not a list of tuples or lists), ``list_plot`` forms a list of tuples ``(i, x_i)`` where ``i`` goes from 0 to ``len(data)-1`` and ``x_i`` is the ``i``-th data value, and puts points at those tuple values.
``list_plot`` will plot a list of complex numbers in the obvious way; any numbers for which :func:`CC()<sage.rings.complex_field.ComplexField>` makes sense will work.
``list_plot`` also takes a list of tuples ``(x_i, y_i)`` where ``x_i`` and ``y_i`` are the ``i``-th values representing the ``x``- and ``y``-values, respectively.
If given a dictionary, ``list_plot`` interprets the keys as `x`-values and the values as `y`-values.
The ``plotjoined=True`` option tells ``list_plot`` to plot a line joining all the data.
For other keyword options that the ``list_plot`` function can take, refer to :func:`~sage.plot.plot.plot`.
It is possible to pass empty dictionaries, lists, or tuples to ``list_plot``. Doing so will plot nothing (returning an empty plot).
EXAMPLES::
sage: list_plot([i^2 for i in range(5)]) # long time Graphics object consisting of 1 graphics primitive
.. PLOT::
g = list_plot([i**2 for i in range(5)]) # long time sphinx_plot(g)
Here are a bunch of random red points::
sage: r = [(random(),random()) for _ in range(20)] sage: list_plot(r, color='red') Graphics object consisting of 1 graphics primitive
.. PLOT::
r = [(random(),random()) for _ in range(20)] g = list_plot(r, color='red') sphinx_plot(g)
This gives all the random points joined in a purple line::
sage: list_plot(r, plotjoined=True, color='purple') Graphics object consisting of 1 graphics primitive
.. PLOT::
r = [(random(),random()) for _ in range(20)] g = list_plot(r, plotjoined=True, color='purple') sphinx_plot(g)
You can provide a numpy array.::
sage: import numpy sage: list_plot(numpy.arange(10)) Graphics object consisting of 1 graphics primitive
.. PLOT::
import numpy g = list_plot(numpy.arange(10)) sphinx_plot(g)
::
sage: list_plot(numpy.array([[1,2], [2,3], [3,4]])) Graphics object consisting of 1 graphics primitive
.. PLOT::
import numpy g = list_plot(numpy.array([[1,2], [2,3], [3,4]])) sphinx_plot(g)
Plot a list of complex numbers::
sage: list_plot([1, I, pi + I/2, CC(.25, .25)]) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = list_plot([1, I, pi + I/2, CC(.25, .25)]) sphinx_plot(g)
::
sage: list_plot([exp(I*theta) for theta in [0, .2..pi]]) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = list_plot([exp(I*theta) for theta in srange(0,pi,0.2)]) sphinx_plot(g)
Note that if your list of complex numbers are all actually real, they get plotted as real values, so this
::
sage: list_plot([CDF(1), CDF(1/2), CDF(1/3)]) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = list_plot([CDF(1), CDF(1/2), CDF(1/3)]) sphinx_plot(g)
is the same as ``list_plot([1, 1/2, 1/3])`` -- it produces a plot of the points `(0,1)`, `(1,1/2)`, and `(2,1/3)`.
If you have separate lists of `x` values and `y` values which you want to plot against each other, use the ``zip`` command to make a single list whose entries are pairs of `(x,y)` values, and feed the result into ``list_plot``::
sage: x_coords = [cos(t)^3 for t in srange(0, 2*pi, 0.02)] sage: y_coords = [sin(t)^3 for t in srange(0, 2*pi, 0.02)] sage: list_plot(list(zip(x_coords, y_coords))) Graphics object consisting of 1 graphics primitive
.. PLOT::
x_coords = [cos(t)**3 for t in srange(0, 2*pi, 0.02)] y_coords = [sin(t)**3 for t in srange(0, 2*pi, 0.02)] g = list_plot(list(zip(x_coords, y_coords))) sphinx_plot(g)
If instead you try to pass the two lists as separate arguments, you will get an error message::
sage: list_plot(x_coords, y_coords) Traceback (most recent call last): ... TypeError: The second argument 'plotjoined' should be boolean (True or False). If you meant to plot two lists 'x' and 'y' against each other, use 'list_plot(list(zip(x,y)))'.
