Plotting

Sage can produce two-dimensional and three-dimensional plots.

Two-dimensional Plots

In two dimensions, Sage can draw circles, lines, and polygons; plots of functions in rectangular coordinates; and also polar plots, contour plots and vector field plots. We present examples of some of these here. For more examples of plotting with Sage, see Solving Differential Equations and Maxima, and also the Sage Constructions documentation.

This command produces a yellow circle of radius 1, centered at the origin:

sage: circle((0,0), 1, rgbcolor=(1,1,0))
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> circle((Integer(0),Integer(0)), Integer(1), rgbcolor=(Integer(1),Integer(1),Integer(0)))
Graphics object consisting of 1 graphics primitive

circle((0,0), 1, rgbcolor=(1,1,0))

You can also produce a filled circle:

sage: circle((0,0), 1, rgbcolor=(1,1,0), fill=True)
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> circle((Integer(0),Integer(0)), Integer(1), rgbcolor=(Integer(1),Integer(1),Integer(0)), fill=True)
Graphics object consisting of 1 graphics primitive

circle((0,0), 1, rgbcolor=(1,1,0), fill=True)

You can also create a circle by assigning it to a variable; this does not plot it:

sage: c = circle((0,0), 1, rgbcolor=(1,1,0))

>>> from sage.all import *
>>> c = circle((Integer(0),Integer(0)), Integer(1), rgbcolor=(Integer(1),Integer(1),Integer(0)))

c = circle((0,0), 1, rgbcolor=(1,1,0))

To plot it, use c.show() or show(c), as follows:

sage: c.show()

>>> from sage.all import *
>>> c.show()

c.show()

Alternatively, evaluating c.save('filename.png') will save the plot to the given file.

Now, these ‘circles’ look more like ellipses because the axes are scaled differently. You can fix this:

sage: c.show(aspect_ratio=1)

>>> from sage.all import *
>>> c.show(aspect_ratio=Integer(1))

c.show(aspect_ratio=1)

The command show(c, aspect_ratio=1) accomplishes the same thing, or you can save the picture using c.save('filename.png', aspect_ratio=1).

It’s easy to plot basic functions:

sage: plot(cos, (-5,5))
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> plot(cos, (-Integer(5),Integer(5)))
Graphics object consisting of 1 graphics primitive

plot(cos, (-5,5))

Once you specify a variable name, you can create parametric plots also:

sage: x = var('x')
sage: parametric_plot((cos(x),sin(x)^3),(x,0,2*pi),rgbcolor=hue(0.6))
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> x = var('x')
>>> parametric_plot((cos(x),sin(x)**Integer(3)),(x,Integer(0),Integer(2)*pi),rgbcolor=hue(RealNumber('0.6')))
Graphics object consisting of 1 graphics primitive

x = var('x')
parametric_plot((cos(x),sin(x)^3),(x,0,2*pi),rgbcolor=hue(0.6))

It’s important to notice that the axes of the plots will only intersect if the origin is in the viewing range of the graph, and that with sufficiently large values scientific notation may be used:

sage: plot(x^2,(x,300,500))
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> plot(x**Integer(2),(x,Integer(300),Integer(500)))
Graphics object consisting of 1 graphics primitive

plot(x^2,(x,300,500))

You can combine several plots by adding them:

sage: x = var('x')
sage: p1 = parametric_plot((cos(x),sin(x)),(x,0,2*pi),rgbcolor=hue(0.2))
sage: p2 = parametric_plot((cos(x),sin(x)^2),(x,0,2*pi),rgbcolor=hue(0.4))
sage: p3 = parametric_plot((cos(x),sin(x)^3),(x,0,2*pi),rgbcolor=hue(0.6))
sage: show(p1+p2+p3, axes=false)

