Tutorial for Advanced 2d Plotting

This Sage document is one of the tutorials developed for the MAA PREP Workshop “Sage: Using Open-Source Mathematics Software with Undergraduates” (funding provided by NSF DUE 0817071). It is licensed under the Creative Commons Attribution-ShareAlike 3.0 license (CC BY-SA).

Thanks to Sage’s integration of projects like matplotlib, Sage has comprehensive two-dimensional plotting capabilities. This worksheet consists of the following sections:

This tutorial assumes that one is familiar with the basics of Sage, such as evaluating a cell by clicking the “evaluate” link, or by pressing Shift-Enter (hold down Shift while pressing the Enter key).

Graphing Functions and Plotting Curves

Cartesian Plots

A simple quadratic is easy.

sage: plot(x^2, (x,-2,2))

You can combine “plot objects” by adding them.

sage: regular = plot(x^2, (x,-2,2), color= 'purple')
sage: skinny = plot(4*x^2, (x,-2,2), color = 'green')
sage: regular + skinny

Problem : Plot a green \(y=\sin(x)\) together with a red \(y=2\,\cos(x)\). (Hint: you can use pi as part of your range.)

Boundaries of a plot can be specified, in addition to the overall size.

sage: plot(1+e^(-x^2), xmin=-2, xmax=2, ymin=0, ymax=2.5, figsize=10)

Problem : Plot \(y=5+3\,\sin(4x)\) with suitable boundaries.

You can add lots of extra information.

sage: exponential = plot(1+e^(-x^2), xmin=-2, xmax=2, ymin=0, ymax=2.5)
sage: max_line = plot(2, xmin=-2, xmax=2, linestyle='-.', color = 'red')
sage: min_line = plot(1, xmin=-2, xmax=2, linestyle=':', color = 'red')
sage: exponential + max_line + min_line

You can fill regions with transparent color, and thicken the curve. This example uses several options to fine-tune our graphic.

sage: exponential = plot(1+e^(-x^2), xmin=-2, xmax=2, ymin=0, ymax=2.5, fill=0.5, fillcolor='grey', fillalpha=0.3)
sage: min_line = plot(1, xmin=-2, xmax=2, linestyle='-', thickness= 6, color = 'red')
sage: exponential + min_line
sage: sum([plot(x^n,(x,0,1),color=rainbow(5)[n]) for n in [0..4]])

Problem : Create a plot showing the cross-section area for the following solid of revolution problem: Consider the area bounded by \(y=x^2-3x+6\) and the line \(y=4\). Find the volume created by rotating this area around the line \(y=1\).

Parametric Plots

A parametric plot needs a list of two functions of the parameter; in Sage, we use square brackets to delimit the list. Notice also that we must declare t as a variable first. Because the graphic is slightly wider than it is tall, we use the aspect_ratio option (such options are called keywords ) to ensure the axes are correct for how we want to view this object.

sage: t = var('t')
sage: parametric_plot([cos(t) + 3 * cos(t/9), sin(t) - 3 * sin(t/9)], (t, 0, 18*pi), fill = True, aspect_ratio=1)

Problem : These parametric equations will create a hypocycloid.

\[x(t)=17\cos(t)+3\cos(17t/3)\]
\[y(t)=17\sin(t)-3\sin(17t/3)\]

Create this as a parametric plot.

Sage automatically plots a 2d or 3d plot, and a curve or a surface, depending on how many variables and coordinates you specify.

sage: t = var('t')
sage: parametric_plot((t^2,sin(t)), (t,0,pi))
sage: parametric_plot((t^2,sin(t),cos(t)), (t,0,pi))
sage: r = var('r')
sage: parametric_plot((t^2,sin(r*t),cos(r*t)), (t,0,pi),(r,-1,1))

Polar Plots

Sage can also do polar plots.

sage: polar_plot(2 + 2*cos(x), (x, 0, 2*pi), color=hue(0.5), thickness=4)

Although they aren’t essential, many of these examples try to demonstrate things like coloring, fills, and shading to give you a sense of the possibilities.

More than one polar curve can be specified in a list (square brackets). Notice the automatic graded shading of the fill color.

sage: t = var('t')
sage: polar_plot([cos(4*t) + 1.5,  0.5 * cos(4*t) + 2.5], (t, 0, 2*pi),\
...     color='black', thickness=2, fill=True, fillcolor='orange')

Problem: Create a plot for the following problem. Find the area that is inside the circle \(r=2\), but outside the cardiod \(2+2\cos(\theta)\).

