Plotting¶
Sage can plot using matplotlib, openmath, gnuplot, or surf but only matplotlib and openmath are included with Sage in the standard distribution. For surf examples, see Plotting surfaces.
Plotting in Sage can be done in many different ways. You can plot a
function (in 2 or 3 dimensions) or a set of points (in 2-D only)
via gnuplot, you can plot a solution to a differential equation via
Maxima (which in turn calls gnuplot or openmath), or you can use
Singular’s interface with the plotting package surf (which does not
come with Sage). gnuplot does not have an implicit plotting
command, so if you want to plot a curve or surface using an
implicit plot, it is best to use the Singular’s interface to surf,
as described in chapter ch:AG, Algebraic geometry.
Plotting functions in 2D¶
The default plotting method in uses the excellent matplotlib
package.
To view any of these, type P.save("<path>/myplot.png") and then
open it in a graphics viewer such as gimp.
You can plot piecewise-defined functions:
sage: f1 = 1
sage: f2 = 1-x
sage: f3 = exp(x)
sage: f4 = sin(2*x)
sage: f = piecewise([((0,1),f1), ((1,2),f2), ((2,3),f3), ((3,10),f4)])
sage: f.plot(x,0,10)
Graphics object consisting of 1 graphics primitive
>>> from sage.all import *
>>> f1 = Integer(1)
>>> f2 = Integer(1)-x
>>> f3 = exp(x)
>>> f4 = sin(Integer(2)*x)
>>> f = piecewise([((Integer(0),Integer(1)),f1), ((Integer(1),Integer(2)),f2), ((Integer(2),Integer(3)),f3), ((Integer(3),Integer(10)),f4)])
>>> f.plot(x,Integer(0),Integer(10))
Graphics object consisting of 1 graphics primitive
f1 = 1 f2 = 1-x f3 = exp(x) f4 = sin(2*x) f = piecewise([((0,1),f1), ((1,2),f2), ((2,3),f3), ((3,10),f4)]) f.plot(x,0,10)
Other function plots can be produced as well:
A red plot of the Jacobi elliptic function
\(\text{sn}(x,2)\), \(-3<x<3\) (do not type the
....::
sage: L = [(i/100.0, maxima.eval('jacobi_sn (%s/100.0,2.0)'%i))
....: for i in range(-300,300)]
sage: show(line(L, rgbcolor=(3/4,1/4,1/8)))
>>> from sage.all import *
>>> L = [(i/RealNumber('100.0'), maxima.eval('jacobi_sn (%s/100.0,2.0)'%i))
... for i in range(-Integer(300),Integer(300))]
>>> show(line(L, rgbcolor=(Integer(3)/Integer(4),Integer(1)/Integer(4),Integer(1)/Integer(8))))
L = [(i/100.0, maxima.eval('jacobi_sn (%s/100.0,2.0)'%i))
for i in range(-300,300)]
show(line(L, rgbcolor=(3/4,1/4,1/8)))
A red plot of \(J\)-Bessel function \(J_2(x)\), \(0<x<10\):
sage: L = [(i/10.0, maxima.eval('bessel_j (2,%s/10.0)'%i)) for i in range(100)]
sage: show(line(L, rgbcolor=(3/4,1/4,5/8)))
>>> from sage.all import *
>>> L = [(i/RealNumber('10.0'), maxima.eval('bessel_j (2,%s/10.0)'%i)) for i in range(Integer(100))]
>>> show(line(L, rgbcolor=(Integer(3)/Integer(4),Integer(1)/Integer(4),Integer(5)/Integer(8))))
L = [(i/10.0, maxima.eval('bessel_j (2,%s/10.0)'%i)) for i in range(100)]
show(line(L, rgbcolor=(3/4,1/4,5/8)))
A purple plot of the Riemann zeta function \(\zeta(1/2 + it)\), \(0<t<30\):
sage: I = CDF.0
sage: show(line([zeta(1/2 + k*I/6) for k in range(180)], rgbcolor=(3/4,1/2,5/8)))
>>> from sage.all import *
>>> I = CDF.gen(0)
>>> show(line([zeta(Integer(1)/Integer(2) + k*I/Integer(6)) for k in range(Integer(180))], rgbcolor=(Integer(3)/Integer(4),Integer(1)/Integer(2),Integer(5)/Integer(8))))
I = CDF.0 show(line([zeta(1/2 + k*I/6) for k in range(180)], rgbcolor=(3/4,1/2,5/8)))
Plotting curves¶
To plot a curve in Sage, you can use Singular and surf (http://surf.sourceforge.net/, also available as an experimental package) or use matplotlib (included with Sage).