Dictionaries with numeric keys and values can be plotted::
sage: list_plot({22: 3365, 27: 3295, 37: 3135, 42: 3020, 47: 2880, 52: 2735, 57: 2550}) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = list_plot({22: 3365, 27: 3295, 37: 3135, 42: 3020, 47: 2880, 52: 2735, 57: 2550}) sphinx_plot(g)
Plotting in logarithmic scale is possible for 2D list plots. There are two different syntaxes available::
sage: yl = [2**k for k in range(20)] sage: list_plot(yl, scale='semilogy') # long time # log axis on vertical Graphics object consisting of 1 graphics primitive
.. PLOT::
yl = [2**k for k in range(20)] g = list_plot(yl, scale='semilogy') # long time # log axis on vertical sphinx_plot(g)
::
sage: list_plot_semilogy(yl) # same Graphics object consisting of 1 graphics primitive
.. warning::
If ``plotjoined`` is ``False`` then the axis that is in log scale must have all points strictly positive. For instance, the following plot will show no points in the figure since the points in the horizontal axis starts from `(0,1)`. Further, matplotlib will display a user warning.
::
sage: list_plot(yl, scale='loglog') # both axes are log doctest:warning ... Graphics object consisting of 1 graphics primitive
Instead this will work. We drop the point `(0,1)`.::
sage: list_plot(list(zip(range(1,len(yl)), yl[1:])), scale='loglog') # long time Graphics object consisting of 1 graphics primitive
We use :func:`list_plot_loglog` and plot in a different base.::
sage: list_plot_loglog(list(zip(range(1,len(yl)), yl[1:])), base=2) # long time Graphics object consisting of 1 graphics primitive
.. PLOT::
yl = [2**k for k in range(20)] g = list_plot_loglog(list(zip(range(1,len(yl)), yl[1:])), base=2) # long time sphinx_plot(g)
We can also change the scale of the axes in the graphics just before displaying::
sage: G = list_plot(yl) # long time sage: G.show(scale=('semilogy', 2)) # long time
TESTS:
We check to see whether elements of the Symbolic Ring are properly handled; see :trac:`16378` ::
sage: list_plot([1+I, 2+I]) Graphics object consisting of 1 graphics primitive
sage: list_plot([1+I, 2, CC(3+I)]) Graphics object consisting of 1 graphics primitive
sage: list_plot([2, SR(1), CC(1+i)]) Graphics object consisting of 1 graphics primitive
We check to see that the x/y min/max data are set correctly::
sage: d = list_plot([(100,100), (120, 120)]).get_minmax_data() sage: d['xmin'] 100.0 sage: d['ymin'] 100.0 """ return Graphics() "(True or False). If you meant to plot two lists 'x' " "and 'y' against each other, use 'list_plot(list(zip(x,y)))'.") list_data = sorted(list(iteritems(data))) else: # or tuple or numpy array, or an element of CC, CDF # We also want to avoid doing CC(data[0]) here since it will go # through if data[0] is really a tuple and every element of the # data will be converted to a complex and later converted back to # a tuple. # So, the only other check we need to do is whether data[0] is an # element of the Symbolic Ring.
else: # Assume we have complex-valued input and plot real and imaginary parts. # Need to catch IndexError because if data is, say, [(0, 1), (1, I)], # point3d() throws an IndexError on the (0,1) before it ever # gets to (1, I). # if we get here, we already did "list(enumerate(data))", # so look at z[1] in inner list return line(data, **kwargs) else:
#------------------------ Graphs on log scale ---------------------------# @options(base=10) def plot_loglog(funcs, *args, **kwds): """ Plot graphics in 'loglog' scale, that is, both the horizontal and the vertical axes will be in logarithmic scale.
INPUT:
- ``base`` -- (default: 10) the base of the logarithm. This must be greater than 1. The base can be also given as a list or tuple ``(basex, basey)``. ``basex`` sets the base of the logarithm along the horizontal axis and ``basey`` sets the base along the vertical axis.
