def least_quadratic_nonresidue(p):
"""
Returns the smallest positive integer quadratic non-residue in Z/pZ for primes p>2.
EXAMPLES::
sage: least_quadratic_nonresidue(5)
2
sage: [least_quadratic_nonresidue(p) for p in prime_range(3,100)]
[2, 2, 3, 2, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 2, 2, 2, 7, 5, 3, 2, 3, 5]
TESTS:
Raises an error if input is a positive composite integer.
::
sage: least_quadratic_nonresidue(20)
Traceback (most recent call last):
...
ValueError: Oops! p must be a prime number > 2.
Raises an error if input is 2. This is because every integer is a
quadratic residue modulo 2.
::
sage: least_quadratic_nonresidue(2)
Traceback (most recent call last):
...
ValueError: Oops! There are no quadratic non-residues in Z/2Z.
"""
from sage.functions.all import floor
p1 = abs(p)
## Deal with the prime p = 2 and |p| <= 1.
if p1 == 2:
raise ValueError("Oops! There are no quadratic non-residues in Z/2Z.")
if p1 < 2:
raise ValueError("Oops! p must be a prime number > 2.")
## Find the smallest non-residue mod p
## For 7/8 of primes the answer is 2, 3 or 5:
if p%8 in (3,5):
return ZZ(2)
if p%12 in (5,7):
return ZZ(3)
if p%5 in (2,3):
return ZZ(5)
## default case (first needed for p=71):
if not p.is_prime():
raise ValueError("Oops! p must be a prime number > 2.")
from sage.misc.misc import xsrange
for r in xsrange(7,p):
if legendre_symbol(r, p) == -1:
return ZZ(r)
def _parametric_plot3d_curve(f, urange, plot_points, **kwds):
r"""
This function is used internally by the
``parametric_plot3d`` command.
"""
from sage.plot.misc import setup_for_eval_on_grid
g, ranges = setup_for_eval_on_grid(f, [urange], plot_points)
f_x,f_y,f_z = g
w = [(f_x(u), f_y(u), f_z(u)) for u in xsrange(*ranges[0], include_endpoint=True)]
return line3d(w, **kwds)
def _parametric_plot3d_curve(f, urange, plot_points, **kwds):
r"""
This function is used internally by the
``parametric_plot3d`` command.
"""
from sage.plot.misc import setup_for_eval_on_grid
g, ranges = setup_for_eval_on_grid(f, [urange], plot_points)
f_x, f_y, f_z = g
w = [(f_x(u), f_y(u), f_z(u))
for u in xsrange(*ranges[0], include_endpoint=True)]
return line3d(w, **kwds)
def _parametric_plot3d_curve(f, urange, plot_points, **kwds):
r"""
Return a parametric three-dimensional space curve.
This function is used internally by the
:func:`parametric_plot3d` command.
There are two ways this function is invoked by
:func:`parametric_plot3d`.
- ``parametric_plot3d([f_x, f_y, f_z], (u_min,
u_max))``:
`f_x, f_y, f_z` are three functions and
`u_{\min}` and `u_{\max}` are real numbers
- ``parametric_plot3d([f_x, f_y, f_z], (u, u_min,
u_max))``:
`f_x, f_y, f_z` can be viewed as functions of
`u`
INPUT:
- ``f`` - a 3-tuple of functions or expressions, or vector of size 3
- ``urange`` - a 2-tuple (u_min, u_max) or a 3-tuple
(u, u_min, u_max)
- ``plot_points`` - (default: "automatic", which is 75) initial
number of sample points in each parameter; an integer.
EXAMPLES:
We demonstrate each of the two ways of calling this. See
:func:`parametric_plot3d` for many more examples.
We do the first one with a lambda function, which creates a
callable Python function that sends `u` to `u/10`::
sage: parametric_plot3d( (sin, cos, lambda u: u/10), (0, 20)) # indirect doctest
Graphics3d Object
Now we do the same thing with symbolic expressions::
sage: u = var('u')
sage: parametric_plot3d( (sin(u), cos(u), u/10), (u, 0, 20))
Graphics3d Object
"""
from sage.plot.misc import setup_for_eval_on_grid
g, ranges = setup_for_eval_on_grid(f, [urange], plot_points)
f_x, f_y, f_z = g
w = [(f_x(u), f_y(u), f_z(u))
for u in xsrange(*ranges[0], include_endpoint=True)]
return line3d(w, **kwds)
def _parametric_plot3d_curve(f, urange, plot_points, **kwds):
r"""
Return a parametric three-dimensional space curve.
This function is used internally by the
:func:`parametric_plot3d` command.
There are two ways this function is invoked by
:func:`parametric_plot3d`.
- ``parametric_plot3d([f_x, f_y, f_z], (u_min,
u_max))``:
`f_x, f_y, f_z` are three functions and
`u_{\min}` and `u_{\max}` are real numbers
- ``parametric_plot3d([f_x, f_y, f_z], (u, u_min,
u_max))``:
`f_x, f_y, f_z` can be viewed as functions of
`u`
INPUT:
- ``f`` - a 3-tuple of functions or expressions, or vector of size 3
- ``urange`` - a 2-tuple (u_min, u_max) or a 3-tuple
(u, u_min, u_max)
- ``plot_points`` - (default: "automatic", which is 75) initial
number of sample points in each parameter; an integer.
EXAMPLES:
We demonstrate each of the two ways of calling this. See
:func:`parametric_plot3d` for many more examples.
We do the first one with a lambda function, which creates a
callable Python function that sends `u` to `u/10`::
sage: parametric_plot3d( (sin, cos, lambda u: u/10), (0, 20)) # indirect doctest
Graphics3d Object
Now we do the same thing with symbolic expressions::
sage: u = var('u')
sage: parametric_plot3d( (sin(u), cos(u), u/10), (u, 0, 20))
Graphics3d Object
"""
from sage.plot.misc import setup_for_eval_on_grid
g, ranges = setup_for_eval_on_grid(f, [urange], plot_points)
f_x,f_y,f_z = g
w = [(f_x(u), f_y(u), f_z(u)) for u in xsrange(*ranges[0], include_endpoint=True)]
return line3d(w, **kwds)
def f(self):
r"""
Returns the set between ``self.start`` and ``self.stop``.
EXAMPLES::
sage: from sage.sets.set_from_iterator import DummyExampleForPicklingTest
sage: d = DummyExampleForPicklingTest()
sage: d.f()
{10, 11, 12, 13, 14, ...}
sage: d.start = 4
sage: d.stop = 200
sage: d.f()
{4, 5, 6, 7, 8, ...}
"""
from sage.misc.misc import xsrange
return xsrange(self.start, self.stop)
def f(self):
r"""
Returns the set between ``self.start`` and ``self.stop``.