Interactive Demonstration

It may be of interest to see all these things put together in a very nice pedagogical graphic. Even though this is fairly advanced, and so you may want to skip the code, it is not as difficult as you might think to put together.

sage: html('<h2>Sine and unit circle (by Jurgis Pralgauskis)</h2> inspired by <a href="http://www.youtube.com/watch?v=Ohp6Okk_tww&feature=related">this video</a>' )
sage: # http://www.sagemath.org/doc/reference/sage/plot/plot.html
sage: radius = 100 # scale for radius of "unit" circle
sage: graph_params = dict(xmin = -2*radius,    xmax = 360,
...                      ymin = -(radius+30), ymax = radius+30,
...                      aspect_ratio=1,
...                      axes = False
...                      )
sage: def sine_and_unit_circle( angle=30, instant_show = True, show_pi=True ):
...      ccenter_x, ccenter_y = -radius, 0  # center of cirlce on real coords
...      sine_x = angle # the big magic to sync both graphs :)
...      current_y = circle_y = sine_y = radius * sin(angle*pi/180)
...      circle_x = ccenter_x + radius * cos(angle*pi/180)
...      graph = Graphics()
...      # we'll put unit circle and sine function on the same graph
...      # so there will be some coordinate mangling ;)
...      # CIRCLE
...      unit_circle = circle((ccenter_x, ccenter_y), radius, color="#ccc")
...      # SINE
...      x = var('x')
...      sine = plot( radius * sin(x*pi/180) , (x, 0, 360), color="#ccc" )
...      graph += unit_circle + sine
...      # CIRCLE axis
...      # x axis
...      graph +=  arrow( [-2*radius, 0], [0, 0], color = "#666" )
...      graph += text("$(1, 0)$",  [-16, 16],  color = "#666")
...      # circle y axis
...      graph +=  arrow( [ccenter_x,-radius], [ccenter_x, radius], color = "#666" )
...      graph += text("$(0, 1)$",  [ccenter_x, radius+15],  color = "#666")
...      # circle center
...      graph += text("$(0, 0)$",  [ccenter_x, 0],  color = "#666")
...      # SINE x axis
...      graph +=  arrow( [0,0], [360, 0], color = "#000" )
...      # let's set tics
...      # or http://aghitza.org/posts/tweak_labels_and_ticks_in_2d_plots_using_matplotlib/
...      # or wayt for http://trac.sagemath.org/sage_trac/ticket/1431
...      # ['$-\pi/3$', '$2\pi/3$', '$5\pi/3$']
...      for x in range(0, 361, 30):
...          graph += point( [x, 0] )
...          angle_label = ".  $%3d^{\circ}$ " % x
...          if show_pi: angle_label += " $(%s\pi) $"% x/180
...          graph += text(angle_label,  [x, 0], rotation=-90,
...          vertical_alignment='top', fontsize=8, color="#000" )
...      # CURRENT VALUES
...      # SINE -- y
...      graph +=  arrow( [sine_x,0], [sine_x, sine_y], width=1, arrowsize=3)
...      graph +=  arrow( [circle_x,0], [circle_x, circle_y], width=1, arrowsize=3)
...      graph +=  line(([circle_x, current_y], [sine_x, current_y]), rgbcolor="#0F0", linestyle = "--", alpha=0.5)
...      # LABEL on sine
...      graph += text("$(%d^{\circ}, %3.2f)$"%(sine_x, float(current_y)/radius),  [sine_x+30, current_y],  color = "#0A0")
...      # ANGLE -- x
...      # on sine
...      graph += arrow( [0,0], [sine_x, 0], width=1, arrowsize=1, color='red')
...      # on circle
...      graph += disk( (ccenter_x, ccenter_y), float(radius)/4, (0, angle*pi/180), color='red', fill=False, thickness=1)
...      graph +=  arrow( [ccenter_x, ccenter_y], [circle_x, circle_y],
...                    rgbcolor="#cccccc", width=1, arrowsize=1)
...      if instant_show:
...          show (graph,  **graph_params)
...      return graph
sage: #####################
sage: # make Interaction
sage: ######################
sage: @interact
sage: def _( angle = slider([0..360], default=30, step_size=5,
...            label="Pasirinkite kampą:    ", display_value=True) ):
...       sine_and_unit_circle(angle, show_pi = False)

Plotting Data

Plotting Data Points

Sometimes one wishes to simply plot data. Here, we demonstrate several ways of plotting points and data via the simple approximation to the Fibonacci numbers given by

\[F_n=\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right)^n\; ,\]

which is quite good after about \(n=5\).

