Bases: sage.plot.primitive.GraphicPrimitive
Primitive class for the contour plot graphics type. See contour_plot? for help actually doing contour plots.
INPUT:
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
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
contour_plot takes a function of two variables,
and plots contour lines of the function over the specified
xrange and yrange as demonstrated below.
contour_plot(f, (xmin, xmax), (ymin, ymax), ...)
INPUT:
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.
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.
legend_label – the label for this item in the legend
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 ,
:
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
Returns the equation rewritten as a symbolic function to give negative values when True, positive when False.
EXAMPLES:
sage: from sage.plot.contour_plot import equify
sage: var('x, y')
(x, y)
sage: equify(x^2 < 2)
x^2 - 2
sage: equify(x^2 > 2)
-x^2 + 2
sage: equify(x*y > 1)
-x*y + 1
sage: equify(y > 0)
-y
sage: f=equify(lambda x,y: x>y)
sage: f(1,2)
1
sage: f(2,1)
-1
implicit_plot takes a function of two variables,
and plots the curve
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:
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
.
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 sage.plot.colors; try colors? to see them all.
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 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
I can do the same thing, but using a callable function so I don’t need to explicitly define the variables in the ranges, and filling the inside:
sage: f(x,y) = x^2 + y^2 - 2
sage: implicit_plot(f, (-3, 3), (-3, 3),fill=True)
Graphics object consisting of 1 graphics primitive
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
And 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
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
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
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.show(frame=False)
We can define a level- 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
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
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
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 1 graphics primitive
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
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
region_plot takes a boolean function of two variables,
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:
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 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