matplotlib¶
Here are several examples. To view them, type
p.save("<path>/my_plot.png") (where <path> is a directory path
which you have write permissions to where you want to save the
plot) and view it in a viewer (such as GIMP).
A blue conchoid of Nicomedes:
sage: L = [[1+5*cos(pi/2+pi*i/100), tan(pi/2+pi*i/100)*
....: (1+5*cos(pi/2+pi*i/100))] for i in range(1,100)]
sage: line(L, rgbcolor=(1/4,1/8,3/4))
Graphics object consisting of 1 graphics primitive
>>> from sage.all import *
>>> L = [[Integer(1)+Integer(5)*cos(pi/Integer(2)+pi*i/Integer(100)), tan(pi/Integer(2)+pi*i/Integer(100))*
... (Integer(1)+Integer(5)*cos(pi/Integer(2)+pi*i/Integer(100)))] for i in range(Integer(1),Integer(100))]
>>> line(L, rgbcolor=(Integer(1)/Integer(4),Integer(1)/Integer(8),Integer(3)/Integer(4)))
Graphics object consisting of 1 graphics primitive
L = [[1+5*cos(pi/2+pi*i/100), tan(pi/2+pi*i/100)*
(1+5*cos(pi/2+pi*i/100))] for i in range(1,100)]
line(L, rgbcolor=(1/4,1/8,3/4))
A blue hypotrochoid (3 leaves):
sage: n = 4; h = 3; b = 2
sage: L = [[n*cos(pi*i/100)+h*cos((n/b)*pi*i/100),
....: n*sin(pi*i/100)-h*sin((n/b)*pi*i/100)] for i in range(200)]
sage: line(L, rgbcolor=(1/4,1/4,3/4))
Graphics object consisting of 1 graphics primitive
>>> from sage.all import *
>>> n = Integer(4); h = Integer(3); b = Integer(2)
>>> L = [[n*cos(pi*i/Integer(100))+h*cos((n/b)*pi*i/Integer(100)),
... n*sin(pi*i/Integer(100))-h*sin((n/b)*pi*i/Integer(100))] for i in range(Integer(200))]
>>> line(L, rgbcolor=(Integer(1)/Integer(4),Integer(1)/Integer(4),Integer(3)/Integer(4)))
Graphics object consisting of 1 graphics primitive
n = 4; h = 3; b = 2
L = [[n*cos(pi*i/100)+h*cos((n/b)*pi*i/100),
n*sin(pi*i/100)-h*sin((n/b)*pi*i/100)] for i in range(200)]
line(L, rgbcolor=(1/4,1/4,3/4))
A blue hypotrochoid (4 leaves):
sage: n = 6; h = 5; b = 2
sage: L = [[n*cos(pi*i/100)+h*cos((n/b)*pi*i/100),
....: n*sin(pi*i/100)-h*sin((n/b)*pi*i/100)] for i in range(200)]
sage: line(L, rgbcolor=(1/4,1/4,3/4))
Graphics object consisting of 1 graphics primitive
>>> from sage.all import *
>>> n = Integer(6); h = Integer(5); b = Integer(2)
>>> L = [[n*cos(pi*i/Integer(100))+h*cos((n/b)*pi*i/Integer(100)),
... n*sin(pi*i/Integer(100))-h*sin((n/b)*pi*i/Integer(100))] for i in range(Integer(200))]
>>> line(L, rgbcolor=(Integer(1)/Integer(4),Integer(1)/Integer(4),Integer(3)/Integer(4)))
Graphics object consisting of 1 graphics primitive
n = 6; h = 5; b = 2
L = [[n*cos(pi*i/100)+h*cos((n/b)*pi*i/100),
n*sin(pi*i/100)-h*sin((n/b)*pi*i/100)] for i in range(200)]
line(L, rgbcolor=(1/4,1/4,3/4))
A red limaçon of Pascal:
sage: L = [[sin(pi*i/100)+sin(pi*i/50),-(1+cos(pi*i/100)+cos(pi*i/50))]