- ``funcs`` -- any Sage object which is acceptable to the :func:`plot`.
For all other inputs, look at the documentation of :func:`plot`.
EXAMPLES::
sage: plot_loglog(exp, (1,10)) # plot in loglog scale with base 10 Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot_loglog(exp, (1,10)) # plot in loglog scale with base 10 sphinx_plot(g)
::
sage: plot_loglog(exp, (1,10), base=2.1) # long time # with base 2.1 on both axes Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot_loglog(exp, (1,10), base=2.1) # long time # with base 2.1 on both axes sphinx_plot(g)
::
sage: plot_loglog(exp, (1,10), base=(2,3)) Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot_loglog(exp, (1,10), base=(2,3)) sphinx_plot(g)
"""
@options(base=10) def plot_semilogx(funcs, *args, **kwds): """ Plot graphics in 'semilogx' scale, that is, the horizontal axis will be in logarithmic scale.
INPUT:
- ``base`` -- (default: 10) the base of the logarithm. This must be greater than 1.
- ``funcs`` -- any Sage object which is acceptable to the :func:`plot`.
For all other inputs, look at the documentation of :func:`plot`.
EXAMPLES::
sage: plot_semilogx(exp, (1,10)) # long time # plot in semilogx scale, base 10 Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot_semilogx(exp, (1,10)) # long time # plot in semilogx scale, base 10 sphinx_plot(g)
::
sage: plot_semilogx(exp, (1,10), base=2) # with base 2 Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot_semilogx(exp, (1,10), base=2) # with base 2 sphinx_plot(g)
"""
@options(base=10) def plot_semilogy(funcs, *args, **kwds): """ Plot graphics in 'semilogy' scale, that is, the vertical axis will be in logarithmic scale.
INPUT:
- ``base`` -- (default: 10) the base of the logarithm. This must be greater than 1.
- ``funcs`` -- any Sage object which is acceptable to the :func:`plot`.
For all other inputs, look at the documentation of :func:`plot`.
EXAMPLES::
sage: plot_semilogy(exp, (1,10)) # long time # plot in semilogy scale, base 10 Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot_semilogy(exp, (1,10)) # long time # plot in semilogy scale, base 10 sphinx_plot(g)
::
sage: plot_semilogy(exp, (1,10), base=2) # long time # with base 2 Graphics object consisting of 1 graphics primitive
.. PLOT::
g = plot_semilogy(exp, (1,10), base=2) # long time # with base 2 sphinx_plot(g)
"""
@options(base=10) def list_plot_loglog(data, plotjoined=False, **kwds): """ Plot the ``data`` in 'loglog' scale, that is, both the horizontal and the vertical axes will be in logarithmic scale.
INPUT:
- ``base`` -- (default: 10) the base of the logarithm. This must be greater than 1. The base can be also given as a list or tuple ``(basex, basey)``. ``basex`` sets the base of the logarithm along the horizontal axis and ``basey`` sets the base along the vertical axis.
For all other inputs, look at the documentation of :func:`list_plot`.
EXAMPLES::
sage: yl = [5**k for k in range(10)]; xl = [2**k for k in range(10)] sage: list_plot_loglog(list(zip(xl, yl))) # long time # plot in loglog scale with base 10 Graphics object consisting of 1 graphics primitive
.. PLOT::
yl = [5**k for k in range(10)] xl = [2**k for k in range(10)] g = list_plot_loglog(list(zip(xl, yl))) # long time # plot in loglog scale with base 10 sphinx_plot(g)
::
sage: list_plot_loglog(list(zip(xl, yl)), base=2.1) # long time # with base 2.1 on both axes Graphics object consisting of 1 graphics primitive
.. PLOT::
yl = [5**k for k in range(10)] xl = [2**k for k in range(10)] g = list_plot_loglog(list(zip(xl, yl)), base=2.1) # long time # with base 2.1 on both axes sphinx_plot(g)
::
sage: list_plot_loglog(list(zip(xl, yl)), base=(2,5)) # long time Graphics object consisting of 1 graphics primitive
.. warning::
If ``plotjoined`` is ``False`` then the axis that is in log scale must have all points strictly positive. For instance, the following plot will show no points in the figure since the points in the horizontal axis starts from `(0,1)`.