First, we notice that the Fibonacci numbers are built in.

sage: fibonacci_sequence(6)
<generator object fibonacci_sequence at ...>
sage: list(fibonacci_sequence(6))
[0, 1, 1, 2, 3, 5]

The enumerate command is useful for taking a list and coordinating it with the counting numbers.

sage: list(enumerate(fibonacci_sequence(6)))
[(0, 0), (1, 1), (2, 1), (3, 2), (4, 3), (5, 5)]

So we just define the numbers and coordinate pairs we are about to plot.

sage: fibonacci = list(enumerate(fibonacci_sequence(6)))
sage: f(n)=(1/sqrt(5))*((1+sqrt(5))/2)^n
sage: asymptotic = [(i, f(i)) for i in range(6)]
sage: fibonacci
[(0, 0), (1, 1), (2, 1), (3, 2), (4, 3), (5, 5)]
sage: asymptotic
[(0, 1/5*sqrt(5)), (1, 1/10*sqrt(5)*(sqrt(5) + 1)), (2, 1/20*sqrt(5)*(sqrt(5) + 1)^2), (3, 1/40*sqrt(5)*(sqrt(5) + 1)^3), (4, 1/80*sqrt(5)*(sqrt(5) + 1)^4), (5, 1/160*sqrt(5)*(sqrt(5) + 1)^5)]

Now we can plot not just the two sets of points, but also use several of the documented options for plotting points. Those coming from other systems may prefer list_plot.

sage: fib_plot=list_plot(fibonacci, color='red', pointsize=30)
sage: asy_plot = list_plot(asymptotic, marker='D',color='black',thickness=2,plotjoined=True)
sage: show(fib_plot+asy_plot, aspect_ratio=1)

Other options include line, points, and scatter_plot. Having the choice of markers for different data is particularly helpful for generating publishable graphics.

sage: fib_plot=scatter_plot(fibonacci, facecolor='red', marker='o',markersize=40)
sage: asy_plot = line(asymptotic, marker='D',color='black',thickness=2)
sage: show(fib_plot+asy_plot, aspect_ratio=1)

Contour-type Plots

Contour Plots

Contour plotting can be very useful when trying to get a handle on multivariable functions, as well as modeling. The basic syntax is essentially the same as for 3D plotting - simply an extension of the 2D plotting syntax.

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

We can change colors, specify contours, label curves, and many other things. When there are many levels, the colorbar keyword becomes quite useful for keeping track of them. Notice that, as opposed to many other options, it can only be True or False (corresponding to whether it appears or does not appear).

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

This example is fairly self-explanatory, but demonstrates the power of formatting, labeling, and the wide variety of built-in color gradations (colormaps or cmap). The strange-looking construction corresponding to label_fmt is a Sage/Python data type called a dictionary , and turns out to be useful for more advanced Sage use; it consists of pairs connected by a colon, all inside curly braces.

sage: contour_plot(f, (x,-pi,pi), (y,-pi,pi), contours=[-4,0,4], fill=False,\
...     cmap='cool', labels=True, label_inline=True, label_fmt={-4:"low", 0:"medium", 4: "hi"}, label_colors='black')

Implicit plots are a special type of contour plot (they just plot the zero contour).

sage: f(x,y)
-x^3 + y^2 - x + 1
sage: implicit_plot(f(x,y)==0,(x,-pi,pi),(y,-pi,pi))

A density plot is like a contour plot, but without discrete levels.

sage: density_plot(f, (x, -2, 2), (y, -2, 2))

Sometimes contour plots can be a little misleading (which makes for a great classroom discussion about the problems of ignorantly relying on technology). Here we combine a density plot and contour plot to show even better what is happening with the function.

sage: density_plot(f,(x,-2,2),(y,-2,2))+contour_plot(f,(x,-2,2),(y,-2,2),fill=False,labels=True,label_inline=True,cmap='jet')

It can be worth getting familiar with the various options for different plots, especially if you will be doing a lot of them in a given worksheet or pedagogical situation.

Here are the options for contour plots.

  • They are given as an “attribute” - no parentheses - of the contour_plot object.
  • They are given as a dictionary (see the programming tutorial).
sage: contour_plot.options
{'labels': False, 'linestyles': None, 'region': None, 'axes': False, 'plot_points': 100, 'linewidths': None, 'colorbar': False, 'contours': None, 'aspect_ratio': 1, 'legend_label': None, 'frame': True, 'fill': True}

Let’s change it so that all future contour plots don’t have the fill. That’s how some of us might use them in a class. We’ll also check that the change happened.

sage: contour_plot.options["fill"]=False
sage: contour_plot.options
{'labels': False, 'linestyles': None, 'region': None, 'axes': False, 'plot_points': 100, 'linewidths': None, 'colorbar': False, 'contours': None, 'aspect_ratio': 1, 'legend_label': None, 'frame': True, 'fill': False}

And it works!