....: for i in range(-100,101)]
sage: line(L, rgbcolor=(1,1/4,1/2))
Graphics object consisting of 1 graphics primitive
>>> from sage.all import *
>>> L = [[sin(pi*i/Integer(100))+sin(pi*i/Integer(50)),-(Integer(1)+cos(pi*i/Integer(100))+cos(pi*i/Integer(50)))]
... for i in range(-Integer(100),Integer(101))]
>>> line(L, rgbcolor=(Integer(1),Integer(1)/Integer(4),Integer(1)/Integer(2)))
Graphics object consisting of 1 graphics primitive
L = [[sin(pi*i/100)+sin(pi*i/50),-(1+cos(pi*i/100)+cos(pi*i/50))]
for i in range(-100,101)]
line(L, rgbcolor=(1,1/4,1/2))
A light green trisectrix of Maclaurin:
sage: L = [[2*(1-4*cos(-pi/2+pi*i/100)^2),10*tan(-pi/2+pi*i/100)*
....: (1-4*cos(-pi/2+pi*i/100)^2)] for i in range(1,100)]
sage: line(L, rgbcolor=(1/4,1,1/8))
Graphics object consisting of 1 graphics primitive
>>> from sage.all import *
>>> L = [[Integer(2)*(Integer(1)-Integer(4)*cos(-pi/Integer(2)+pi*i/Integer(100))**Integer(2)),Integer(10)*tan(-pi/Integer(2)+pi*i/Integer(100))*
... (Integer(1)-Integer(4)*cos(-pi/Integer(2)+pi*i/Integer(100))**Integer(2))] for i in range(Integer(1),Integer(100))]
>>> line(L, rgbcolor=(Integer(1)/Integer(4),Integer(1),Integer(1)/Integer(8)))
Graphics object consisting of 1 graphics primitive
L = [[2*(1-4*cos(-pi/2+pi*i/100)^2),10*tan(-pi/2+pi*i/100)*
(1-4*cos(-pi/2+pi*i/100)^2)] for i in range(1,100)]
line(L, rgbcolor=(1/4,1,1/8))
A green lemniscate of Bernoulli (we omit i==100 since that would give a 0 division error):
sage: v = [(1/cos(-pi/2+pi*i/100), tan(-pi/2+pi*i/100)) for i in range(1,200) if i!=100 ]
sage: L = [(a/(a^2+b^2), b/(a^2+b^2)) for a,b in v]
sage: line(L, rgbcolor=(1/4,3/4,1/8))
Graphics object consisting of 1 graphics primitive
>>> from sage.all import *
>>> v = [(Integer(1)/cos(-pi/Integer(2)+pi*i/Integer(100)), tan(-pi/Integer(2)+pi*i/Integer(100))) for i in range(Integer(1),Integer(200)) if i!=Integer(100) ]
>>> L = [(a/(a**Integer(2)+b**Integer(2)), b/(a**Integer(2)+b**Integer(2))) for a,b in v]
>>> line(L, rgbcolor=(Integer(1)/Integer(4),Integer(3)/Integer(4),Integer(1)/Integer(8)))
Graphics object consisting of 1 graphics primitive
v = [(1/cos(-pi/2+pi*i/100), tan(-pi/2+pi*i/100)) for i in range(1,200) if i!=100 ] L = [(a/(a^2+b^2), b/(a^2+b^2)) for a,b in v] line(L, rgbcolor=(1/4,3/4,1/8))
surf¶
In particular, since surf is only available on a UNIX type OS
(and is not included with Sage), plotting using the commands below
in Sage is only available on such an OS. Incidentally, surf is
included with several popular Linux distributions.
sage: s = singular.eval
sage: s('LIB "surf.lib";')
...
sage: s("ring rr0 = 0,(x1,x2),dp;")
''
sage: s("ideal I = x1^3 - x2^2;")
''
sage: s("plot(I);")
...
