Coverage for local/lib/python2.7/site-packages/sage/plot/contour_plot.py : 84%

""" 

Contour Plots 

""" 

#***************************************************************************** 

# Copyright (C) 2006 Alex Clemesha <clemesha@gmail.com>, 

# William Stein <wstein@gmail.com>, 

# 2008 Mike Hansen <mhansen@gmail.com>, 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.plot.primitive import GraphicPrimitive 

from sage.misc.decorators import options, suboptions 

from sage.plot.colors import rgbcolor, get_cmap 

from sage.arith.srange import xsrange 

import operator 

class ContourPlot(GraphicPrimitive): 

""" 

Primitive class for the contour plot graphics type. See 

``contour_plot?`` for help actually doing contour plots. 

INPUT: 

- ``xy_data_array`` - list of lists giving evaluated values of the function 

on the grid 

- ``xrange`` - tuple of 2 floats indicating range for horizontal direction 

- ``yrange`` - tuple of 2 floats indicating range for vertical direction 

- ``options`` - dict of valid plot options to pass to constructor 

EXAMPLES: 

Note this should normally be used indirectly via ``contour_plot``:: 

sage: from sage.plot.contour_plot import ContourPlot 

sage: C = ContourPlot([[1,3],[2,4]], (1,2), (2,3), options={}) 

sage: C 

ContourPlot defined by a 2 x 2 data grid 

sage: C.xrange 

(1, 2) 

TESTS: 

We test creating a contour plot:: 

sage: x,y = var('x,y') 

sage: contour_plot(x^2-y^3+10*sin(x*y), (x,-4,4), (y,-4,4), 

....: plot_points=121, cmap='hsv') 

Graphics object consisting of 1 graphics primitive 

""" 

def __init__(self, xy_data_array, xrange, yrange, options): 

""" 

Initializes base class ContourPlot. 

EXAMPLES:: 

sage: x,y = var('x,y') 

sage: C = contour_plot(x^2-y^3+10*sin(x*y), (x,-4,4), (y,-4,4), 

....: plot_points=121, cmap='hsv') 

sage: C[0].xrange 

(-4.0, 4.0) 

sage: C[0].options()['plot_points'] 

121 

""" 

self.xrange = xrange 

self.yrange = yrange 

self.xy_data_array = xy_data_array 

self.xy_array_row = len(xy_data_array) 

self.xy_array_col = len(xy_data_array[0]) 

GraphicPrimitive.__init__(self, options) 

def get_minmax_data(self): 

""" 

Returns a dictionary with the bounding box data. 

EXAMPLES:: 

sage: x,y = var('x,y') 

sage: f(x,y) = x^2 + y^2 

sage: d = contour_plot(f, (3,6), (3,6))[0].get_minmax_data() 

sage: d['xmin'] 

3.0 

sage: d['ymin'] 

3.0 

""" 

from sage.plot.plot import minmax_data 

return minmax_data(self.xrange, self.yrange, dict=True) 

def _allowed_options(self): 

""" 

Return the allowed options for the ContourPlot class. 

EXAMPLES:: 

sage: x,y = var('x,y') 

sage: C = contour_plot(x^2 - y^2, (x,-2,2), (y,-2,2)) 

sage: isinstance(C[0]._allowed_options(), dict) 

True 

""" 

return {'plot_points': 'How many points to use for plotting precision', 

'cmap': """the name of a predefined colormap, 

a list of colors, or an instance of a 

matplotlib Colormap. Type: import matplotlib.cm; 

matplotlib.cm.datad.keys() 

for available colormap names.""", 

'colorbar': "Include a colorbar indicating the levels", 

'colorbar_options': "a dictionary of options for colorbars", 

'fill': 'Fill contours or not', 

'legend_label': 'The label for this item in the legend.', 

'contours': """Either an integer specifying the number of 

contour levels, or a sequence of numbers giving 

the actual contours to use.""", 

'linewidths': 'the width of the lines to be plotted', 

'linestyles': 'the style of the lines to be plotted', 

'labels': 'show line labels or not', 

'label_options': 'a dictionary of options for the labels', 

'zorder': 'The layer level in which to draw'} 

def _repr_(self): 

""" 

String representation of ContourPlot primitive. 