::
sage: yl = [2**k for k in range(20)] sage: list_plot_loglog(yl) Graphics object consisting of 1 graphics primitive
Instead this will work. We drop the point `(0,1)`.::
sage: list_plot_loglog(list(zip(range(1,len(yl)), yl[1:]))) Graphics object consisting of 1 graphics primitive
"""
@options(base=10) def list_plot_semilogx(data, plotjoined=False, **kwds): """ Plot ``data`` in 'semilogx' scale, that is, the horizontal axis will be in logarithmic scale.
INPUT:
- ``base`` -- (default: 10) the base of the logarithm. This must be greater than 1.
For all other inputs, look at the documentation of :func:`list_plot`.
EXAMPLES::
sage: yl = [2**k for k in range(12)] sage: list_plot_semilogx(list(zip(yl,yl))) Graphics object consisting of 1 graphics primitive
.. PLOT::
yl = [2**k for k in range(12)] g = list_plot_semilogx(list(zip(yl,yl))) sphinx_plot(g)
.. warning::
If ``plotjoined`` is ``False`` then the horizontal axis must have all points strictly positive. Otherwise the plot will come up empty. For instance the following plot contains a point at `(0,1)`.
::
sage: yl = [2**k for k in range(12)] sage: list_plot_semilogx(yl) # plot empty due to (0,1) Graphics object consisting of 1 graphics primitive
We remove `(0,1)` to fix this.::
sage: list_plot_semilogx(list(zip(range(1, len(yl)), yl[1:]))) Graphics object consisting of 1 graphics primitive
::
sage: list_plot_semilogx([(1,2),(3,4),(3,-1),(25,3)], base=2) # with base 2 Graphics object consisting of 1 graphics primitive
.. PLOT::
g = list_plot_semilogx([(1,2),(3,4),(3,-1),(25,3)], base=2) # with base 2 sphinx_plot(g)
"""
@options(base=10) def list_plot_semilogy(data, plotjoined=False, **kwds): """ Plot ``data`` in 'semilogy' scale, that is, the vertical axis will be in logarithmic scale.
INPUT:
- ``base`` -- (default: 10) the base of the logarithm. This must be greater than 1.
For all other inputs, look at the documentation of :func:`list_plot`.
EXAMPLES::
sage: yl = [2**k for k in range(12)] sage: list_plot_semilogy(yl) # plot in semilogy scale, base 10 Graphics object consisting of 1 graphics primitive
.. PLOT::
yl = [2**k for k in range(12)] g = list_plot_semilogy(yl) # plot in semilogy scale, base 10 sphinx_plot(g)
.. warning::
If ``plotjoined`` is ``False`` then the vertical axis must have all points strictly positive. Otherwise the plot will come up empty. For instance the following plot contains a point at `(1,0)`. Further, matplotlib will display a user warning.
::
sage: xl = [2**k for k in range(12)]; yl = range(len(xl)) sage: list_plot_semilogy(list(zip(xl,yl))) # plot empty due to (1,0) doctest:warning ... Graphics object consisting of 1 graphics primitive
We remove `(1,0)` to fix this.::
sage: list_plot_semilogy(list(zip(xl[1:],yl[1:]))) Graphics object consisting of 1 graphics primitive
::
sage: list_plot_semilogy([2, 4, 6, 8, 16, 31], base=2) # with base 2 Graphics object consisting of 1 graphics primitive
.. PLOT::
g = list_plot_semilogy([2, 4, 6, 8, 16, 31], base=2) # with base 2 sphinx_plot(g)
"""
def to_float_list(v): """ Given a list or tuple or iterable v, coerce each element of v to a float and make a list out of the result.