EXAMPLES::
sage: from sage.sets.set_from_iterator import DummyExampleForPicklingTest
sage: d = DummyExampleForPicklingTest()
sage: d.f()
{10, 11, 12, 13, 14, ...}
sage: d.start = 4
sage: d.stop = 200
sage: d.f()
{4, 5, 6, 7, 8, ...}
"""
from sage.misc.misc import xsrange
return xsrange(self.start, self.stop)
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
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
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
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
::
sage: contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi))
Graphics object consisting of 1 graphics primitive
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
::
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
::
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
::
sage: contour_plot(f, (-2, 2), (-2, 2), contours=(1.0,), fill=False)
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(x-y^2,(x,-5,5),(y,-3,3),contours=[-4,0,1])
Graphics object consisting of 1 graphics primitive
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
::
sage: contour_plot(f, (-2,2), (-2,2), fill=False, linestyles='dashdot')
Graphics object consisting of 1 graphics primitive
::
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
::
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
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
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
::
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
::
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
::
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
::
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
::
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
::
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
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
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
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), colorbar=True,colorbar_orientation='horizontal')
Graphics object consisting of 1 graphics primitive
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[-2,-1,4],colorbar=True)
Graphics object consisting of 1 graphics primitive
::
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
::
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
::
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
::
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
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
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
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
::
sage: contour_plot(f, (0, pi), (0, pi)).show(axes=True) # These are equivalent
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
TESTS:
To check that ticket 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
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),
label=options['legend_label'])
else:
CSF = subplot.contourf(self.xy_data_array,
contours,
cmap=cmap,
extent=(x0, x1, y0, y1),
extend='both',
label=options['legend_label'])
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(l, 'long') for l 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,
label=options['legend_label'])
else:
CS = subplot.contour(self.xy_data_array,
contours,
cmap=cmap,
extent=(x0, x1, y0, y1),
linewidths=linewidths,
linestyles=linestyles,
label=options['legend_label'])
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)
def region_plot(f, xrange, yrange, plot_points, incol, outcol, bordercol, borderstyle, borderwidth,**options):
r"""
``region_plot`` takes a boolean function of two variables, `f(x,y)`
and plots the region where f is True over the specified
``xrange`` and ``yrange`` as demonstrated below.
``region_plot(f, (xmin, xmax), (ymin, ymax), ...)``
INPUT:
- ``f`` -- a boolean 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)``
- ``plot_points`` -- integer (default: 100); number of points to plot
in each direction of the grid
- ``incol`` -- a color (default: ``'blue'``), the color inside the region
- ``outcol`` -- a color (default: ``'white'``), the color of the outside
of the region
If any of these options are specified, the border will be shown as indicated,
otherwise it is only implicit (with color ``incol``) as the border of the
inside of the region.
- ``bordercol`` -- a color (default: ``None``), the color of the border
(``'black'`` if ``borderwidth`` or ``borderstyle`` is specified but not ``bordercol``)
- ``borderstyle`` -- string (default: 'solid'), one of 'solid', 'dashed', 'dotted', 'dashdot'
- ``borderwidth`` -- integer (default: None), the width of the border in pixels
- ``legend_label`` -- the label for this item in the legend
EXAMPLES:
Here we plot a simple function of two variables::
sage: x,y = var('x,y')
sage: region_plot(cos(x^2+y^2) <= 0, (x, -3, 3), (y, -3, 3))
Here we play with the colors::
sage: region_plot(x^2+y^3 < 2, (x, -2, 2), (y, -2, 2), incol='lightblue', bordercol='gray')
An even more complicated plot, with dashed borders::
sage: region_plot(sin(x)*sin(y) >= 1/4, (x,-10,10), (y,-10,10), incol='yellow', bordercol='black', borderstyle='dashed', plot_points=250)
A disk centered at the origin::
sage: region_plot(x^2+y^2<1, (x,-1,1), (y,-1,1))
A plot with more than one condition (all conditions must be true for the statement to be true)::
sage: region_plot([x^2+y^2<1, x<y], (x,-2,2), (y,-2,2))
Since it doesn't look very good, let's increase ``plot_points``::
sage: region_plot([x^2+y^2<1, x<y], (x,-2,2), (y,-2,2), plot_points=400)
To get plots where only one condition needs to be true, use a function.
Using lambda functions, we definitely need the extra ``plot_points``::
sage: region_plot(lambda x,y: x^2+y^2<1 or x<y, (x,-2,2), (y,-2,2), plot_points=400)
The first quadrant of the unit circle::
sage: region_plot([y>0, x>0, x^2+y^2<1], (x,-1.1, 1.1), (y,-1.1, 1.1), plot_points = 400)
Here is another plot, with a huge border::
sage: region_plot(x*(x-1)*(x+1)+y^2<0, (x, -3, 2), (y, -3, 3), incol='lightblue', bordercol='gray', borderwidth=10, plot_points=50)
If we want to keep only the region where x is positive::
sage: region_plot([x*(x-1)*(x+1)+y^2<0, x>-1], (x, -3, 2), (y, -3, 3), incol='lightblue', plot_points=50)
Here we have a cut circle::
sage: region_plot([x^2+y^2<4, x>-1], (x, -2, 2), (y, -2, 2), incol='lightblue', bordercol='gray', plot_points=200)
The first variable range corresponds to the horizontal axis and
the second variable range corresponds to the vertical axis::
sage: s,t=var('s,t')
sage: region_plot(s>0,(t,-2,2),(s,-2,2))
::
sage: region_plot(s>0,(s,-2,2),(t,-2,2))
"""
from sage.plot.plot import Graphics
from sage.plot.misc import setup_for_eval_on_grid
import numpy
if not isinstance(f, (list, tuple)):
f = [f]
f = [equify(g) for g in f]
g, ranges = setup_for_eval_on_grid(f, [xrange, yrange], plot_points)
xrange,yrange=[r[:2] for r in ranges]
xy_data_arrays = numpy.asarray([[[func(x, y) for x in xsrange(*ranges[0], include_endpoint=True)]
for y in xsrange(*ranges[1], include_endpoint=True)]
for func in g],dtype=float)
xy_data_array=numpy.abs(xy_data_arrays.prod(axis=0))