EXAMPLES:: 

sage: x,y = var('x,y') 

sage: C = contour_plot(x^2 - y^2, (x,-2,2), (y,-2,2)) 

sage: c = C[0]; c 

ContourPlot defined by a 100 x 100 data grid 

""" 

return "ContourPlot defined by a %s x %s data grid"%(self.xy_array_row, self.xy_array_col) 

def _render_on_subplot(self, subplot): 

""" 

TESTS: 

A somewhat random plot, but fun to look at:: 

sage: x,y = var('x,y') 

sage: contour_plot(x^2 - y^3 + 10*sin(x*y), (x,-4,4), (y,-4,4), 

....: plot_points=121, cmap='hsv') 

Graphics object consisting of 1 graphics primitive 

""" 

from sage.rings.integer import Integer 

options = self.options() 

fill = options['fill'] 

contours = options['contours'] 

if 'cmap' in options: 

cmap = get_cmap(options['cmap']) 

elif fill or contours is None: 

cmap = get_cmap('gray') 

else: 

if isinstance(contours, (int, Integer)): 

cmap = get_cmap([(i,i,i) for i in xsrange(0,1,1/contours)]) 

else: 

l = Integer(len(contours)) 

cmap = get_cmap([(i,i,i) for i in xsrange(0,1,1/l)]) 

x0, x1 = float(self.xrange[0]), float(self.xrange[1]) 

y0, y1 = float(self.yrange[0]), float(self.yrange[1]) 

if isinstance(contours, (int, Integer)): 

contours = int(contours) 

CSF = None 

if fill: 

if contours is None: 

CSF = subplot.contourf(self.xy_data_array, cmap=cmap, 

extent=(x0, x1, y0, y1)) 

else: 

CSF = subplot.contourf(self.xy_data_array, contours, cmap=cmap, 

extent=(x0, x1, y0, y1), extend='both') 

linewidths = options.get('linewidths', None) 

if isinstance(linewidths, (int, Integer)): 

linewidths = int(linewidths) 

elif isinstance(linewidths, (list, tuple)): 

linewidths = tuple(int(x) for x in linewidths) 

from sage.plot.misc import get_matplotlib_linestyle 

linestyles = options.get('linestyles', None) 

if isinstance(linestyles, (list, tuple)): 

linestyles = [get_matplotlib_linestyle(i, 'long') for i in linestyles] 

else: 

linestyles = get_matplotlib_linestyle(linestyles, 'long') 

if contours is None: 

CS = subplot.contour(self.xy_data_array, cmap=cmap, 

extent=(x0, x1, y0, y1), 

linewidths=linewidths, linestyles=linestyles) 

else: 

CS = subplot.contour(self.xy_data_array, contours, cmap=cmap, 

extent=(x0, x1, y0, y1), 

linewidths=linewidths, linestyles=linestyles) 

if options.get('labels', False): 

label_options = options['label_options'] 

label_options['fontsize'] = int(label_options['fontsize']) 

if fill and label_options is None: 

label_options['inline'] = False 

subplot.clabel(CS, **label_options) 

if options.get('colorbar', False): 

colorbar_options = options['colorbar_options'] 

from matplotlib import colorbar 

cax, kwds = colorbar.make_axes_gridspec(subplot, **colorbar_options) 

if CSF is None: 

cb = colorbar.Colorbar(cax, CS, **kwds) 

else: 

cb = colorbar.Colorbar(cax, CSF, **kwds) 

cb.add_lines(CS) 

@suboptions('colorbar', orientation='vertical', format=None, spacing=None) 

@suboptions('label', fontsize=9, colors='blue', inline=None, inline_spacing=3, 

fmt="%1.2f") 

@options(plot_points=100, fill=True, contours=None, linewidths=None, 

linestyles=None, labels=False, frame=True, axes=False, colorbar=False, 

legend_label=None, aspect_ratio=1, region=None) 

def contour_plot(f, xrange, yrange, **options): 

r""" 

``contour_plot`` takes a function of two variables, `f(x,y)` 

and plots contour lines of the function over the specified 

``xrange`` and ``yrange`` as demonstrated below. 

``contour_plot(f, (xmin,xmax), (ymin,ymax), ...)`` 

INPUT: 

- ``f`` -- a function of two variables 

- ``(xmin,xmax)`` -- 2-tuple, the range of ``x`` values OR 3-tuple 

``(x,xmin,xmax)`` 

- ``(ymin,ymax)`` -- 2-tuple, the range of ``y`` values OR 3-tuple 

``(y,ymin,ymax)`` 

The following inputs must all be passed in as named parameters: 

- ``plot_points`` -- integer (default: 100); number of points to plot 

in each direction of the grid. For old computers, 25 is fine, but 

should not be used to verify specific intersection points. 

- ``fill`` -- bool (default: ``True``), whether to color in the area 

between contour lines 

- ``cmap`` -- a colormap (default: ``'gray'``), the name of 

a predefined colormap, a list of colors or an instance of a matplotlib 

Colormap. Type: ``import matplotlib.cm; matplotlib.cm.datad.keys()`` 

for available colormap names. 

- ``contours`` -- integer or list of numbers (default: ``None``): 

If a list of numbers is given, then this specifies the contour levels 

to use. If an integer is given, then this many contour lines are 

used, but the exact levels are determined automatically. If ``None`` 

is passed (or the option is not given), then the number of contour 

lines is determined automatically, and is usually about 5. 