>>> from sage.all import *
>>> x = var('x')
>>> p1 = parametric_plot((cos(x),sin(x)),(x,Integer(0),Integer(2)*pi),rgbcolor=hue(RealNumber('0.2')))
>>> p2 = parametric_plot((cos(x),sin(x)**Integer(2)),(x,Integer(0),Integer(2)*pi),rgbcolor=hue(RealNumber('0.4')))
>>> p3 = parametric_plot((cos(x),sin(x)**Integer(3)),(x,Integer(0),Integer(2)*pi),rgbcolor=hue(RealNumber('0.6')))
>>> show(p1+p2+p3, axes=false)

x = var('x')
p1 = parametric_plot((cos(x),sin(x)),(x,0,2*pi),rgbcolor=hue(0.2))
p2 = parametric_plot((cos(x),sin(x)^2),(x,0,2*pi),rgbcolor=hue(0.4))
p3 = parametric_plot((cos(x),sin(x)^3),(x,0,2*pi),rgbcolor=hue(0.6))
show(p1+p2+p3, axes=false)

A good way to produce filled-in shapes is to produce a list of points (L in the example below) and then use the polygon command to plot the shape with boundary formed by those points. For example, here is a green deltoid:

sage: L = [[-1+cos(pi*i/100)*(1+cos(pi*i/100)),
....:     2*sin(pi*i/100)*(1-cos(pi*i/100))] for i in range(200)]
sage: p = polygon(L, rgbcolor=(1/8,3/4,1/2))
sage: p
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> L = [[-Integer(1)+cos(pi*i/Integer(100))*(Integer(1)+cos(pi*i/Integer(100))),
...     Integer(2)*sin(pi*i/Integer(100))*(Integer(1)-cos(pi*i/Integer(100)))] for i in range(Integer(200))]
>>> p = polygon(L, rgbcolor=(Integer(1)/Integer(8),Integer(3)/Integer(4),Integer(1)/Integer(2)))
>>> p
Graphics object consisting of 1 graphics primitive

L = [[-1+cos(pi*i/100)*(1+cos(pi*i/100)),
    2*sin(pi*i/100)*(1-cos(pi*i/100))] for i in range(200)]
p = polygon(L, rgbcolor=(1/8,3/4,1/2))
p

Type show(p, axes=false) to see this without any axes.

You can add text to a plot:

sage: L = [[6*cos(pi*i/100)+5*cos((6/2)*pi*i/100),
....:     6*sin(pi*i/100)-5*sin((6/2)*pi*i/100)] for i in range(200)]
sage: p = polygon(L, rgbcolor=(1/8,1/4,1/2))
sage: t = text("hypotrochoid", (5,4), rgbcolor=(1,0,0))
sage: show(p+t)

>>> from sage.all import *
>>> L = [[Integer(6)*cos(pi*i/Integer(100))+Integer(5)*cos((Integer(6)/Integer(2))*pi*i/Integer(100)),
...     Integer(6)*sin(pi*i/Integer(100))-Integer(5)*sin((Integer(6)/Integer(2))*pi*i/Integer(100))] for i in range(Integer(200))]
>>> p = polygon(L, rgbcolor=(Integer(1)/Integer(8),Integer(1)/Integer(4),Integer(1)/Integer(2)))
>>> t = text("hypotrochoid", (Integer(5),Integer(4)), rgbcolor=(Integer(1),Integer(0),Integer(0)))
>>> show(p+t)

L = [[6*cos(pi*i/100)+5*cos((6/2)*pi*i/100),
    6*sin(pi*i/100)-5*sin((6/2)*pi*i/100)] for i in range(200)]
p = polygon(L, rgbcolor=(1/8,1/4,1/2))
t = text("hypotrochoid", (5,4), rgbcolor=(1,0,0))
show(p+t)

Calculus teachers draw the following plot frequently on the board: not just one branch of arcsin but rather several of them: i.e., the plot of \(y=\sin(x)\) for \(x\) between \(-2\pi\) and \(2\pi\), flipped about the 45 degree line. The following Sage commands construct this:

sage: v = [(sin(x),x) for x in srange(-2*float(pi),2*float(pi),0.1)]
sage: line(v)
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> v = [(sin(x),x) for x in srange(-Integer(2)*float(pi),Integer(2)*float(pi),RealNumber('0.1'))]
>>> line(v)
Graphics object consisting of 1 graphics primitive

v = [(sin(x),x) for x in srange(-2*float(pi),2*float(pi),0.1)]
line(v)