# Now we need to set entries to negative iff all
# functions were negative at that point.
neg_indices = (xy_data_arrays<0).all(axis=0)
xy_data_array[neg_indices]=-xy_data_array[neg_indices]
from matplotlib.colors import ListedColormap
incol = rgbcolor(incol)
outcol = rgbcolor(outcol)
cmap = ListedColormap([incol, outcol])
cmap.set_over(outcol)
cmap.set_under(incol)
g = Graphics()
g._set_extra_kwds(Graphics._extract_kwds_for_show(options, ignore=['xmin', 'xmax']))
g.add_primitive(ContourPlot(xy_data_array, xrange,yrange,
dict(contours=[-1e307, 0, 1e307], cmap=cmap, fill=True, **options)))
if bordercol or borderstyle or borderwidth:
cmap = [rgbcolor(bordercol)] if bordercol else ['black']
linestyles = [borderstyle] if borderstyle else None
linewidths = [borderwidth] if borderwidth else None
g.add_primitive(ContourPlot(xy_data_array, xrange, yrange,
dict(linestyles=linestyles, linewidths=linewidths,
contours=[0], cmap=[bordercol], fill=False, **options)))
return g
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, or dotted. 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
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))
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)
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)
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))
::
sage: contour_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi))
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))
::
sage: contour_plot(f, (-2, 2), (-2, 2), contours=2, cmap=[(1,0,0), (0,1,0), (0,0,1)])
::
sage: contour_plot(f, (-2, 2), (-2, 2), contours=(0.1, 1.0, 1.2, 1.4), cmap='hsv')
::
sage: contour_plot(f, (-2, 2), (-2, 2), contours=(1.0,), fill=False)
::
sage: contour_plot(x-y^2,(x,-5,5),(y,-3,3),contours=[-4,0,1])
We can change the style of the lines::
sage: contour_plot(f, (-2,2), (-2,2), fill=False, linewidths=10)
::
sage: contour_plot(f, (-2,2), (-2,2), fill=False, linestyles='dashdot')
::
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
::
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
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)
::
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
::
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
::
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
::
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
::
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
::
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
We can change the color of the labels if so desired::
sage: contour_plot(f, (-2,2), (-2,2), labels=True, label_colors='red')
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)
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), colorbar=True,colorbar_orientation='horizontal')
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[-2,-1,4],colorbar=True)
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[-2,-1,4],colorbar=True,colorbar_spacing='uniform')
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), contours=[0,2,3,6],colorbar=True,colorbar_format='%.3f')
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), labels=True,label_colors='red',contours=[0,2,3,6],colorbar=True)
::
sage: contour_plot(f, (x,-3,3), (y,-3,3), cmap='winter', contours=20, fill=False, colorbar=True)
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))
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)
::
sage: contour_plot(f, (0, pi), (0, pi)).show(axes=True) # These are equivalent
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)
TESTS:
To check that ticket 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)
"""
from sage.plot.plot import Graphics
from sage.plot.misc import setup_for_eval_on_grid
g, ranges = setup_for_eval_on_grid([f], [xrange, yrange], options['plot_points'])
g = g[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)]
g = Graphics()
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
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')
"""
from sage.rings.integer import Integer
options = self.options()
fill = options['fill']
contours = options['contours']
if options.has_key('cmap'):
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), label=options['legend_label'])
else:
CSF=subplot.contourf(self.xy_data_array, contours, cmap=cmap, extent=(x0,x1,y0,y1),extend='both', label=options['legend_label'])
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)
linestyles = options.get('linestyles',None)
if contours is None:
CS = subplot.contour(self.xy_data_array, cmap=cmap, extent=(x0,x1,y0,y1),
linewidths=linewidths, linestyles=linestyles, label=options['legend_label'])
else:
CS = subplot.contour(self.xy_data_array, contours, cmap=cmap, extent=(x0,x1,y0,y1),
linewidths=linewidths, linestyles=linestyles, label=options['legend_label'])
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)
def region_plot(f, xrange, yrange, plot_points, incol, outcol, bordercol, borderstyle, borderwidth,**options):
r"""
``region_plot`` takes a boolean function of two variables, `f(x,y)`
and plots the region where f is True over the specified
``xrange`` and ``yrange`` as demonstrated below.
``region_plot(f, (xmin, xmax), (ymin, ymax), ...)``
INPUT:
- ``f`` -- a boolean 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)``
- ``plot_points`` -- integer (default: 100); number of points to plot
in each direction of the grid
- ``incol`` -- a color (default: ``'blue'``), the color inside the region
- ``outcol`` -- a color (default: ``'white'``), the color of the outside
of the region
If any of these options are specified, the border will be shown as indicated,
otherwise it is only implicit (with color ``incol``) as the border of the
inside of the region.
- ``bordercol`` -- a color (default: ``None``), the color of the border
(``'black'`` if ``borderwidth`` or ``borderstyle`` is specified but not ``bordercol``)
- ``borderstyle`` -- string (default: 'solid'), one of 'solid', 'dashed', 'dotted', 'dashdot'
- ``borderwidth`` -- integer (default: None), the width of the border in pixels
- ``legend_label`` -- the label for this item in the legend
EXAMPLES:
Here we plot a simple function of two variables::
sage: x,y = var('x,y')
sage: region_plot(cos(x^2+y^2) <= 0, (x, -3, 3), (y, -3, 3))
Here we play with the colors::
sage: region_plot(x^2+y^3 < 2, (x, -2, 2), (y, -2, 2), incol='lightblue', bordercol='gray')
An even more complicated plot, with dashed borders::
sage: region_plot(sin(x)*sin(y) >= 1/4, (x,-10,10), (y,-10,10), incol='yellow', bordercol='black', borderstyle='dashed', plot_points=250)
A disk centered at the origin::
sage: region_plot(x^2+y^2<1, (x,-1,1), (y,-1,1))
A plot with more than one condition (all conditions must be true for the statement to be true)::
sage: region_plot([x^2+y^2<1, x<y], (x,-2,2), (y,-2,2))
Since it doesn't look very good, let's increase plot_points::
sage: region_plot([x^2+y^2<1, x<y], (x,-2,2), (y,-2,2), plot_points=400)
To get plots where only one condition needs to be true, use a function::
sage: region_plot(lambda x,y: x^2+y^2<1 or x<y, (x,-2,2), (y,-2,2))
The first quadrant of the unit circle::
sage: region_plot([y>0, x>0, x^2+y^2<1], (x,-1.1, 1.1), (y,-1.1, 1.1), plot_points = 400)
Here is another plot, with a huge border::
sage: region_plot(x*(x-1)*(x+1)+y^2<0, (x, -3, 2), (y, -3, 3), incol='lightblue', bordercol='gray', borderwidth=10, plot_points=50)
If we want to keep only the region where x is positive::
sage: region_plot([x*(x-1)*(x+1)+y^2<0, x>-1], (x, -3, 2), (y, -3, 3), incol='lightblue', plot_points=50)
Here we have a cut circle::
sage: region_plot([x^2+y^2<4, x>-1], (x, -2, 2), (y, -2, 2), incol='lightblue', bordercol='gray', plot_points=200)
The first variable range corresponds to the horizontal axis and
the second variable range corresponds to the vertical axis::
sage: s,t=var('s,t')
sage: region_plot(s>0,(t,-2,2),(s,-2,2))
::
sage: region_plot(s>0,(s,-2,2),(t,-2,2))
"""
from sage.plot.plot import Graphics
from sage.plot.misc import setup_for_eval_on_grid
import numpy
if not isinstance(f, (list, tuple)):
f = [f]
f = [equify(g) for g in f]
g, ranges = setup_for_eval_on_grid(f, [xrange, yrange], plot_points)
xrange,yrange=[r[:2] for r in ranges]
xy_data_arrays = numpy.asarray([[[func(x, y) for x in xsrange(*ranges[0], include_endpoint=True)]
for y in xsrange(*ranges[1], include_endpoint=True)]
for func in g],dtype=float)
xy_data_array=numpy.abs(xy_data_arrays.prod(axis=0))