- ``linewidths`` -- integer or list of integer (default: None), if 

a single integer all levels will be of the width given, 

otherwise the levels will be plotted with the width in the order 

given. If the list is shorter than the number of contours, then 

the widths will be repeated cyclically. 

- ``linestyles`` -- string or list of strings (default: None), the 

style of the lines to be plotted, one of: ``"solid"``, ``"dashed"``, 

``"dashdot"``, ``"dotted"``, respectively ``"-"``, ``"--"``, 

``"-."``, ``":"``. If the list is shorter than the number of 

contours, then the styles will be repeated cyclically. 

- ``labels`` -- boolean (default: False) Show level labels or not. 

The following options are to adjust the style and placement of 

labels, they have no effect if no labels are shown. 

- ``label_fontsize`` -- integer (default: 9), the font size of 

the labels. 

- ``label_colors`` -- string or sequence of colors (default: 

None) If a string, gives the name of a single color with which 

to draw all labels. If a sequence, gives the colors of the 

labels. A color is a string giving the name of one or a 

3-tuple of floats. 

- ``label_inline`` -- boolean (default: False if fill is True, 

otherwise True), controls whether the underlying contour is 

removed or not. 

- ``label_inline_spacing`` -- integer (default: 3), When inline, 

this is the amount of contour that is removed from each side, 

in pixels. 

- ``label_fmt`` -- a format string (default: "%1.2f"), this is 

used to get the label text from the level. This can also be a 

dictionary with the contour levels as keys and corresponding 

text string labels as values. It can also be any callable which 

returns a string when called with a numeric contour level. 

- ``colorbar`` -- boolean (default: False) Show a colorbar or not. 

The following options are to adjust the style and placement of 

colorbars. They have no effect if a colorbar is not shown. 

- ``colorbar_orientation`` -- string (default: 'vertical'), 

controls placement of the colorbar, can be either 'vertical' 

or 'horizontal' 

- ``colorbar_format`` -- a format string, this is used to format 

the colorbar labels. 

- ``colorbar_spacing`` -- string (default: 'proportional'). If 

'proportional', make the contour divisions proportional to 

values. If 'uniform', space the colorbar divisions uniformly, 

without regard for numeric values. 

- ``legend_label`` -- the label for this item in the legend 

- ``region`` - (default: None) If region is given, it must be a function 

of two variables. Only segments of the surface where region(x,y) 

returns a number >0 will be included in the plot. 

EXAMPLES: 

Here we plot a simple function of two variables. Note that 

since the input function is an expression, we need to explicitly 

declare the variables in 3-tuples for the range:: 

sage: x,y = var('x,y') 

sage: contour_plot(cos(x^2 + y^2), (x,-4,4), (y,-4,4)) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

g = contour_plot(cos(x**2 + y**2), (x,-4,4), (y,-4,4)) 

sphinx_plot(g) 

Here we change the ranges and add some options:: 

sage: x,y = var('x,y') 

sage: contour_plot((x^2) * cos(x*y), (x,-10,5), (y,-5,5), fill=False, plot_points=150) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

g = contour_plot((x**2) * cos(x*y), (x,-10,5), (y,-5,5), fill=False, plot_points=150) 

sphinx_plot(g) 

An even more complicated plot:: 

sage: x,y = var('x,y') 

sage: contour_plot(sin(x^2+y^2) * cos(x) * sin(y), (x,-4,4), (y,-4,4), plot_points=150) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

g = contour_plot(sin(x**2+y**2) * cos(x) * sin(y), (x,-4,4), (y,-4,4),plot_points=150) 

sphinx_plot(g) 

Some elliptic curves, but with symbolic endpoints. In the first 

example, the plot is rotated 90 degrees because we switch the 

variables `x`, `y`:: 

sage: x,y = var('x,y') 

sage: contour_plot(y^2 + 1 - x^3 - x, (y,-pi,pi), (x,-pi,pi)) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

g = contour_plot(y**2 + 1 - x**3 - x, (y,-pi,pi), (x,-pi,pi)) 

sphinx_plot(g) 

:: 

sage: contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi)) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

g = contour_plot(y**2 + 1 - x**3 - x, (x,-pi,pi), (y,-pi,pi)) 

sphinx_plot(g) 

We can play with the contour levels:: 

sage: x,y = var('x,y') 

sage: f(x,y) = x^2 + y^2 

sage: contour_plot(f, (-2,2), (-2,2)) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

def f(x,y): return x**2 + y**2 

g = contour_plot(f, (-2,2), (-2,2)) 

sphinx_plot(g) 

:: 

sage: contour_plot(f, (-2,2), (-2,2), contours=2, cmap=[(1,0,0), (0,1,0), (0,0,1)]) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

def f(x,y): return x**2 + y**2 

g = contour_plot(f, (-2, 2), (-2, 2), contours=2, cmap=[(1,0,0), (0,1,0), (0,0,1)]) 

sphinx_plot(g) 