EXAMPLES::
sage: from sage.plot.plot import to_float_list sage: to_float_list([1,1/2,3]) [1.0, 0.5, 3.0] """
def reshape(v, n, m): """ Helper function for creating graphics arrays.
The input array is flattened and turned into an `n\times m` array, with blank graphics object padded at the end, if necessary.
INPUT:
- ``v`` - a list of lists or tuples
- ``n, m`` - integers
OUTPUT:
A list of lists of graphics objects
EXAMPLES::
sage: L = [plot(sin(k*x),(x,-pi,pi)) for k in range(10)] sage: graphics_array(L,3,4) # long time (up to 4s on sage.math, 2012) Graphics Array of size 3 x 4
::
sage: M = [[plot(sin(k*x),(x,-pi,pi)) for k in range(3)],[plot(cos(j*x),(x,-pi,pi)) for j in [3..5]]] sage: graphics_array(M,6,1) # long time (up to 4s on sage.math, 2012) Graphics Array of size 6 x 1
TESTS::
sage: L = [plot(sin(k*x),(x,-pi,pi)) for k in [1..3]] sage: graphics_array(L,0,-1) # indirect doctest Traceback (most recent call last): ... ValueError: array sizes must be positive """ return [[G]*m]*n
# a list of lists -- flatten it v = sum([list(x) for x in v], [])
# Now v should be a single list. # First, make it have the right length.
# Next, create a list of lists out of it.
def graphics_array(array, nrows=None, ncols=None): r""" ``graphics_array`` take a list of lists (or tuples) of graphics objects and plots them all on one canvas (single plot).
INPUT:
- ``array`` -- a list of lists or tuples. The graphics objects to combine into a graphics array.
- ``nrows, ncols`` -- (optional) integers. If both are given then the input array is flattened and turned into an ``nrows`` x ``ncols`` array, with blank graphics objects padded at the end, if necessary. If only one is specified, the other is chosen automatically.
EXAMPLES: Make some plots of `\sin` functions::
sage: f(x) = sin(x) sage: g(x) = sin(2*x) sage: h(x) = sin(4*x) sage: p1 = plot(f,(-2*pi,2*pi),color=hue(0.5)) # long time sage: p2 = plot(g,(-2*pi,2*pi),color=hue(0.9)) # long time sage: p3 = parametric_plot((f,g),(0,2*pi),color=hue(0.6)) # long time sage: p4 = parametric_plot((f,h),(0,2*pi),color=hue(1.0)) # long time
Now make a graphics array out of the plots::
sage: graphics_array(((p1,p2),(p3,p4))) # long time Graphics Array of size 2 x 2
.. PLOT::
def f(x): return sin(x) def g(x): return sin(2*x) def h(x): return sin(4*x) p1 = plot(f,(-2*pi,2*pi),color=hue(0.5)) # long time p2 = plot(g,(-2*pi,2*pi),color=hue(0.9)) # long time p3 = parametric_plot((f,g),(0,2*pi),color=hue(0.6)) # long time p4 = parametric_plot((f,h),(0,2*pi),color=hue(1.0)) # long time g = graphics_array(((p1,p2),(p3,p4))) # long time sphinx_plot(g)
One can also name the array, and then use :meth:`~sage.plot.graphics.GraphicsArray.show` or :meth:`~sage.plot.graphics.GraphicsArray.save`::
sage: ga = graphics_array(((p1,p2),(p3,p4))) # long time sage: ga.show() # long time
Here we give only one row::
sage: p1 = plot(sin,(-4,4)) sage: p2 = plot(cos,(-4,4)) sage: g = graphics_array([p1, p2]); print(g) Graphics Array of size 1 x 2 sage: g.show()
.. PLOT::
p1 = plot(sin,(-4,4)) p2 = plot(cos,(-4,4)) g = graphics_array([p1, p2]) sphinx_plot(g)
It is possible to use ``figsize`` to change the size of the plot as a whole::
sage: L = [plot(sin(k*x),(x,-pi,pi)) for k in [1..3]] sage: G = graphics_array(L) sage: G.show(figsize=[5,3]) # smallish and compact sage: G.show(figsize=[10,20]) # bigger and tall and thin; long time (2s on sage.math, 2012) sage: G.show(figsize=8) # figure as a whole is a square
Specifying only the number of rows or the number of columns computes the other dimension automatically::
sage: ga = graphics_array([plot(sin)] * 10, nrows=3) sage: ga.nrows(), ga.ncols() (3, 4) sage: ga = graphics_array([plot(sin)] * 10, ncols=3) sage: ga.nrows(), ga.ncols() (4, 3) """ # TODO: refactor the whole array flattening and reshaping into a class else: # nrows is None xor ncols is None else: length = sum(map(len, array)) else: assert False
def var_and_list_of_values(v, plot_points): """ INPUT:
- ``v`` - (v0, v1) or (var, v0, v1); if the former return the range of values between v0 and v1 taking plot_points steps; if var is given, also return var.