# Now we need to set entries to negative iff all
# functions were negative at that point.
neg_indices = (xy_data_arrays<0).all(axis=0)
xy_data_array[neg_indices]=-xy_data_array[neg_indices]
from matplotlib.colors import ListedColormap
incol = rgbcolor(incol)
outcol = rgbcolor(outcol)
cmap = ListedColormap([incol, outcol])
cmap.set_over(outcol)
cmap.set_under(incol)
g = Graphics()
g._set_extra_kwds(Graphics._extract_kwds_for_show(options, ignore=['xmin', 'xmax']))
g.add_primitive(ContourPlot(xy_data_array, xrange,yrange,
dict(contours=[-1e307, 0, 1e307], cmap=cmap, fill=True, **options)))
if bordercol or borderstyle or borderwidth:
cmap = [rgbcolor(bordercol)] if bordercol else ['black']
linestyles = [borderstyle] if borderstyle else None
linewidths = [borderwidth] if borderwidth else None
g.add_primitive(ContourPlot(xy_data_array, xrange, yrange,
dict(linestyles=linestyles, linewidths=linewidths,
contours=[0], cmap=[bordercol], fill=False, **options)))
return g
def density_plot(f, xrange, yrange, **options):
r"""
``density_plot`` takes a function of two variables, `f(x,y)`
and plots the height of of the function over the specified
``xrange`` and ``yrange`` as demonstrated below.
``density_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: 25); number of points to plot
in each direction of the grid
- ``cmap`` -- a colormap (type ``cmap_help()`` for more information).
- ``interpolation`` -- string (default: ``'catrom'``), the interpolation
method to use: ``'bilinear'``, ``'bicubic'``, ``'spline16'``,
``'spline36'``, ``'quadric'``, ``'gaussian'``, ``'sinc'``,
``'bessel'``, ``'mitchell'``, ``'lanczos'``, ``'catrom'``,
``'hermite'``, ``'hanning'``, ``'hamming'``, ``'kaiser'``
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: density_plot(sin(x)*sin(y), (x, -2, 2), (y, -2, 2))
Here we change the ranges and add some options; note that here
``f`` is callable (has variables declared), so we can use 2-tuple ranges::
sage: x,y = var('x,y')
sage: f(x,y) = x^2*cos(x*y)
sage: density_plot(f, (x,-10,5), (y, -5,5), interpolation='sinc', plot_points=100)
An even more complicated plot::
sage: x,y = var('x,y')
sage: density_plot(sin(x^2 + y^2)*cos(x)*sin(y), (x, -4, 4), (y, -4, 4), cmap='jet', plot_points=100)
This should show a "spotlight" right on the origin::
sage: x,y = var('x,y')
sage: density_plot(1/(x^10+y^10), (x, -10, 10), (y, -10, 10))
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: density_plot(y^2 + 1 - x^3 - x, (y,-pi,pi), (x,-pi,pi))
::
sage: density_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi))
Extra options will get passed on to show(), as long as they are valid::
sage: density_plot(log(x) + log(y), (x, 1, 10), (y, 1, 10), dpi=20)
::
sage: density_plot(log(x) + log(y), (x, 1, 10), (y, 1, 10)).show(dpi=20) # These are equivalent
"""
from sage.plot.all import Graphics
from sage.plot.misc import setup_for_eval_on_grid
g, ranges = setup_for_eval_on_grid([f], [xrange, yrange], options["plot_points"])
g = g[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)
]
g = Graphics()
g._set_extra_kwds(Graphics._extract_kwds_for_show(options, ignore=["xmin", "xmax"]))
g.add_primitive(DensityPlot(xy_data_array, xrange, yrange, options))
return g
def plot_vector_field(f_g, xrange, yrange, **options):
r"""
``plot_vector_field`` takes two functions of two variables xvar and yvar
(for instance, if the variables are `x` and `y`, take `(f(x,y), g(x,y))`)
and plots vector arrows of the function over the specified ranges, with
xrange being of xvar between xmin and xmax, and yrange similarly (see below).