:: 

sage: contour_plot(f, (-2,2), (-2,2), 

....: contours=(0.1,1.0,1.2,1.4), cmap='hsv') 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

def f(x,y): return x**2 + y**2 

g = contour_plot(f, (-2,2), (-2,2), contours=(0.1,1.0,1.2,1.4), cmap='hsv') 

sphinx_plot(g) 

:: 

sage: contour_plot(f, (-2,2), (-2,2), contours=(1.0,), fill=False) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

def f(x,y): return x**2 + y**2 

g = contour_plot(f, (-2,2), (-2,2), contours=(1.0,), fill=False) 

sphinx_plot(g) 

:: 

sage: contour_plot(x - y^2, (x,-5,5), (y,-3,3), contours=[-4,0,1]) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

g = contour_plot(x - y**2, (x,-5,5), (y,-3,3), contours=[-4,0,1]) 

sphinx_plot(g) 

We can change the style of the lines:: 

sage: contour_plot(f, (-2,2), (-2,2), fill=False, linewidths=10) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

def f(x,y): return x**2 + y**2 

g = contour_plot(f, (-2,2), (-2,2), fill=False, linewidths=10) 

sphinx_plot(g) 

:: 

sage: contour_plot(f, (-2,2), (-2,2), fill=False, linestyles='dashdot') 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

def f(x,y): return x**2 + y**2 

g = contour_plot(f, (-2,2), (-2,2), fill=False, linestyles='dashdot') 

sphinx_plot(g) 

:: 

sage: P = contour_plot(x^2 - y^2, (x,-3,3), (y,-3,3), 

....: contours=[0,1,2,3,4], linewidths=[1,5], 

....: linestyles=['solid','dashed'], fill=False) 

sage: P 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

P = contour_plot(x**2 - y**2, (x,-3,3), (y,-3,3), 

contours=[0,1,2,3,4], linewidths=[1,5], 

linestyles=['solid','dashed'], fill=False) 

sphinx_plot(P) 

:: 

sage: P = contour_plot(x^2 - y^2, (x,-3,3), (y,-3,3), 

....: contours=[0,1,2,3,4], linewidths=[1,5], 

....: linestyles=['solid','dashed']) 

sage: P 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

P = contour_plot(x**2 - y**2, (x,-3,3), (y,-3,3), 

contours=[0,1,2,3,4], linewidths=[1,5], 

linestyles=['solid','dashed']) 

sphinx_plot(P) 

:: 

sage: P = contour_plot(x^2 - y^2, (x,-3,3), (y,-3,3), 

....: contours=[0,1,2,3,4], linewidths=[1,5], 

....: linestyles=['-',':']) 

sage: P 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

P = contour_plot(x**2 - y**2, (x,-3,3), (y,-3,3), 

contours=[0,1,2,3,4], linewidths=[1,5], 

linestyles=['-',':']) 

sphinx_plot(P) 

We can add labels and play with them:: 

sage: contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), 

....: fill=False, cmap='hsv', labels=True) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

P = contour_plot(y**2 + 1 - x**3 - x, (x,-pi,pi), (y,-pi,pi), 

fill=False, cmap='hsv', labels=True) 

sphinx_plot(P) 

:: 

sage: P=contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), 

....: fill=False, cmap='hsv', 

....: labels=True, label_fmt="%1.0f", 

....: label_colors='black') 

sage: P 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

P=contour_plot(y**2 + 1 - x**3 - x, (x,-pi,pi), (y,-pi,pi), 

fill=False, cmap='hsv', 

labels=True, label_fmt="%1.0f", 

label_colors='black') 

sphinx_plot(P) 

:: 

sage: P = contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), 

....: fill=False, cmap='hsv', labels=True, 

....: contours=[-4,0,4], 

....: label_fmt={-4:"low", 0:"medium", 4: "hi"}, 

....: label_colors='black') 

sage: P 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

P = contour_plot(y**2 + 1 - x**3 - x, (x,-pi,pi), (y,-pi,pi), 

fill=False, cmap='hsv', labels=True, 

contours=[-4,0,4], 

label_fmt={-4:"low", 0:"medium", 4: "hi"}, 

label_colors='black') 

sphinx_plot(P) 

:: 

sage: P = contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), 

....: fill=False, cmap='hsv', labels=True, 

....: contours=[-4,0,4], label_fmt=lambda x: "$z=%s$"%x, 

....: label_colors='black', label_inline=True, 

....: label_fontsize=12) 

sage: P 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

P = contour_plot(y**2 + 1 - x**3 - x, (x,-pi,pi), (y,-pi,pi), 

fill=False, cmap='hsv', labels=True, 

contours=[-4,0,4], label_fmt=lambda x: "$z=%s$"%x, 

label_colors='black', label_inline=True, 

label_fontsize=12) 

sphinx_plot(P) 