- ``plot_points`` - integer = 2 (the endpoints)
OUTPUT:
- ``var`` - a variable or None
- ``list`` - a list of floats
EXAMPLES::
sage: from sage.plot.plot import var_and_list_of_values sage: var_and_list_of_values((var('theta'), 2, 5), 5) doctest:...: DeprecationWarning: var_and_list_of_values is deprecated. Please use sage.plot.misc.setup_for_eval_on_grid; note that that function has slightly different calling and return conventions which make it more generally applicable See http://trac.sagemath.org/7008 for details. (theta, [2.0, 2.75, 3.5, 4.25, 5.0]) sage: var_and_list_of_values((2, 5), 5) (None, [2.0, 2.75, 3.5, 4.25, 5.0]) sage: var_and_list_of_values((var('theta'), 2, 5), 2) (theta, [2.0, 5.0]) sage: var_and_list_of_values((2, 5), 2) (None, [2.0, 5.0]) """ raise ValueError("plot_points must be greater than 1") raise TypeError("v must be a tuple or list") else: raise ValueError("parametric value range must be a list or tuple of length 2 or 3.")
else:
def setup_for_eval_on_grid(v, xrange, yrange, plot_points): """ This function is deprecated. Please use ``sage.plot.misc.setup_for_eval_on_grid`` instead. Please note that that function has slightly different calling and return conventions which make it more generally applicable.
INPUT:
- ``v`` - a list of functions
- ``xrange`` - 2 or 3 tuple (if 3, first is a variable)
- ``yrange`` - 2 or 3 tuple
- ``plot_points`` - a positive integer
OUTPUT:
- ``g`` - tuple of fast callable functions
- ``xstep`` - step size in xdirection
- ``ystep`` - step size in ydirection
- ``xrange`` - tuple of 2 floats
- ``yrange`` - tuple of 2 floats
EXAMPLES::
sage: x,y = var('x,y') sage: sage.plot.plot.setup_for_eval_on_grid([x^2 + y^2], (x,0,5), (y,0,pi), 11) doctest:...: DeprecationWarning: sage.plot.plot.setup_for_eval_on_grid is deprecated. Please use sage.plot.misc.setup_for_eval_on_grid; note that that function has slightly different calling and return conventions which make it more generally applicable See http://trac.sagemath.org/7008 for details. ([<sage.ext... object at ...>], 0.5, 0.3141592653589793, (0.0, 5.0), (0.0, 3.141592653589793))
We always plot at least two points; one at the beginning and one at the end of the ranges.
::
sage: sage.plot.plot.setup_for_eval_on_grid([x^2+y^2], (x,0,1), (y,-1,1), 1) ([<sage.ext... object at ...>], 1.0, 2.0, (0.0, 1.0), (-1.0, 1.0))
"""
def minmax_data(xdata, ydata, dict=False): """ Returns the minimums and maximums of xdata and ydata.
If dict is False, then minmax_data returns the tuple (xmin, xmax, ymin, ymax); otherwise, it returns a dictionary whose keys are 'xmin', 'xmax', 'ymin', and 'ymax' and whose values are the corresponding values.