``plot_vector_field((f, g), (xvar, xmin, xmax), (yvar, ymin, ymax))``
EXAMPLES:
Plot some vector fields involving sin and cos::
sage: x,y = var('x y')
sage: plot_vector_field((sin(x), cos(y)), (x,-3,3), (y,-3,3))
Graphics object consisting of 1 graphics primitive
::
sage: plot_vector_field(( y, (cos(x)-2)*sin(x)), (x,-pi,pi), (y,-pi,pi))
Graphics object consisting of 1 graphics primitive
Plot a gradient field::
sage: u,v = var('u v')
sage: f = exp(-(u^2+v^2))
sage: plot_vector_field(f.gradient(), (u,-2,2), (v,-2,2), color='blue')
Graphics object consisting of 1 graphics primitive
Plot two orthogonal vector fields::
sage: x,y = var('x,y')
sage: a=plot_vector_field((x,y), (x,-3,3),(y,-3,3),color='blue')
sage: b=plot_vector_field((y,-x),(x,-3,3),(y,-3,3),color='red')
sage: show(a+b)
We ignore function values that are infinite or NaN::
sage: x,y = var('x,y')
sage: plot_vector_field( (-x/sqrt(x^2+y^2), -y/sqrt(x^2+y^2)), (x, -10, 10), (y, -10, 10))
Graphics object consisting of 1 graphics primitive
::
sage: x,y = var('x,y')
sage: plot_vector_field( (-x/sqrt(x+y), -y/sqrt(x+y)), (x, -10, 10), (y, -10, 10))
Graphics object consisting of 1 graphics primitive
Extra options will get passed on to show(), as long as they are valid::
sage: plot_vector_field((x, y), (x, -2, 2), (y, -2, 2), xmax=10)
Graphics object consisting of 1 graphics primitive
sage: plot_vector_field((x, y), (x, -2, 2), (y, -2, 2)).show(xmax=10) # These are equivalent
"""
(f, g) = f_g
from sage.plot.all import Graphics
from sage.plot.misc import setup_for_eval_on_grid
z, ranges = setup_for_eval_on_grid([f, g], [xrange, yrange],
options['plot_points'])
f, g = z
xpos_array, ypos_array, xvec_array, yvec_array = [], [], [], []
for x in xsrange(*ranges[0], include_endpoint=True):
for y in xsrange(*ranges[1], include_endpoint=True):
xpos_array.append(x)
ypos_array.append(y)
xvec_array.append(f(x, y))
yvec_array.append(g(x, y))
import numpy
xvec_array = numpy.ma.masked_invalid(numpy.array(xvec_array, dtype=float))
yvec_array = numpy.ma.masked_invalid(numpy.array(yvec_array, dtype=float))
g = Graphics()
g._set_extra_kwds(Graphics._extract_kwds_for_show(options))
g.add_primitive(
PlotField(xpos_array, ypos_array, xvec_array, yvec_array, options))
return g
def region_plot(f, xrange, yrange, plot_points, incol, outcol, bordercol, borderstyle, borderwidth, alpha, **options):
r"""
``region_plot`` takes a boolean function of two variables, `f(x,y)`
and plots the region where f is True over the specified
``xrange`` and ``yrange`` as demonstrated below.
``region_plot(f, (xmin, xmax), (ymin, ymax), ...)``
INPUT:
- ``f`` -- a boolean function or a list of boolean functions 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)``
- ``plot_points`` -- integer (default: 100); number of points to plot
in each direction of the grid
- ``incol`` -- a color (default: ``'blue'``), the color inside the region
- ``outcol`` -- a color (default: ``None``), the color of the outside
of the region
If any of these options are specified, the border will be shown as indicated,
otherwise it is only implicit (with color ``incol``) as the border of the
inside of the region.
- ``bordercol`` -- a color (default: ``None``), the color of the border
(``'black'`` if ``borderwidth`` or ``borderstyle`` is specified but not ``bordercol``)
- ``borderstyle`` -- string (default: 'solid'), one of ``'solid'``,
``'dashed'``, ``'dotted'``, ``'dashdot'``, respectively ``'-'``,
``'--'``, ``':'``, ``'-.'``.
- ``borderwidth`` -- integer (default: None), the width of the border in pixels
- ``alpha`` -- (default: 1) How transparent the fill is. A number between 0 and 1.
- ``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:
Here we plot a simple function of two variables::
sage: x,y = var('x,y')
sage: region_plot(cos(x^2+y^2) <= 0, (x, -3, 3), (y, -3, 3))
Graphics object consisting of 1 graphics primitive
Here we play with the colors::
sage: region_plot(x^2+y^3 < 2, (x, -2, 2), (y, -2, 2), incol='lightblue', bordercol='gray')
Graphics object consisting of 2 graphics primitives
An even more complicated plot, with dashed borders::
sage: region_plot(sin(x)*sin(y) >= 1/4, (x,-10,10), (y,-10,10), incol='yellow', bordercol='black', borderstyle='dashed', plot_points=250)
Graphics object consisting of 2 graphics primitives
A disk centered at the origin::
sage: region_plot(x^2+y^2<1, (x,-1,1), (y,-1,1))
Graphics object consisting of 1 graphics primitive
A plot with more than one condition (all conditions must be true for the statement to be true)::
sage: region_plot([x^2+y^2<1, x<y], (x,-2,2), (y,-2,2))
Graphics object consisting of 1 graphics primitive
Since it doesn't look very good, let's increase ``plot_points``::
sage: region_plot([x^2+y^2<1, x<y], (x,-2,2), (y,-2,2), plot_points=400)
Graphics object consisting of 1 graphics primitive
To get plots where only one condition needs to be true, use a function.
Using lambda functions, we definitely need the extra ``plot_points``::
sage: region_plot(lambda x,y: x^2+y^2<1 or x<y, (x,-2,2), (y,-2,2), plot_points=400)
Graphics object consisting of 1 graphics primitive
The first quadrant of the unit circle::
sage: region_plot([y>0, x>0, x^2+y^2<1], (x,-1.1, 1.1), (y,-1.1, 1.1), plot_points = 400)
Graphics object consisting of 1 graphics primitive
Here is another plot, with a huge border::
sage: region_plot(x*(x-1)*(x+1)+y^2<0, (x, -3, 2), (y, -3, 3), incol='lightblue', bordercol='gray', borderwidth=10, plot_points=50)
Graphics object consisting of 2 graphics primitives
If we want to keep only the region where x is positive::
sage: region_plot([x*(x-1)*(x+1)+y^2<0, x>-1], (x, -3, 2), (y, -3, 3), incol='lightblue', plot_points=50)
Graphics object consisting of 1 graphics primitive
Here we have a cut circle::
sage: region_plot([x^2+y^2<4, x>-1], (x, -2, 2), (y, -2, 2), incol='lightblue', bordercol='gray', plot_points=200)
Graphics object consisting of 2 graphics primitives
The first variable range corresponds to the horizontal axis and
the second variable range corresponds to the vertical axis::
sage: s,t=var('s,t')
sage: region_plot(s>0,(t,-2,2),(s,-2,2))
Graphics object consisting of 1 graphics primitive
::
sage: region_plot(s>0,(s,-2,2),(t,-2,2))
Graphics object consisting of 1 graphics primitive
An example of a region plot in 'loglog' scale::
sage: region_plot(x^2+y^2<100, (x,1,10), (y,1,10), scale='loglog')
Graphics object consisting of 1 graphics primitive
TESTS:
To check that :trac:`16907` is fixed::
sage: x, y = var('x, y')
sage: disc1 = region_plot(x^2+y^2 < 1, (x, -1, 1), (y, -1, 1), alpha=0.5)
sage: disc2 = region_plot((x-0.7)^2+(y-0.7)^2 < 0.5, (x, -2, 2), (y, -2, 2), incol='red', alpha=0.5)
sage: disc1 + disc2
Graphics object consisting of 2 graphics primitives
To check that :trac:`18286` is fixed::
sage: x, y = var('x, y')
sage: region_plot([x == 0], (x, -1, 1), (y, -1, 1))
Graphics object consisting of 1 graphics primitive
sage: region_plot([x^2+y^2==1, x<y], (x, -1, 1), (y, -1, 1))
Graphics object consisting of 1 graphics primitive
"""
from sage.plot.all import Graphics
from sage.plot.misc import setup_for_eval_on_grid
from sage.symbolic.expression import is_Expression
from warnings import warn
import numpy
if not isinstance(f, (list, tuple)):
f = [f]
feqs = [equify(g) for g in f if is_Expression(g) and g.operator() is operator.eq and not equify(g).is_zero()]
f = [equify(g) for g in f if not (is_Expression(g) and g.operator() is operator.eq)]
neqs = len(feqs)
if neqs > 1:
warn("There are at least 2 equations; If the region is degenerated to points, plotting might show nothing.")