:: 

sage: P = contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), 

....: fill=False, cmap='hsv', labels=True, 

....: label_fontsize=18) 

sage: P 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

P = contour_plot(y**2 + 1 - x**3 - x, (x,-pi,pi), (y,-pi,pi), 

fill=False, cmap='hsv', labels=True, 

label_fontsize=18) 

sphinx_plot(P) 

:: 

sage: P = contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), 

....: fill=False, cmap='hsv', labels=True, 

....: label_inline_spacing=1) 

sage: P 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

P = contour_plot(y**2 + 1 - x**3 - x, (x,-pi,pi), (y,-pi,pi), 

fill=False, cmap='hsv', labels=True, 

label_inline_spacing=1) 

sphinx_plot(P) 

:: 

sage: P = contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi), 

....: fill=False, cmap='hsv', labels=True, 

....: label_inline=False) 

sage: P 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

P = contour_plot(y**2 + 1 - x**3 - x, (x,-pi,pi), (y,-pi,pi), 

fill=False, cmap='hsv', labels=True, 

label_inline=False) 

sphinx_plot(P) 

We can change the color of the labels if so desired:: 

sage: contour_plot(f, (-2,2), (-2,2), labels=True, label_colors='red') 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

def f(x,y): return x**2 + y**2 

g = contour_plot(f, (-2,2), (-2,2), labels=True, label_colors='red') 

sphinx_plot(g) 

We can add a colorbar as well:: 

sage: f(x, y)=x^2-y^2 

sage: contour_plot(f, (x,-3,3), (y,-3,3), colorbar=True) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

def f(x,y): return x**2 + y**2 

g = contour_plot(f, (x,-3,3), (y,-3,3), colorbar=True) 

sphinx_plot(g) 

:: 

sage: contour_plot(f, (x,-3,3), (y,-3,3), colorbar=True, colorbar_orientation='horizontal') 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

def f(x,y): return x**2 + y**2 

g = contour_plot(f, (x,-3,3), (y,-3,3), colorbar=True, colorbar_orientation='horizontal') 

sphinx_plot(g) 

:: 

sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[-2,-1,4], colorbar=True) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

def f(x,y): return x**2 + y**2 

g = contour_plot(f, (x,-3,3), (y,-3,3), contours=[-2,-1,4], 

colorbar=True) 

sphinx_plot(g) 

:: 

sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[-2,-1,4], 

....: colorbar=True, colorbar_spacing='uniform') 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

def f(x,y): return x**2 + y**2 

g = contour_plot(f, (x,-3,3), (y,-3,3), contours=[-2,-1,4], 

colorbar=True, colorbar_spacing='uniform') 

sphinx_plot(g) 

:: 

sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[0,2,3,6], 

....: colorbar=True, colorbar_format='%.3f') 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

def f(x,y): return x**2 + y**2 

g = contour_plot(f, (x,-3,3), (y,-3,3), contours=[0,2,3,6], 

colorbar=True, colorbar_format='%.3f') 

sphinx_plot(g) 

:: 

sage: contour_plot(f, (x,-3,3), (y,-3,3), labels=True, 

....: label_colors='red', contours=[0,2,3,6], 

....: colorbar=True) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

def f(x,y): return x**2 + y**2 

g = contour_plot(f, (x,-3,3), (y,-3,3), labels=True, 

label_colors='red', contours=[0,2,3,6], 

colorbar=True) 

sphinx_plot(g) 

:: 

sage: contour_plot(f, (x,-3,3), (y,-3,3), cmap='winter', 

....: contours=20, fill=False, colorbar=True) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

def f(x,y): return x**2 + y**2 

g = contour_plot(f, (x,-3,3), (y,-3,3), cmap='winter', 

contours=20, fill=False, colorbar=True) 

sphinx_plot(g) 

This should plot concentric circles centered at the origin:: 

sage: x,y = var('x,y') 

sage: contour_plot(x^2 + y^2-2,(x,-1,1), (y,-1,1)) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

g = contour_plot(x**2 + y**2-2,(x,-1,1), (y,-1,1)) 

sphinx_plot(g) 

Extra options will get passed on to show(), as long as they are valid:: 

sage: f(x,y) = cos(x) + sin(y) 

sage: contour_plot(f, (0,pi), (0,pi), axes=True) 

Graphics object consisting of 1 graphics primitive 

:: 

sage: contour_plot(f, (0,pi), (0,pi)).show(axes=True) # These are equivalent 

.. PLOT:: 

x,y = var('x,y') 

def f(x,y): return cos(x) + sin(y) 

g = contour_plot(f, (0,pi), (0,pi), axes=True) 

sphinx_plot(g) 