EXAMPLES::
sage: from sage.plot.plot import minmax_data sage: minmax_data([], []) (-1, 1, -1, 1) sage: minmax_data([-1, 2], [4, -3]) (-1, 2, -3, 4) sage: d = minmax_data([-1, 2], [4, -3], dict=True) sage: list(sorted(d.items())) [('xmax', 2), ('xmin', -1), ('ymax', 4), ('ymin', -3)] """ 'ymin':ymin, 'ymax':ymax} else:
def adaptive_refinement(f, p1, p2, adaptive_tolerance=0.01, adaptive_recursion=5, level=0): r""" The adaptive refinement algorithm for plotting a function ``f``. See the docstring for plot for a description of the algorithm.
INPUT:
- ``f`` - a function of one variable
- ``p1, p2`` - two points to refine between
- ``adaptive_recursion`` - (default: 5) how many levels of recursion to go before giving up when doing adaptive refinement. Setting this to 0 disables adaptive refinement.
- ``adaptive_tolerance`` - (default: 0.01) how large a relative difference should be before the adaptive refinement code considers it significant; see documentation for generate_plot_points for more information. See the documentation for :func:`plot` for more information on how the adaptive refinement algorithm works.
OUTPUT:
- ``list`` - a list of points to insert between ``p1`` and ``p2`` to get a better linear approximation between them
TESTS::
sage: from sage.plot.plot import adaptive_refinement sage: adaptive_refinement(sin, (0,0), (pi,0), adaptive_tolerance=0.01, adaptive_recursion=0) [] sage: adaptive_refinement(sin, (0,0), (pi,0), adaptive_tolerance=0.01) [(0.125*pi, 0.3826834323650898), (0.1875*pi, 0.5555702330196022), (0.25*pi, 0.7071067811865475), (0.3125*pi, 0.8314696123025452), (0.375*pi, 0.9238795325112867), (0.4375*pi, 0.9807852804032304), (0.5*pi, 1.0), (0.5625*pi, 0.9807852804032304), (0.625*pi, 0.9238795325112867), (0.6875*pi, 0.8314696123025455), (0.75*pi, 0.7071067811865476), (0.8125*pi, 0.5555702330196022), (0.875*pi, 0.3826834323650899)]
This shows that lowering ``adaptive_tolerance`` and raising ``adaptive_recursion`` both increase the number of subdivision points, though which one creates more points is heavily dependent upon the function being plotted.
::
sage: x = var('x') sage: f(x) = sin(1/x) sage: n1 = len(adaptive_refinement(f, (0,0), (pi,0), adaptive_tolerance=0.01)); n1 15 sage: n2 = len(adaptive_refinement(f, (0,0), (pi,0), adaptive_recursion=10, adaptive_tolerance=0.01)); n2 79 sage: n3 = len(adaptive_refinement(f, (0,0), (pi,0), adaptive_tolerance=0.001)); n3 26 """
# give up for this branch
# give up for this branch
# this distance calculation is not perfect. adaptive_tolerance=adaptive_tolerance, adaptive_recursion=adaptive_recursion, level=level+1) \ + [(x, y)] + \ adaptive_refinement(f, (x, y), p2, adaptive_tolerance=adaptive_tolerance, adaptive_recursion=adaptive_recursion, level=level+1) else:
def generate_plot_points(f, xrange, plot_points=5, adaptive_tolerance=0.01, adaptive_recursion=5, randomize=True, initial_points=None): r""" Calculate plot points for a function f in the interval xrange. The adaptive refinement algorithm is also automatically invoked with a *relative* adaptive tolerance of adaptive_tolerance; see below.
INPUT:
- ``f`` - a function of one variable
- ``p1, p2`` - two points to refine between
- ``plot_points`` - (default: 5) the minimal number of plot points. (Note however that in any actual plot a number is passed to this, with default value 200.)
- ``adaptive_recursion`` - (default: 5) how many levels of recursion to go before giving up when doing adaptive refinement. Setting this to 0 disables adaptive refinement.
- ``adaptive_tolerance`` - (default: 0.01) how large the relative difference should be before the adaptive refinement code considers it significant. If the actual difference is greater than adaptive_tolerance*delta, where delta is the initial subinterval size for the given xrange and plot_points, then the algorithm will consider it significant.