>>> from sage.all import *
>>> s = singular.eval
>>> s('LIB "surf.lib";')
...
>>> s("ring rr0 = 0,(x1,x2),dp;")
''
>>> s("ideal I = x1^3 - x2^2;")
''
>>> s("plot(I);")
...
s = singular.eval
s('LIB "surf.lib";')
s("ring rr0 = 0,(x1,x2),dp;")
s("ideal I = x1^3 - x2^2;")
s("plot(I);")
Press q with the surf window active to exit from surf and return to
Sage.
You can save this plot as a surf script. In the surf window which
pops up, just choose file, save as, etc.. (Type q or select
file, quit, to close the window.)
The plot produced is omitted but the gentle reader is encouraged to try it out.
openmath¶
Openmath is a TCL/Tk GUI plotting program written by W. Schelter.
The following command plots the function \(\cos(2x)+2e^{-x}\)
sage: maxima.plot2d('cos(2*x) + 2*exp(-x)','[x,0,1]', # not tested (pops up a window)
....: '[plot_format,openmath]')
>>> from sage.all import *
>>> maxima.plot2d('cos(2*x) + 2*exp(-x)','[x,0,1]', # not tested (pops up a window)
... '[plot_format,openmath]')
maxima.plot2d('cos(2*x) + 2*exp(-x)','[x,0,1]', # not tested (pops up a window)
'[plot_format,openmath]')
(Mac OS X users: Note that these openmath commands were run in a
session of started in an xterm shell, not using the standard Mac
Terminal application.)
sage: maxima.eval('load("plotdf");')
'".../share/maxima.../share/dynamics/plotdf.lisp"'
sage: maxima.eval('plotdf(x+y,[trajectory_at,2,-0.1]); ') # not tested
>>> from sage.all import *
>>> maxima.eval('load("plotdf");')
'".../share/maxima.../share/dynamics/plotdf.lisp"'
>>> maxima.eval('plotdf(x+y,[trajectory_at,2,-0.1]); ') # not tested
maxima.eval('load("plotdf");')
maxima.eval('plotdf(x+y,[trajectory_at,2,-0.1]); ') # not tested
This plots a direction field (the plotdf Maxima package was also written by W. Schelter.)
A 2D plot of several functions:
sage: maxima.plot2d('[x,x^2,x^3]','[x,-1,1]','[plot_format,openmath]') # not tested
>>> from sage.all import *
>>> maxima.plot2d('[x,x^2,x^3]','[x,-1,1]','[plot_format,openmath]') # not tested
maxima.plot2d('[x,x^2,x^3]','[x,-1,1]','[plot_format,openmath]') # not tested
Openmath also does 3D plots of surfaces of the form \(z=f(x,y)\), as \(x\) and \(y\) range over a rectangle. For example, here is a “live” 3D plot which you can move with your mouse:
sage: maxima.plot3d ("sin(x^2 + y^2)", "[x, -3, 3]", "[y, -3, 3]", # not tested
....: '[plot_format, openmath]')
>>> from sage.all import *
>>> maxima.plot3d ("sin(x^2 + y^2)", "[x, -3, 3]", "[y, -3, 3]", # not tested
... '[plot_format, openmath]')
maxima.plot3d ("sin(x^2 + y^2)", "[x, -3, 3]", "[y, -3, 3]", # not tested
'[plot_format, openmath]')
By rotating this suitably, you can view the contour plot.