One can also plot over a reduced region:: 

sage: contour_plot(x**2 - y**2, (x,-2,2), (y,-2,2), region=x - y, plot_points=300) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

g = contour_plot(x**2 - y**2, (x,-2,2), (y,-2,2), region=x - y, 

plot_points=300) 

sphinx_plot(g) 

Note that with ``fill=False`` and grayscale contours, there is the 

possibility of confusion between the contours and the axes, so use 

``fill=False`` together with ``axes=True`` with caution:: 

sage: contour_plot(f, (-pi,pi), (-pi,pi), fill=False, axes=True) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x,y = var('x,y') 

def f(x,y): return cos(x) + sin(y) 

g = contour_plot(f, (-pi,pi), (-pi,pi), fill=False, axes=True) 

sphinx_plot(g) 

TESTS: 

To check that :trac:`5221` is fixed, note that this has three curves, not 

two:: 

sage: x,y = var('x,y') 

sage: contour_plot(x - y^2, (x,-5,5), (y,-3,3), 

....: contours=[-4,-2,0], fill=False) 

Graphics object consisting of 1 graphics primitive 

""" 

from sage.plot.all import Graphics 

from sage.plot.misc import setup_for_eval_on_grid 

region = options.pop('region') 

ev = [f] if region is None else [f, region] 

F, ranges = setup_for_eval_on_grid(ev, [xrange, yrange], options['plot_points']) 

g = F[0] 

xrange, yrange = [r[:2] for r in ranges] 

xy_data_array = [[g(x, y) for x in xsrange(*ranges[0], include_endpoint=True)] 

for y in xsrange(*ranges[1], include_endpoint=True)] 

if region is not None: 

import numpy 

xy_data_array = numpy.ma.asarray(xy_data_array, dtype=float) 

m = F[1] 

mask = numpy.asarray([[m(x, y) <= 0 

for x in xsrange(*ranges[0], 

include_endpoint=True)] 

for y in xsrange(*ranges[1], 

include_endpoint=True)], 

dtype=bool) 

xy_data_array[mask] = numpy.ma.masked 

g = Graphics() 

# Reset aspect_ratio to 'automatic' in case scale is 'semilog[xy]'. 

# Otherwise matplotlib complains. 

scale = options.get('scale', None) 

if isinstance(scale, (list, tuple)): 

scale = scale[0] 

if scale == 'semilogy' or scale == 'semilogx': 

options['aspect_ratio'] = 'automatic' 

g._set_extra_kwds(Graphics._extract_kwds_for_show(options, ignore=['xmin', 'xmax'])) 

g.add_primitive(ContourPlot(xy_data_array, xrange, yrange, options)) 

return g 

@options(plot_points=150, contours=(0,), fill=False, cmap=["blue"]) 

def implicit_plot(f, xrange, yrange, **options): 

r""" 

``implicit_plot`` takes a function of two variables, `f(x, y)` 

and plots the curve `f(x,y) = 0` over the specified 

``xrange`` and ``yrange`` as demonstrated below. 

``implicit_plot(f, (xmin,xmax), (ymin,ymax), ...)`` 

``implicit_plot(f, (x,xmin,xmax), (y,ymin,ymax), ...)`` 

INPUT: 

- ``f`` -- a function of two variables or equation in two variables 

- ``(xmin,xmax)`` -- 2-tuple, the range of ``x`` 

values or ``(x,xmin,xmax)`` 

- ``(ymin,ymax)`` -- 2-tuple, the range of ``y`` 

values or ``(y,ymin,ymax)`` 

The following inputs must all be passed in as named parameters: 

- ``plot_points`` -- integer (default: 150); number of points to plot 

in each direction of the grid 

- ``fill`` -- boolean (default: ``False``); if ``True``, fill the region 

`f(x, y) < 0`. 

- ``fillcolor`` -- string (default: ``'blue'``), the color of the region 

where `f(x,y) < 0` if ``fill = True``. Colors are defined in 

:mod:`sage.plot.colors`; try ``colors?`` to see them all. 

- ``linewidth`` -- integer (default: None), if a single integer all levels 

will be of the width given, otherwise the levels will be plotted with the 

widths in the order given. 

- ``linestyle`` -- string (default: None), the style of the line to be 

plotted, one of: ``"solid"``, ``"dashed"``, ``"dashdot"`` or 

``"dotted"``, respectively ``"-"``, ``"--"``, ``"-."``, or ``":"``. 

- ``color`` -- string (default: ``'blue'``), the color of the plot. Colors 

are defined in :mod:`sage.plot.colors`; try ``colors?`` to see them all. 

If ``fill = True``, then this sets only the color of the border of the 

plot. See ``fillcolor`` for setting the color of the fill region. 