Since the tangent function has a larger range than sine, if you use the same trick to plot the inverse tangent, you should change the minimum and maximum coordinates for the x-axis:

sage: v = [(tan(x),x) for x in srange(-2*float(pi),2*float(pi),0.01)]
sage: show(line(v), xmin=-20, xmax=20)

>>> from sage.all import *
>>> v = [(tan(x),x) for x in srange(-Integer(2)*float(pi),Integer(2)*float(pi),RealNumber('0.01'))]
>>> show(line(v), xmin=-Integer(20), xmax=Integer(20))

v = [(tan(x),x) for x in srange(-2*float(pi),2*float(pi),0.01)]
show(line(v), xmin=-20, xmax=20)

Sage also computes polar plots, contour plots and vector field plots (for special types of functions). Here is an example of a contour plot:

sage: f = lambda x,y: cos(x*y)
sage: contour_plot(f, (-4, 4), (-4, 4))
Graphics object consisting of 1 graphics primitive

>>> from sage.all import *
>>> f = lambda x,y: cos(x*y)
>>> contour_plot(f, (-Integer(4), Integer(4)), (-Integer(4), Integer(4)))
Graphics object consisting of 1 graphics primitive

f = lambda x,y: cos(x*y)
contour_plot(f, (-4, 4), (-4, 4))

Three-Dimensional Plots

Sage can also be used to create three-dimensional plots. In both the notebook and the REPL, these plots will be displayed by default using the open source package [ThreeJS], which supports interactively rotating and zooming the figure with the mouse.

Use plot3d to graph a function of the form \(f(x, y) = z\):

sage: x, y = var('x,y')
sage: plot3d(x^2 + y^2, (x,-2,2), (y,-2,2))
Graphics3d Object

>>> from sage.all import *
>>> x, y = var('x,y')
>>> plot3d(x**Integer(2) + y**Integer(2), (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)))
Graphics3d Object

x, y = var('x,y')
plot3d(x^2 + y^2, (x,-2,2), (y,-2,2))

Alternatively, you can use parametric_plot3d to graph a parametric surface where each of \(x, y, z\) is determined by a function of one or two variables (the parameters, typically \(u\) and \(v\)). The previous plot can be expressed parametrically as follows:

sage: u, v = var('u, v')
sage: f_x(u, v) = u
sage: f_y(u, v) = v
sage: f_z(u, v) = u^2 + v^2
sage: parametric_plot3d([f_x, f_y, f_z], (u, -2, 2), (v, -2, 2))
Graphics3d Object

>>> from sage.all import *
>>> u, v = var('u, v')
>>> __tmp__=var("u,v"); f_x = symbolic_expression(u).function(u,v)
>>> __tmp__=var("u,v"); f_y = symbolic_expression(v).function(u,v)
>>> __tmp__=var("u,v"); f_z = symbolic_expression(u**Integer(2) + v**Integer(2)).function(u,v)
>>> parametric_plot3d([f_x, f_y, f_z], (u, -Integer(2), Integer(2)), (v, -Integer(2), Integer(2)))
Graphics3d Object

u, v = var('u, v')
f_x(u, v) = u
f_y(u, v) = v
f_z(u, v) = u^2 + v^2
parametric_plot3d([f_x, f_y, f_z], (u, -2, 2), (v, -2, 2))

The third way to plot a 3D surface in Sage is implicit_plot3d, which graphs a contour of a function like \(f(x, y, z) = 0\) (this defines a set of points). We graph a sphere using the classical formula:

sage: x, y, z = var('x, y, z')
sage: implicit_plot3d(x^2 + y^2 + z^2 - 4, (x,-2, 2), (y,-2, 2), (z,-2, 2))
Graphics3d Object

>>> from sage.all import *
>>> x, y, z = var('x, y, z')
>>> implicit_plot3d(x**Integer(2) + y**Integer(2) + z**Integer(2) - Integer(4), (x,-Integer(2), Integer(2)), (y,-Integer(2), Integer(2)), (z,-Integer(2), Integer(2)))
Graphics3d Object

x, y, z = var('x, y, z')
implicit_plot3d(x^2 + y^2 + z^2 - 4, (x,-2, 2), (y,-2, 2), (z,-2, 2))

Here are some more examples:

Yellow Whitney’s umbrella:

sage: u, v = var('u,v')
sage: fx = u*v
sage: fy = u
sage: fz = v^2
sage: parametric_plot3d([fx, fy, fz], (u, -1, 1), (v, -1, 1),
....:   frame=False, color="yellow")
Graphics3d Object

>>> from sage.all import *
>>> u, v = var('u,v')
>>> fx = u*v
>>> fy = u
>>> fz = v**Integer(2)
>>> parametric_plot3d([fx, fy, fz], (u, -Integer(1), Integer(1)), (v, -Integer(1), Integer(1)),
...   frame=False, color="yellow")
Graphics3d Object

u, v = var('u,v')
fx = u*v
fy = u
fz = v^2
parametric_plot3d([fx, fy, fz], (u, -1, 1), (v, -1, 1),
  frame=False, color="yellow")

Cross cap:

sage: u, v = var('u,v')
sage: fx = (1+cos(v))*cos(u)
sage: fy = (1+cos(v))*sin(u)
sage: fz = -tanh((2/3)*(u-pi))*sin(v)
sage: parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi),
....:   frame=False, color="red")
Graphics3d Object

>>> from sage.all import *
>>> u, v = var('u,v')
>>> fx = (Integer(1)+cos(v))*cos(u)
>>> fy = (Integer(1)+cos(v))*sin(u)
>>> fz = -tanh((Integer(2)/Integer(3))*(u-pi))*sin(v)
>>> parametric_plot3d([fx, fy, fz], (u, Integer(0), Integer(2)*pi), (v, Integer(0), Integer(2)*pi),
...   frame=False, color="red")
Graphics3d Object

u, v = var('u,v')
fx = (1+cos(v))*cos(u)
fy = (1+cos(v))*sin(u)
fz = -tanh((2/3)*(u-pi))*sin(v)
parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi),
  frame=False, color="red")

Twisted torus:

sage: u, v = var('u,v')
sage: fx = (3+sin(v)+cos(u))*cos(2*v)
sage: fy = (3+sin(v)+cos(u))*sin(2*v)
sage: fz = sin(u)+2*cos(v)
sage: parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi),
....:   frame=False, color="red")
Graphics3d Object

>>> from sage.all import *
>>> u, v = var('u,v')
>>> fx = (Integer(3)+sin(v)+cos(u))*cos(Integer(2)*v)
>>> fy = (Integer(3)+sin(v)+cos(u))*sin(Integer(2)*v)
>>> fz = sin(u)+Integer(2)*cos(v)
>>> parametric_plot3d([fx, fy, fz], (u, Integer(0), Integer(2)*pi), (v, Integer(0), Integer(2)*pi),
...   frame=False, color="red")
Graphics3d Object

u, v = var('u,v')
fx = (3+sin(v)+cos(u))*cos(2*v)
fy = (3+sin(v)+cos(u))*sin(2*v)
fz = sin(u)+2*cos(v)
parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi),
  frame=False, color="red")

Lemniscate:

sage: x, y, z = var('x,y,z')
sage: f(x, y, z) = 4*x^2 * (x^2 + y^2 + z^2 + z) + y^2 * (y^2 + z^2 - 1)
sage: implicit_plot3d(f, (x, -0.5, 0.5), (y, -1, 1), (z, -1, 1))
Graphics3d Object

>>> from sage.all import *
>>> x, y, z = var('x,y,z')
>>> __tmp__=var("x,y,z"); f = symbolic_expression(Integer(4)*x**Integer(2) * (x**Integer(2) + y**Integer(2) + z**Integer(2) + z) + y**Integer(2) * (y**Integer(2) + z**Integer(2) - Integer(1))).function(x,y,z)
>>> implicit_plot3d(f, (x, -RealNumber('0.5'), RealNumber('0.5')), (y, -Integer(1), Integer(1)), (z, -Integer(1), Integer(1)))
Graphics3d Object

x, y, z = var('x,y,z')
f(x, y, z) = 4*x^2 * (x^2 + y^2 + z^2 + z) + y^2 * (y^2 + z^2 - 1)
implicit_plot3d(f, (x, -0.5, 0.5), (y, -1, 1), (z, -1, 1))