feqs = [sum([fn**2 for fn in feqs])]
neqs = 1
if neqs and not bordercol:
bordercol = incol
if not f:
return implicit_plot(feqs[0], xrange, yrange, plot_points=plot_points, fill=False, \
linewidth=borderwidth, linestyle=borderstyle, color=bordercol, **options)
f_all, ranges = setup_for_eval_on_grid(feqs + f, [xrange, yrange], plot_points)
xrange,yrange=[r[:2] for r in ranges]
xy_data_arrays = numpy.asarray([[[func(x, y) for x in xsrange(*ranges[0], include_endpoint=True)]
for y in xsrange(*ranges[1], include_endpoint=True)]
for func in f_all[neqs::]],dtype=float)
xy_data_array=numpy.abs(xy_data_arrays.prod(axis=0))
# Now we need to set entries to negative iff all
# functions were negative at that point.
neg_indices = (xy_data_arrays<0).all(axis=0)
xy_data_array[neg_indices]=-xy_data_array[neg_indices]
from matplotlib.colors import ListedColormap
incol = rgbcolor(incol)
if outcol:
outcol = rgbcolor(outcol)
cmap = ListedColormap([incol, outcol])
cmap.set_over(outcol, alpha=alpha)
else:
outcol = rgbcolor('white')
cmap = ListedColormap([incol, outcol])
cmap.set_over(outcol, alpha=0)
cmap.set_under(incol, alpha=alpha)
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']))
if neqs == 0:
g.add_primitive(ContourPlot(xy_data_array, xrange,yrange,
dict(contours=[-1e-20, 0, 1e-20], cmap=cmap, fill=True, **options)))
else:
mask = numpy.asarray([[elt > 0 for elt in rows] for rows in xy_data_array], dtype=bool)
xy_data_array = numpy.asarray([[f_all[0](x, y) for x in xsrange(*ranges[0], include_endpoint=True)]
for y in xsrange(*ranges[1], include_endpoint=True)], dtype=float)
xy_data_array[mask] = None
if bordercol or borderstyle or borderwidth:
cmap = [rgbcolor(bordercol)] if bordercol else ['black']
linestyles = [borderstyle] if borderstyle else None
linewidths = [borderwidth] if borderwidth else None
g.add_primitive(ContourPlot(xy_data_array, xrange, yrange,
dict(linestyles=linestyles, linewidths=linewidths,
contours=[0], cmap=[bordercol], fill=False, **options)))
return g
def region_plot(f, xrange, yrange, plot_points, incol, outcol, bordercol,
borderstyle, borderwidth, **options):
r"""
``region_plot`` takes a boolean function of two variables, `f(x,y)`
and plots the region where f is True over the specified
``xrange`` and ``yrange`` as demonstrated below.
``region_plot(f, (xmin, xmax), (ymin, ymax), ...)``
INPUT:
- ``f`` -- a boolean 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)``
- ``plot_points`` -- integer (default: 100); number of points to plot
in each direction of the grid
- ``incol`` -- a color (default: ``'blue'``), the color inside the region
- ``outcol`` -- a color (default: ``'white'``), the color of the outside
of the region
If any of these options are specified, the border will be shown as indicated,
otherwise it is only implicit (with color ``incol``) as the border of the
inside of the region.
- ``bordercol`` -- a color (default: ``None``), the color of the border
(``'black'`` if ``borderwidth`` or ``borderstyle`` is specified but not ``bordercol``)
- ``borderstyle`` -- string (default: 'solid'), one of ``'solid'``,
``'dashed'``, ``'dotted'``, ``'dashdot'``, respectively ``'-'``,
``'--'``, ``':'``, ``'-.'``.
- ``borderwidth`` -- integer (default: None), the width of the border in pixels
- ``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:
Here we plot a simple function of two variables::
sage: x,y = var('x,y')
sage: region_plot(cos(x^2+y^2) <= 0, (x, -3, 3), (y, -3, 3))
Graphics object consisting of 1 graphics primitive
Here we play with the colors::
sage: region_plot(x^2+y^3 < 2, (x, -2, 2), (y, -2, 2), incol='lightblue', bordercol='gray')
Graphics object consisting of 2 graphics primitives
An even more complicated plot, with dashed borders::
sage: region_plot(sin(x)*sin(y) >= 1/4, (x,-10,10), (y,-10,10), incol='yellow', bordercol='black', borderstyle='dashed', plot_points=250)
Graphics object consisting of 2 graphics primitives
A disk centered at the origin::
sage: region_plot(x^2+y^2<1, (x,-1,1), (y,-1,1))
Graphics object consisting of 1 graphics primitive
A plot with more than one condition (all conditions must be true for the statement to be true)::
sage: region_plot([x^2+y^2<1, x<y], (x,-2,2), (y,-2,2))
Graphics object consisting of 1 graphics primitive
Since it doesn't look very good, let's increase ``plot_points``::
sage: region_plot([x^2+y^2<1, x<y], (x,-2,2), (y,-2,2), plot_points=400)
Graphics object consisting of 1 graphics primitive
To get plots where only one condition needs to be true, use a function.
