Streamline plots

class sage.plot.streamline_plot.StreamlinePlot(xpos_array, ypos_array, xvec_array, yvec_array, options)

Bases: GraphicPrimitive

Primitive class that initializes the StreamlinePlot graphics type

get_minmax_data()

Return a dictionary with the bounding box data.

EXAMPLES:

Sage

sage: x, y = var('x y')
sage: import numpy  # to ensure numpy 2.0 compatibility
sage: if int(numpy.version.short_version[0]) > 1:
....:     _ = numpy.set_printoptions(legacy="1.25")
sage: d = streamline_plot((.01*x, x+y), (x,10,20), (y,10,20))[0].get_minmax_data()
sage: d['xmin']
10.0
sage: d['ymin']
10.0

Python

>>> from sage.all import *
>>> x, y = var('x y')
>>> import numpy  # to ensure numpy 2.0 compatibility
>>> if int(numpy.version.short_version[Integer(0)]) > Integer(1):
...     _ = numpy.set_printoptions(legacy="1.25")
>>> d = streamline_plot((RealNumber('.01')*x, x+y), (x,Integer(10),Integer(20)), (y,Integer(10),Integer(20)))[Integer(0)].get_minmax_data()
>>> d['xmin']
10.0
>>> d['ymin']
10.0

Sage Live

x, y = var('x y')
import numpy  # to ensure numpy 2.0 compatibility
if int(numpy.version.short_version[0]) > 1:
    _ = numpy.set_printoptions(legacy="1.25")
d = streamline_plot((.01*x, x+y), (x,10,20), (y,10,20))[0].get_minmax_data()
d['xmin']
d['ymin']

sage.plot.streamline_plot.streamline_plot(f_g, xrange, yrange, plot_points=20, density=1.0, frame=True, **options)

Return a streamline plot in a vector field.

streamline_plot can take either one or two functions. Consider two variables $x$ and $y$.

If given two functions $(f(x,y),g(x,y))$, then this function plots streamlines in the vector field over the specified ranges with xrange being of $x$, denoted by xvar below, between xmin and xmax, and yrange similarly (see below).

streamline_plot((f, g), (xvar, xmin, xmax), (yvar, ymin, ymax))

Similarly, if given one function $f(x,y)$, then this function plots streamlines in the slope field $dy/dx=f(x,y)$ over the specified ranges as given above.

PLOT OPTIONS:

• plot_points – (default: 200) the minimal number of plot points

• density – float (default: 1.); controls the closeness of streamlines

• start_points – (optional) list of coordinates of starting points for the streamlines; coordinate pairs can be tuples or lists

EXAMPLES:

Plot some vector fields involving $\sin$ and $\cos$:

Sage

sage: x, y = var('x y')
sage: streamline_plot((sin(x), cos(y)), (x,-3,3), (y,-3,3))
Graphics object consisting of 1 graphics primitive

Python

>>> from sage.all import *
>>> x, y = var('x y')
>>> streamline_plot((sin(x), cos(y)), (x,-Integer(3),Integer(3)), (y,-Integer(3),Integer(3)))
Graphics object consisting of 1 graphics primitive

Sage Live

x, y = var('x y')
streamline_plot((sin(x), cos(y)), (x,-3,3), (y,-3,3))

Sage

sage: streamline_plot((y, (cos(x)-2) * sin(x)), (x,-pi,pi), (y,-pi,pi))
Graphics object consisting of 1 graphics primitive

Python

>>> from sage.all import *
>>> streamline_plot((y, (cos(x)-Integer(2)) * sin(x)), (x,-pi,pi), (y,-pi,pi))
Graphics object consisting of 1 graphics primitive

Sage Live

streamline_plot((y, (cos(x)-2) * sin(x)), (x,-pi,pi), (y,-pi,pi))

We increase the density of the plot:

Sage

sage: streamline_plot((y, (cos(x)-2) * sin(x)),
....:                 (x,-pi,pi), (y,-pi,pi), density=2)
Graphics object consisting of 1 graphics primitive