- ``legend_label`` -- the label for this item in the legend 

- ``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. 

EXAMPLES: 

A simple circle with a radius of 2. Note that 

since the input function is an expression, we need to explicitly 

declare the variables in 3-tuples for the range:: 

sage: var("x y") 

(x, y) 

sage: implicit_plot(x^2 + y^2 - 2, (x,-3,3), (y,-3,3)) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x, y =var("x y") 

g = implicit_plot(x**2 + y**2 - 2, (x,-3,3), (y,-3,3)) 

sphinx_plot(g) 

We can do the same thing, but using a callable function so we don't 

need to explicitly define the variables in the ranges. We also fill 

the inside:: 

sage: f(x,y) = x^2 + y^2 - 2 

sage: implicit_plot(f, (-3,3), (-3,3), fill=True, plot_points=500) # long time 

Graphics object consisting of 2 graphics primitives 

.. PLOT:: 

def f(x,y): return x**2 + y**2 - 2 

g = implicit_plot(f, (-3,3), (-3,3), fill=True, plot_points=500) 

sphinx_plot(g) 

The same circle but with a different line width:: 

sage: implicit_plot(f, (-3,3), (-3,3), linewidth=6) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

def f(x,y): return x**2 + y**2 - 2 

g = implicit_plot(f, (-3,3), (-3,3), linewidth=6) 

sphinx_plot(g) 

Again the same circle but this time with a dashdot border:: 

sage: implicit_plot(f, (-3,3), (-3,3), linestyle='dashdot') 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x, y =var("x y") 

def f(x,y): return x**2 + y**2 - 2 

g = implicit_plot(f, (-3,3), (-3,3), linestyle='dashdot') 

sphinx_plot(g) 

The same circle with different line and fill colors:: 

sage: implicit_plot(f, (-3,3), (-3,3), color='red', fill=True, fillcolor='green', 

....: plot_points=500) # long time 

Graphics object consisting of 2 graphics primitives 

.. PLOT:: 

def f(x,y): return x**2 + y**2 - 2 

g = implicit_plot(f, (-3,3), (-3,3), color='red', fill=True, fillcolor='green', 

plot_points=500) 

sphinx_plot(g) 

You can also plot an equation:: 

sage: var("x y") 

(x, y) 

sage: implicit_plot(x^2 + y^2 == 2, (x,-3,3), (y,-3,3)) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x, y =var("x y") 

g = implicit_plot(x**2 + y**2 == 2, (x,-3,3), (y,-3,3)) 

sphinx_plot(g) 

You can even change the color of the plot:: 

sage: implicit_plot(x^2 + y^2 == 2, (x,-3,3), (y,-3,3), color="red") 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x, y =var("x y") 

g = implicit_plot(x**2 + y**2 == 2, (x,-3,3), (y,-3,3), color="red") 

sphinx_plot(g) 

The color of the fill region can be changed:: 

sage: implicit_plot(x**2 + y**2 == 2, (x,-3,3), (y,-3,3), fill=True, fillcolor='red') 

Graphics object consisting of 2 graphics primitives 

.. PLOT:: 

x, y =var("x y") 

g = implicit_plot(x**2 + y**2 == 2, (x,-3,3), (y,-3,3), fill=True, fillcolor="red") 

sphinx_plot(g) 

Here is a beautiful (and long) example which also tests that all 

colors work with this:: 

sage: G = Graphics() 

sage: counter = 0 

sage: for col in colors.keys(): # long time 

....: G += implicit_plot(x^2 + y^2 == 1 + counter*.1, (x,-4,4),(y,-4,4), color=col) 

....: counter += 1 

sage: G # long time 

Graphics object consisting of 148 graphics primitives 

.. PLOT:: 

x, y = var("x y") 

G = Graphics() 

counter = 0 

for col in colors.keys(): 

G += implicit_plot(x**2 + y**2 == 1 + counter*.1, (x,-4,4), (y,-4,4), color=col) 

counter += 1 

sphinx_plot(G) 

We can define a level-`n` approximation of the boundary of the 

Mandelbrot set:: 

sage: def mandel(n): 

....: c = polygen(CDF, 'c') 

....: z = 0 

....: for i in range(n): 

....: z = z*z + c 

....: def f(x,y): 

....: val = z(CDF(x, y)) 

....: return val.norm() - 4 

....: return f 

The first-level approximation is just a circle:: 

sage: implicit_plot(mandel(1), (-3,3), (-3,3)) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

def mandel(n): 

c = polygen(CDF, 'c') 

z = 0 

for i in range(n): 

z = z*z + c 

def f(x,y): 

val = z(CDF(x, y)) 

return val.norm() - 4 

return f 

g = implicit_plot(mandel(1), (-3,3), (-3,3)) 

sphinx_plot(g) 

A third-level approximation starts to get interesting:: 

sage: implicit_plot(mandel(3), (-2,1), (-1.5,1.5)) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

def mandel(n): 

c = polygen(CDF, 'c') 

z = 0 

for i in range(n): 

z = z*z + c 

def f(x,y): 

val = z(CDF(x, y)) 

return val.norm() - 4 

return f 

g = implicit_plot(mandel(3), (-2,1), (-1.5,1.5)) 

sphinx_plot(g) 

The seventh-level approximation is a degree 64 polynomial, and 

``implicit_plot`` does a pretty good job on this part of the curve. 