- ``initial_points`` - (default: None) a list of points that should be evaluated.
OUTPUT:
- a list of points (x, f(x)) in the interval xrange, which approximate the function f.
TESTS::
sage: from sage.plot.plot import generate_plot_points sage: generate_plot_points(sin, (0, pi), plot_points=2, adaptive_recursion=0) [(0.0, 0.0), (3.141592653589793, 1.2246...e-16)]
sage: from sage.plot.plot import generate_plot_points sage: generate_plot_points(lambda x: x^2, (0, 6), plot_points=2, adaptive_recursion=0, initial_points=[1,2,3]) [(0.0, 0.0), (1.0, 1.0), (2.0, 4.0), (3.0, 9.0), (6.0, 36.0)]
sage: generate_plot_points(sin(x).function(x), (-pi, pi), randomize=False) [(-3.141592653589793, -1.2246...e-16), (-2.748893571891069, -0.3826834323650899), (-2.356194490192345, -0.707106781186547...), (-2.1598449493429825, -0.831469612302545...), (-1.9634954084936207, -0.9238795325112867), (-1.7671458676442586, -0.9807852804032304), (-1.5707963267948966, -1.0), (-1.3744467859455345, -0.9807852804032304), (-1.1780972450961724, -0.9238795325112867), (-0.9817477042468103, -0.831469612302545...), (-0.7853981633974483, -0.707106781186547...), (-0.39269908169872414, -0.3826834323650898), (0.0, 0.0), (0.39269908169872414, 0.3826834323650898), (0.7853981633974483, 0.707106781186547...), (0.9817477042468103, 0.831469612302545...), (1.1780972450961724, 0.9238795325112867), (1.3744467859455345, 0.9807852804032304), (1.5707963267948966, 1.0), (1.7671458676442586, 0.9807852804032304), (1.9634954084936207, 0.9238795325112867), (2.1598449493429825, 0.831469612302545...), (2.356194490192345, 0.707106781186547...), (2.748893571891069, 0.3826834323650899), (3.141592653589793, 1.2246...e-16)]
This shows that lowering adaptive_tolerance and raising adaptive_recursion both increase the number of subdivision points. (Note that which creates more points is heavily dependent on the particular function plotted.)
::
sage: x = var('x') sage: f(x) = sin(1/x) sage: [len(generate_plot_points(f, (-pi, pi), plot_points=16, adaptive_tolerance=i, randomize=False)) for i in [0.01, 0.001, 0.0001]] [97, 161, 275]
sage: [len(generate_plot_points(f, (-pi, pi), plot_points=16, adaptive_recursion=i, randomize=False)) for i in [5, 10, 15]] [97, 499, 2681] """
# Slightly randomize the interior sample points if # randomize is true
# add initial points
# nan != nan continue else:
# nan != nan continue except (ArithmeticError, TypeError, ValueError) as msg: pass else: exceptions += 1 exception_indices.append(i) else:
# calls adaptive refinement
adaptive_tolerance=adaptive_tolerance, adaptive_recursion=adaptive_recursion):
# Old imports required for unpickling old pickles from .line import line, line2d, Line as GraphicPrimitive_Line from .arrow import arrow, Arrow as GraphicPrimitive_Arrow from .bar_chart import bar_chart, BarChart as GraphicPrimitive_BarChart from .disk import disk, Disk as GraphicPrimitive_Disk from .point import point, points, point2d, Point as GraphicPrimitive_Point from .matrix_plot import matrix_plot, MatrixPlot as GraphicPrimitive_MatrixPlot from .plot_field import plot_vector_field, plot_slope_field, PlotField as GraphicPrimitive_PlotField from .text import text, Text as GraphicPrimitive_Text from .polygon import polygon, Polygon as GraphicPrimitive_Polygon from .circle import circle, Circle as GraphicPrimtive_Circle from .contour_plot import contour_plot, implicit_plot, ContourPlot as GraphicPrimitive_ContourPlot |