Using lambda functions, we definitely need the extra ``plot_points``::
sage: region_plot(lambda x,y: x^2+y^2<1 or x<y, (x,-2,2), (y,-2,2), plot_points=400)
Graphics object consisting of 1 graphics primitive
The first quadrant of the unit circle::
sage: region_plot([y>0, x>0, x^2+y^2<1], (x,-1.1, 1.1), (y,-1.1, 1.1), plot_points = 400)
Graphics object consisting of 1 graphics primitive
Here is another plot, with a huge border::
sage: region_plot(x*(x-1)*(x+1)+y^2<0, (x, -3, 2), (y, -3, 3), incol='lightblue', bordercol='gray', borderwidth=10, plot_points=50)
Graphics object consisting of 2 graphics primitives
If we want to keep only the region where x is positive::
sage: region_plot([x*(x-1)*(x+1)+y^2<0, x>-1], (x, -3, 2), (y, -3, 3), incol='lightblue', plot_points=50)
Graphics object consisting of 1 graphics primitive
Here we have a cut circle::
sage: region_plot([x^2+y^2<4, x>-1], (x, -2, 2), (y, -2, 2), incol='lightblue', bordercol='gray', plot_points=200)
Graphics object consisting of 2 graphics primitives
The first variable range corresponds to the horizontal axis and
the second variable range corresponds to the vertical axis::
sage: s,t=var('s,t')
sage: region_plot(s>0,(t,-2,2),(s,-2,2))
Graphics object consisting of 1 graphics primitive
::
sage: region_plot(s>0,(s,-2,2),(t,-2,2))
Graphics object consisting of 1 graphics primitive
An example of a region plot in 'loglog' scale::
sage: region_plot(x^2+y^2<100, (x,1,10), (y,1,10), scale='loglog')
Graphics object consisting of 1 graphics primitive
"""
from sage.plot.all import Graphics
from sage.plot.misc import setup_for_eval_on_grid
import numpy
if not isinstance(f, (list, tuple)):
f = [f]
f = [equify(g) for g in f]
g, ranges = setup_for_eval_on_grid(f, [xrange, yrange], plot_points)
xrange, yrange = [r[:2] for r in ranges]
xy_data_arrays = numpy.asarray(
[[[func(x, y) for x in xsrange(*ranges[0], include_endpoint=True)]
for y in xsrange(*ranges[1], include_endpoint=True)] for func in g],
dtype=float)
xy_data_array = numpy.abs(xy_data_arrays.prod(axis=0))
# Now we need to set entries to negative iff all
# functions were negative at that point.
neg_indices = (xy_data_arrays < 0).all(axis=0)
xy_data_array[neg_indices] = -xy_data_array[neg_indices]
from matplotlib.colors import ListedColormap
incol = rgbcolor(incol)
outcol = rgbcolor(outcol)
cmap = ListedColormap([incol, outcol])
cmap.set_over(outcol)
cmap.set_under(incol)
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,
dict(contours=[-1e307, 0, 1e307], cmap=cmap, fill=True,
**options)))
if bordercol or borderstyle or borderwidth:
cmap = [rgbcolor(bordercol)] if bordercol else ['black']
linestyles = [borderstyle] if borderstyle else None
linewidths = [borderwidth] if borderwidth else None
g.add_primitive(
ContourPlot(
xy_data_array, xrange, yrange,
dict(linestyles=linestyles,
linewidths=linewidths,
contours=[0],
cmap=[bordercol],
fill=False,
**options)))
return g
def plot_vector_field(f_g, xrange, yrange, **options):
r"""
``plot_vector_field`` takes two functions of two variables xvar and yvar
(for instance, if the variables are `x` and `y`, take `(f(x,y), g(x,y))`)
and plots vector arrows of the function over the specified ranges, with
xrange being of xvar between xmin and xmax, and yrange similarly (see below).
``plot_vector_field((f, g), (xvar, xmin, xmax), (yvar, ymin, ymax))``
EXAMPLES:
Plot some vector fields involving sin and cos::
sage: x,y = var('x y')
sage: plot_vector_field((sin(x), cos(y)), (x,-3,3), (y,-3,3))
Graphics object consisting of 1 graphics primitive
::
sage: plot_vector_field(( y, (cos(x)-2)*sin(x)), (x,-pi,pi), (y,-pi,pi))
Graphics object consisting of 1 graphics primitive
Plot a gradient field::
sage: u,v = var('u v')
sage: f = exp(-(u^2+v^2))
sage: plot_vector_field(f.gradient(), (u,-2,2), (v,-2,2), color='blue')
Graphics object consisting of 1 graphics primitive
Plot two orthogonal vector fields::
sage: x,y = var('x,y')
sage: a=plot_vector_field((x,y), (x,-3,3),(y,-3,3),color='blue')
sage: b=plot_vector_field((y,-x),(x,-3,3),(y,-3,3),color='red')
sage: show(a+b)
We ignore function values that are infinite or NaN::
sage: x,y = var('x,y')
sage: plot_vector_field( (-x/sqrt(x^2+y^2), -y/sqrt(x^2+y^2)), (x, -10, 10), (y, -10, 10))
Graphics object consisting of 1 graphics primitive
::
sage: x,y = var('x,y')
sage: plot_vector_field( (-x/sqrt(x+y), -y/sqrt(x+y)), (x, -10, 10), (y, -10, 10))
Graphics object consisting of 1 graphics primitive
Extra options will get passed on to show(), as long as they are valid::
sage: plot_vector_field((x, y), (x, -2, 2), (y, -2, 2), xmax=10)
Graphics object consisting of 1 graphics primitive
sage: plot_vector_field((x, y), (x, -2, 2), (y, -2, 2)).show(xmax=10) # These are equivalent
"""
(f, g) = f_g
from sage.plot.all import Graphics
from sage.plot.misc import setup_for_eval_on_grid
z, ranges = setup_for_eval_on_grid([f,g], [xrange, yrange], options['plot_points'])
f,g = z
xpos_array, ypos_array, xvec_array, yvec_array = [],[],[],[]
for x in xsrange(*ranges[0], include_endpoint=True):
for y in xsrange(*ranges[1], include_endpoint=True):
xpos_array.append(x)
ypos_array.append(y)
xvec_array.append(f(x,y))
yvec_array.append(g(x,y))
import numpy
xvec_array = numpy.ma.masked_invalid(numpy.array(xvec_array, dtype=float))
yvec_array = numpy.ma.masked_invalid(numpy.array(yvec_array, dtype=float))
g = Graphics()
g._set_extra_kwds(Graphics._extract_kwds_for_show(options))
g.add_primitive(PlotField(xpos_array, ypos_array, xvec_array, yvec_array, options))
return g
sage: x,y = var('x,y')
sage: plot_vector_field( (-x/sqrt(x+y), -y/sqrt(x+y)), (x, -10, 10), (y, -10, 10))
Extra options will get passed on to show(), as long as they are valid::
sage: plot_vector_field((x, y), (x, -2, 2), (y, -2, 2), xmax=10)
sage: plot_vector_field((x, y), (x, -2, 2), (y, -2, 2)).show(xmax=10) # These are equivalent
"""
from sage.plot.plot import Graphics
from sage.plot.misc import setup_for_eval_on_grid
z, ranges = setup_for_eval_on_grid([f, g], [xrange, yrange],
options['plot_points'])
f, g = z
xpos_array, ypos_array, xvec_array, yvec_array = [], [], [], []
for x in xsrange(*ranges[0], include_endpoint=True):
for y in xsrange(*ranges[1], include_endpoint=True):
xpos_array.append(x)
ypos_array.append(y)
xvec_array.append(f(x, y))
yvec_array.append(g(x, y))
import numpy
xvec_array = numpy.ma.masked_invalid(numpy.array(xvec_array, dtype=float))
yvec_array = numpy.ma.masked_invalid(numpy.array(yvec_array, dtype=float))
g = Graphics()
g._set_extra_kwds(Graphics._extract_kwds_for_show(options))
g.add_primitive(
PlotField(xpos_array, ypos_array, xvec_array, yvec_array, options))
return g
from sage.misc.decorators import options
@options(plot_points=20)
def plot_vector_field_on_curve( (xf, yf), (x, y), range, **options ):
r"""Plot values of a vector-values function along points of a curve
in the plane.