(``plot_points=200`` looks even better, but it takes over a second.) 

:: 

sage: implicit_plot(mandel(7), (-0.3, 0.05), (-1.15, -0.9), plot_points=50) 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

def mandel(n): 

c = polygen(CDF, 'c') 

z = 0 

for i in range(n): 

z = z*z + c 

def f(x,y): 

val = z(CDF(x, y)) 

return val.norm() - 4 

return f 

g = implicit_plot(mandel(7), (-0.3,0.05), (-1.15,-0.9), 

plot_points=50) 

sphinx_plot(g) 

When making a filled implicit plot using a python function rather than a 

symbolic expression the user should increase the number of plot points to 

avoid artifacts:: 

sage: implicit_plot(lambda x, y: x^2 + y^2 - 2, (x,-3,3), (y,-3,3), 

....: fill=True, plot_points=500) # long time 

Graphics object consisting of 2 graphics primitives 

.. PLOT:: 

x, y = var("x y") 

g = implicit_plot(lambda x, y: x**2 + y**2 - 2, (x,-3,3), (y,-3,3), 

fill=True, plot_points=500) 

sphinx_plot(g) 

An example of an implicit plot on 'loglog' scale:: 

sage: implicit_plot(x^2 + y^2 == 200, (x,1,200), (y,1,200), scale='loglog') 

Graphics object consisting of 1 graphics primitive 

.. PLOT:: 

x, y = var("x y") 

g = implicit_plot(x**2 + y**2 == 200, (x,1,200), (y,1,200), scale='loglog') 

sphinx_plot(g) 

TESTS:: 

sage: f(x,y) = x^2 + y^2 - 2 

sage: implicit_plot(f, (-3,3), (-3,3), fill=5) 

Traceback (most recent call last): 

... 

ValueError: fill=5 is not supported 

To check that :trac:`9654` is fixed:: 

sage: f(x,y) = x^2 + y^2 - 2 

sage: implicit_plot(f, (-3,3), (-3,3), rgbcolor=(1,0,0)) 

Graphics object consisting of 1 graphics primitive 

sage: implicit_plot(f, (-3,3), (-3,3), color='green') 

Graphics object consisting of 1 graphics primitive 

sage: implicit_plot(f, (-3,3), (-3,3), rgbcolor=(1,0,0), color='green') 

Traceback (most recent call last): 

... 

ValueError: only one of color or rgbcolor should be specified 

""" 

from sage.symbolic.expression import is_SymbolicEquation 

if is_SymbolicEquation(f): 

if f.operator() != operator.eq: 

raise ValueError("input to implicit plot must be function or equation") 

f = f.lhs() - f.rhs() 

linewidths = options.pop('linewidth', None) 

linestyles = options.pop('linestyle', None) 

if 'color' in options and 'rgbcolor' in options: 

raise ValueError('only one of color or rgbcolor should be specified') 

elif 'color' in options: 

options['cmap']=[options.pop('color', None)] 

elif 'rgbcolor' in options: 

options['cmap']=[rgbcolor(options.pop('rgbcolor', None))] 

if options['fill'] is True: 

options.pop('fill') 

options.pop('contours', None) 

incol = options.pop('fillcolor', 'blue') 

bordercol = options.pop('cmap', [None])[0] 

from sage.symbolic.expression import is_Expression 

if not is_Expression(f): 

return region_plot(lambda x, y: f(x, y) < 0, xrange, yrange, 

borderwidth=linewidths, borderstyle=linestyles, 

incol=incol, bordercol=bordercol, 

**options) 

else: 

return region_plot(f < 0, xrange, yrange, borderwidth=linewidths, 

borderstyle=linestyles, 

incol=incol, bordercol=bordercol, 

**options) 

elif options['fill'] is False: 

options.pop('fillcolor', None) 

return contour_plot(f, xrange, yrange, linewidths=linewidths, 

linestyles=linestyles, **options) 

else: 

raise ValueError("fill=%s is not supported" % options['fill']) 

@options(plot_points=100, incol='blue', outcol=None, bordercol=None, 

borderstyle=None, borderwidth=None, frame=False, axes=True, 

legend_label=None, aspect_ratio=1, alpha=1) 

def region_plot(f, xrange, yrange, plot_points, incol, outcol, bordercol,