Tachyon 3D plotting¶
The ray-tracing package Tachyon is distributed with Sage. The 3D plots look very nice but tend to take a bit more setting up. Here is an example of a parametric space curve:
sage: f = lambda t: (t,t^2,t^3)
sage: t = Tachyon(camera_center=(5,0,4))
sage: t.texture('t')
sage: t.light((-20,-20,40), 0.2, (1,1,1))
sage: t.parametric_plot(f,-5,5,'t',min_depth=6)
>>> from sage.all import *
>>> f = lambda t: (t,t**Integer(2),t**Integer(3))
>>> t = Tachyon(camera_center=(Integer(5),Integer(0),Integer(4)))
>>> t.texture('t')
>>> t.light((-Integer(20),-Integer(20),Integer(40)), RealNumber('0.2'), (Integer(1),Integer(1),Integer(1)))
>>> t.parametric_plot(f,-Integer(5),Integer(5),'t',min_depth=Integer(6))
f = lambda t: (t,t^2,t^3)
t = Tachyon(camera_center=(5,0,4))
t.texture('t')
t.light((-20,-20,40), 0.2, (1,1,1))
t.parametric_plot(f,-5,5,'t',min_depth=6)
Type t.show() to view this.
Other examples are in the Reference Manual.
gnuplot¶
You must have gnuplot installed to run these commands.
First, here’s way to plot a function: {plot!a function}
sage: maxima.plot2d('sin(x)','[x,-5,5]')
sage: opts = '[gnuplot_term, ps], [gnuplot_out_file, "sin-plot.eps"]'
sage: maxima.plot2d('sin(x)','[x,-5,5]',opts)
sage: opts = '[gnuplot_term, ps], [gnuplot_out_file, "/tmp/sin-plot.eps"]'
sage: maxima.plot2d('sin(x)','[x,-5,5]',opts)
>>> from sage.all import *
>>> maxima.plot2d('sin(x)','[x,-5,5]')
>>> opts = '[gnuplot_term, ps], [gnuplot_out_file, "sin-plot.eps"]'
>>> maxima.plot2d('sin(x)','[x,-5,5]',opts)
>>> opts = '[gnuplot_term, ps], [gnuplot_out_file, "/tmp/sin-plot.eps"]'
>>> maxima.plot2d('sin(x)','[x,-5,5]',opts)
maxima.plot2d('sin(x)','[x,-5,5]')
opts = '[gnuplot_term, ps], [gnuplot_out_file, "sin-plot.eps"]'
maxima.plot2d('sin(x)','[x,-5,5]',opts)
opts = '[gnuplot_term, ps], [gnuplot_out_file, "/tmp/sin-plot.eps"]'
maxima.plot2d('sin(x)','[x,-5,5]',opts)
The eps file is saved by default to the current directory but you may specify a path if you prefer.
Here is an example of a plot of a parametric curve in the plane:
sage: maxima.plot2d_parametric(["sin(t)","cos(t)"], "t",[-3.1,3.1])
sage: opts = '[gnuplot_preamble, "set nokey"], [gnuplot_term, ps],
....: [gnuplot_out_file, "circle-plot.eps"]'
sage: maxima.plot2d_parametric(["sin(t)","cos(t)"], "t", [-3.1,3.1], options=opts)
>>> from sage.all import *
>>> maxima.plot2d_parametric(["sin(t)","cos(t)"], "t",[-RealNumber('3.1'),RealNumber('3.1')])
>>> opts = '[gnuplot_preamble, "set nokey"], [gnuplot_term, ps],
... [gnuplot_out_file, "circle-plot.eps"]'
>>> maxima.plot2d_parametric(["sin(t)","cos(t)"], "t", [-RealNumber('3.1'),RealNumber('3.1')], options=opts)
maxima.plot2d_parametric(["sin(t)","cos(t)"], "t",[-3.1,3.1])
opts = '[gnuplot_preamble, "set nokey"], [gnuplot_term, ps],
[gnuplot_out_file, "circle-plot.eps"]'
maxima.plot2d_parametric(["sin(t)","cos(t)"], "t", [-3.1,3.1], options=opts)
Here is an example of a plot of a parametric surface in 3-space: {plot!a parametric surface}
sage: maxima.plot3d_parametric(["v*sin(u)","v*cos(u)","v"], ["u","v"],
....: [-3.2,3.2],[0,3])