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

We can always access the default options, of course, to remind us.

sage: contour_plot.defaults()
{'labels': False, 'linestyles': None, 'region': None, 'axes': False, 'plot_points': 100, 'linewidths': None, 'colorbar': False, 'contours': None, 'aspect_ratio': 1, 'legend_label': None, 'frame': True, 'fill': True}

Vector fields

The syntax for vector fields is very similar to other multivariate constructions. Notice that the arrows are scaled appropriately, and colored by length in the 3D case.

sage: var('x,y')
(x, y)
sage: plot_vector_field((-y+x,y*x),(x,-3,3),(y,-3,3))
sage: var('x,y,z')
(x, y, z)
sage: plot_vector_field3d((-y,-z,x), (x,-3,3),(y,-3,3),(z,-3,3))

3d vector field plots are ideally viewed with 3d glasses (right-click on the plot and select “Style” and “Stereographic”)

Complex Plots

We can plot functions of complex variables, where the magnitude is indicated by the brightness (black is zero magnitude) and the argument is indicated by the hue (red is a positive real number).

sage: f(z) = exp(z) #z^5 + z - 1 + 1/z
sage: complex_plot(f, (-5,5),(-5,5))

Region plots

These plot where an expression is true, and are useful for plotting inequalities.

sage: region_plot(cos(x^2+y^2) <= 0, (x, -3, 3), (y, -3, 3),aspect_ratio=1)

We can get fancier options as well.

sage: region_plot(sin(x)*sin(y) >= 1/4, (x,-10,10), (y,-10,10), incol='yellow', bordercol='black', borderstyle='dashed', plot_points=250,aspect_ratio=1)

Remember, what command would give full information about the syntax, options, and examples?

Miscellaneous Plot Information

Builtin Graphics Objects

Sage includes a variety of built-in graphics objects. These are particularly useful for adding to one’s plot certain objects which are difficult to describe with equations, but which are basic geometric objects nonetheless. In this section we will try to demonstrate the syntax of some of the most useful of them; for most of the the contextual (remember, append ?) help will give more details.

Points

To make one point, a coordinate pair suffices.

sage: point((3,5))

It doesn’t matter how multiple point are generated; they must go in as input via a list (square brackets). Here, we demonstrate the hard (but naive) and easy (but a little more sophisticated) way to do this.

sage: f(x)=x^2
sage: points([(0,f(0)), (1,f(1)), (2,f(2)), (3,f(3)), (4,f(4))])
sage: points([(x,f(x)) for x in range(5)])

Sage tries to tell how many dimensions you are working in automatically.

sage: f(x,y)=x^2-y^2
sage: points([(x,y,f(x,y)) for x in range(5) for y in range(5)])

Lines

The syntax for lines is the same as that for points, but you get... well, you get connecting lines too!

sage: f(x)=x^2
sage: line([(x,f(x)) for x in range(5)])

Balls

Sage has disks and spheres of various types available. Generally the center and radius are all that is needed, but other options are possible.

sage: circle((0,1),1,aspect_ratio=1)
sage: disk((0,0), 1, (pi, 3*pi/2), color='yellow',aspect_ratio=1)

There are also ellipses and various arcs; see the full plot documentation.

Arrows

sage: arrow((0,0), (1,1))

Polygons

Polygons will try to complete themselves and fill in the interior; otherwise the syntax is fairly self-evident.

sage: polygon([[0,0],[1,1],[1,2]])

Text

In 2d, one can typeset mathematics using the text command. This can be used to fine-tune certain types of labels. Unfortunately, in 3D the text is just text.

sage: text('$\int_0^2 x^2\, dx$', (0.5,2))+plot(x^2,(x,0,2),fill=True)

Saving Plots

We can save 2d plots to many different formats. Sage can determine the format based on the filename for the image.

sage: p=plot(x^2,(x,-1,1))
sage: p

For testing purposes, we use the Sage standard temporary filename; however, you could use any string for a name that you wanted, like "my_plot.png".

sage: name = tmp_filename() # this is a string
sage: png_savename = name+'.png'
sage: p.save(png_savename)

In the notebook, these are usually ready for downloading in little links by the cells.

sage: pdf_savename = name+'.pdf'
sage: p.save(pdf_savename)

Notably, we can export in formats ready for inclusion in web pages.

sage: svg_savename = name+'.svg'
sage: p.save(svg_savename)