Note this function doesn't plot the curve itself."""
from sage.plot.all import Graphics
from sage.misc.misc import xsrange
from sage.plot.plot_field import PlotField
from sage.plot.misc import setup_for_eval_on_grid
zz, rangez = setup_for_eval_on_grid( (x, y, xf, yf), [ range ], options['plot_points'] )
#print 'setup: ', zz, rangez
x, y, xf, yf = zz
xpos_array, ypos_array, xvec_array, yvec_array = [],[],[],[]
for t in xsrange( *rangez[0], include_endpoint=True ):
xpos_array.append( x(t) )
ypos_array.append( y(t) )
xvec_array.append( xf(t) )
yvec_array.append( yf(t) )
import numpy
xvec_array = numpy.ma.masked_invalid(numpy.array(xvec_array, dtype=float))
yvec_array = numpy.ma.masked_invalid(numpy.array(yvec_array, dtype=float))
g = Graphics()
g._set_extra_kwds(Graphics._extract_kwds_for_show(options))
g.add_primitive(PlotField(xpos_array, ypos_array, xvec_array, yvec_array, options))
return g
def density_plot(f, xrange, yrange, **options):
r"""
``density_plot`` takes a function of two variables, `f(x,y)`
and plots the height of of the function over the specified
``xrange`` and ``yrange`` as demonstrated below.
``density_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: 25); number of points to plot
in each direction of the grid
- ``cmap`` -- a colormap (type ``cmap_help()`` for more information).
- ``interpolation`` -- string (default: ``'catrom'``), the interpolation
method to use: ``'bilinear'``, ``'bicubic'``, ``'spline16'``,
``'spline36'``, ``'quadric'``, ``'gaussian'``, ``'sinc'``,
``'bessel'``, ``'mitchell'``, ``'lanczos'``, ``'catrom'``,
``'hermite'``, ``'hanning'``, ``'hamming'``, ``'kaiser'``
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: density_plot(sin(x)*sin(y), (x, -2, 2), (y, -2, 2))
Here we change the ranges and add some options; note that here
``f`` is callable (has variables declared), so we can use 2-tuple ranges::
sage: x,y = var('x,y')
sage: f(x,y) = x^2*cos(x*y)
sage: density_plot(f, (x,-10,5), (y, -5,5), interpolation='sinc', plot_points=100)
An even more complicated plot::
sage: x,y = var('x,y')
sage: density_plot(sin(x^2 + y^2)*cos(x)*sin(y), (x, -4, 4), (y, -4, 4), cmap='jet', plot_points=100)
This should show a "spotlight" right on the origin::
sage: x,y = var('x,y')
sage: density_plot(1/(x^10+y^10), (x, -10, 10), (y, -10, 10))
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: density_plot(y^2 + 1 - x^3 - x, (y,-pi,pi), (x,-pi,pi))
::
sage: density_plot(y^2 + 1 - x^3 - x, (x,-pi,pi), (y,-pi,pi))
Extra options will get passed on to show(), as long as they are valid::
sage: density_plot(log(x) + log(y), (x, 1, 10), (y, 1, 10), dpi=20)
::
sage: density_plot(log(x) + log(y), (x, 1, 10), (y, 1, 10)).show(dpi=20) # These are equivalent
"""
from sage.plot.all import Graphics
from sage.plot.misc import setup_for_eval_on_grid
g, ranges = setup_for_eval_on_grid([f], [xrange, yrange],
options['plot_points'])
g = g[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)]
g = Graphics()
g._set_extra_kwds(
Graphics._extract_kwds_for_show(options, ignore=['xmin', 'xmax']))
g.add_primitive(DensityPlot(xy_data_array, xrange, yrange, options))
return g
sage: x,y = var('x,y')
sage: plot_vector_field( (-x/sqrt(x+y), -y/sqrt(x+y)), (x, -10, 10), (y, -10, 10))
Extra options will get passed on to show(), as long as they are valid::
sage: plot_vector_field((x, y), (x, -2, 2), (y, -2, 2), xmax=10)
sage: plot_vector_field((x, y), (x, -2, 2), (y, -2, 2)).show(xmax=10) # These are equivalent
"""
from sage.plot.plot import Graphics
from sage.plot.misc import setup_for_eval_on_grid
z, ranges = setup_for_eval_on_grid([f,g], [xrange, yrange], options['plot_points'])
f,g = z
xpos_array, ypos_array, xvec_array, yvec_array = [],[],[],[]
for x in xsrange(*ranges[0], include_endpoint=True):
for y in xsrange(*ranges[1], include_endpoint=True):
xpos_array.append(x)
ypos_array.append(y)
xvec_array.append(f(x,y))
yvec_array.append(g(x,y))
import numpy
xvec_array = numpy.ma.masked_invalid(numpy.array(xvec_array, dtype=float))
yvec_array = numpy.ma.masked_invalid(numpy.array(yvec_array, dtype=float))
g = Graphics()
g._set_extra_kwds(Graphics._extract_kwds_for_show(options))
g.add_primitive(PlotField(xpos_array, ypos_array, xvec_array, yvec_array, options))
return g
def plot_slope_field(f, xrange, yrange, **kwds):