sage: opts = '[gnuplot_term, ps], [gnuplot_out_file, "sin-cos-plot.eps"]'
sage: maxima.plot3d_parametric(["v*sin(u)","v*cos(u)","v"], ["u","v"],
....: [-3.2,3.2],[0,3],opts)
>>> from sage.all import *
>>> maxima.plot3d_parametric(["v*sin(u)","v*cos(u)","v"], ["u","v"],
... [-RealNumber('3.2'),RealNumber('3.2')],[Integer(0),Integer(3)])
>>> opts = '[gnuplot_term, ps], [gnuplot_out_file, "sin-cos-plot.eps"]'
>>> maxima.plot3d_parametric(["v*sin(u)","v*cos(u)","v"], ["u","v"],
... [-RealNumber('3.2'),RealNumber('3.2')],[Integer(0),Integer(3)],opts)
maxima.plot3d_parametric(["v*sin(u)","v*cos(u)","v"], ["u","v"],
[-3.2,3.2],[0,3])
opts = '[gnuplot_term, ps], [gnuplot_out_file, "sin-cos-plot.eps"]'
maxima.plot3d_parametric(["v*sin(u)","v*cos(u)","v"], ["u","v"],
[-3.2,3.2],[0,3],opts)
To illustrate how to pass gnuplot options in , here is an example of a plot of a set of points involving the Riemann zeta function \(\zeta(s)\) (computed using Pari but plotted using Maxima and Gnuplot): {plot!points} {Riemann zeta function}
sage: zeta_ptsx = [ (pari(1/2 + i*I/10).zeta().real()).precision(1)
....: for i in range(70,150)]
sage: zeta_ptsy = [ (pari(1/2 + i*I/10).zeta().imag()).precision(1)
....: for i in range(70,150)]
sage: maxima.plot_list(zeta_ptsx, zeta_ptsy) # optional -- pops up a window.
sage: opts='[gnuplot_preamble, "set nokey"], [gnuplot_term, ps],
....: [gnuplot_out_file, "zeta.eps"]'
sage: maxima.plot_list(zeta_ptsx, zeta_ptsy, opts) # optional -- pops up a window.
>>> from sage.all import *
>>> zeta_ptsx = [ (pari(Integer(1)/Integer(2) + i*I/Integer(10)).zeta().real()).precision(Integer(1))
... for i in range(Integer(70),Integer(150))]
>>> zeta_ptsy = [ (pari(Integer(1)/Integer(2) + i*I/Integer(10)).zeta().imag()).precision(Integer(1))
... for i in range(Integer(70),Integer(150))]
>>> maxima.plot_list(zeta_ptsx, zeta_ptsy) # optional -- pops up a window.
>>> opts='[gnuplot_preamble, "set nokey"], [gnuplot_term, ps],
... [gnuplot_out_file, "zeta.eps"]'
>>> maxima.plot_list(zeta_ptsx, zeta_ptsy, opts) # optional -- pops up a window.
zeta_ptsx = [ (pari(1/2 + i*I/10).zeta().real()).precision(1)
for i in range(70,150)]
zeta_ptsy = [ (pari(1/2 + i*I/10).zeta().imag()).precision(1)
for i in range(70,150)]
maxima.plot_list(zeta_ptsx, zeta_ptsy) # optional -- pops up a window.
opts='[gnuplot_preamble, "set nokey"], [gnuplot_term, ps],
[gnuplot_out_file, "zeta.eps"]'
maxima.plot_list(zeta_ptsx, zeta_ptsy, opts) # optional -- pops up a window.
Plotting surfaces¶
To plot a surface in is no different that to plot a curve, though
the syntax is slightly different. In particular, you need to have
surf loaded. {plot!surface using surf}
sage: singular.eval('ring rr1 = 0,(x,y,z),dp;')
''
sage: singular.eval('ideal I(1) = 2x2-1/2x3 +1-y+1;')
''
sage: singular.eval('plot(I(1));')
...
>>> from sage.all import *
>>> singular.eval('ring rr1 = 0,(x,y,z),dp;')
''
>>> singular.eval('ideal I(1) = 2x2-1/2x3 +1-y+1;')
''
>>> singular.eval('plot(I(1));')
...
singular.eval('ring rr1 = 0,(x,y,z),dp;')
singular.eval('ideal I(1) = 2x2-1/2x3 +1-y+1;')
singular.eval('plot(I(1));')