Python

>>> from sage.all import *
>>> streamline_plot((y, (cos(x)-Integer(2)) * sin(x)),
...                 (x,-pi,pi), (y,-pi,pi), density=Integer(2))
Graphics object consisting of 1 graphics primitive

Sage Live

streamline_plot((y, (cos(x)-2) * sin(x)),
                (x,-pi,pi), (y,-pi,pi), density=2)

We ignore function values that are infinite or NaN:

Sage

sage: x, y = var('x y')
sage: streamline_plot((-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

Python

>>> from sage.all import *
>>> x, y = var('x y')
>>> streamline_plot((-x/sqrt(x**Integer(2)+y**Integer(2)), -y/sqrt(x**Integer(2)+y**Integer(2))),
...                 (x,-Integer(10),Integer(10)), (y,-Integer(10),Integer(10)))
Graphics object consisting of 1 graphics primitive

Sage Live

x, y = var('x y')
streamline_plot((-x/sqrt(x^2+y^2), -y/sqrt(x^2+y^2)),
                (x,-10,10), (y,-10,10))

Extra options will get passed on to show(), as long as they are valid:

Sage

sage: streamline_plot((x, y), (x,-2,2), (y,-2,2), xmax=10)
Graphics object consisting of 1 graphics primitive
sage: streamline_plot((x, y), (x,-2,2), (y,-2,2)).show(xmax=10)  # These are equivalent

Python

>>> from sage.all import *
>>> streamline_plot((x, y), (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2)), xmax=Integer(10))
Graphics object consisting of 1 graphics primitive
>>> streamline_plot((x, y), (x,-Integer(2),Integer(2)), (y,-Integer(2),Integer(2))).show(xmax=Integer(10))  # These are equivalent

Sage Live

streamline_plot((x, y), (x,-2,2), (y,-2,2), xmax=10)
streamline_plot((x, y), (x,-2,2), (y,-2,2)).show(xmax=10)  # These are equivalent

We can also construct streamlines in a slope field:

Sage

sage: x, y = var('x y')
sage: streamline_plot((x + y) / sqrt(x^2 + y^2), (x,-3,3), (y,-3,3))
Graphics object consisting of 1 graphics primitive

Python

>>> from sage.all import *
>>> x, y = var('x y')
>>> streamline_plot((x + y) / sqrt(x**Integer(2) + y**Integer(2)), (x,-Integer(3),Integer(3)), (y,-Integer(3),Integer(3)))
Graphics object consisting of 1 graphics primitive

Sage Live

x, y = var('x y')
streamline_plot((x + y) / sqrt(x^2 + y^2), (x,-3,3), (y,-3,3))

We choose some particular points the streamlines pass through:

Sage

sage: pts = [[1, 1], [-2, 2], [1, -3/2]]
sage: g = streamline_plot((x + y) / sqrt(x^2 + y^2),
....:                     (x,-3,3), (y,-3,3), start_points=pts)
sage: g += point(pts, color='red')
sage: g
Graphics object consisting of 2 graphics primitives

Python

>>> from sage.all import *
>>> pts = [[Integer(1), Integer(1)], [-Integer(2), Integer(2)], [Integer(1), -Integer(3)/Integer(2)]]
>>> g = streamline_plot((x + y) / sqrt(x**Integer(2) + y**Integer(2)),
...                     (x,-Integer(3),Integer(3)), (y,-Integer(3),Integer(3)), start_points=pts)
>>> g += point(pts, color='red')
>>> g
Graphics object consisting of 2 graphics primitives

Sage Live

pts = [[1, 1], [-2, 2], [1, -3/2]]
g = streamline_plot((x + y) / sqrt(x^2 + y^2),
                    (x,-3,3), (y,-3,3), start_points=pts)
g += point(pts, color='red')
g

Note

Streamlines currently pass close to start_points but do not necessarily pass directly through them. That is part of the behavior of matplotlib, not an error on your part.

Make live