Plotting fields¶

class sage.plot.plot_field.PlotField(xpos_array, ypos_array, xvec_array, yvec_array, options)[source]¶

Bases: GraphicPrimitive

Primitive class that initializes the PlotField graphics type

get_minmax_data()[source]¶

Return a dictionary with the bounding box data.

EXAMPLES:

Sage

sage: x,y = var('x,y')
sage: d = plot_vector_field((.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')
>>> d = plot_vector_field((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')
d = plot_vector_field((.01*x,x+y), (x,10,20), (y,10,20))[0].get_minmax_data()
d['xmin']
d['ymin']

sage.plot.plot_field.plot_slope_field(f, xrange, yrange, **kwds)[source]¶

plot_slope_field takes a function of two variables xvar and yvar (for instance, if the variables are $x$ and $y$ , take $f (x, y)$ ), and at representative points $(x_{i}, y_{i})$ between xmin, xmax, and ymin, ymax respectively, plots a line with slope $f (x_{i}, y_{i})$ (see below).

plot_slope_field(f, (xvar,xmin,xmax), (yvar,ymin,ymax))

EXAMPLES:

A logistic function modeling population growth:

Sage

sage: x,y = var('x y')
sage: capacity = 3 # thousand
sage: growth_rate = 0.7 # population increases by 70% per unit of time
sage: plot_slope_field(growth_rate * (1-y/capacity) * y, (x,0,5), (y,0,capacity*2))
Graphics object consisting of 1 graphics primitive

Python

>>> from sage.all import *
>>> x,y = var('x y')
>>> capacity = Integer(3) # thousand
>>> growth_rate = RealNumber('0.7') # population increases by 70% per unit of time
>>> plot_slope_field(growth_rate * (Integer(1)-y/capacity) * y, (x,Integer(0),Integer(5)), (y,Integer(0),capacity*Integer(2)))
Graphics object consisting of 1 graphics primitive

Sage Live

x,y = var('x y')
capacity = 3 # thousand
growth_rate = 0.7 # population increases by 70% per unit of time
plot_slope_field(growth_rate * (1-y/capacity) * y, (x,0,5), (y,0,capacity*2))

Plot a slope field involving sin and cos:

Sage

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

Python

>>> from sage.all import *
>>> x,y = var('x y')
>>> plot_slope_field(sin(x+y) + cos(x+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')
plot_slope_field(sin(x+y) + cos(x+y), (x,-3,3), (y,-3,3))

Plot a slope field using a lambda function:

Sage

sage: plot_slope_field(lambda x,y: x + y, (-2,2), (-2,2))
Graphics object consisting of 1 graphics primitive

Python

>>> from sage.all import *
>>> plot_slope_field(lambda x,y: x + y, (-Integer(2),Integer(2)), (-Integer(2),Integer(2)))
Graphics object consisting of 1 graphics primitive

Sage Live

plot_slope_field(lambda x,y: x + y, (-2,2), (-2,2))

sage.plot.plot_field.plot_vector_field(f_g, xrange, yrange, plot_points=20, frame=True, **options)[source]¶

plot_vector_field takes two functions of two variables xvar and yvar (for instance, if the variables are $x$ and $y$ , take $(f (x, y), g (x, y))$ ) and plots vector arrows of the function over the specified ranges, with xrange being of xvar between xmin and xmax, and yrange similarly (see below).

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

EXAMPLES:

Plot some vector fields involving sin and cos:

Sage

sage: x,y = var('x y')
sage: plot_vector_field((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')
>>> plot_vector_field((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')
plot_vector_field((sin(x),cos(y)), (x,-3,3), (y,-3,3))

Sage

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

Python

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

Sage Live

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

Plot a gradient field:

Sage

sage: u, v = var('u v')
sage: f = exp(-(u^2 + v^2))
sage: plot_vector_field(f.gradient(), (u,-2,2), (v,-2,2), color='blue')
Graphics object consisting of 1 graphics primitive

Python

>>> from sage.all import *
>>> u, v = var('u v')
>>> f = exp(-(u**Integer(2) + v**Integer(2)))
>>> plot_vector_field(f.gradient(), (u,-Integer(2),Integer(2)), (v,-Integer(2),Integer(2)), color='blue')
Graphics object consisting of 1 graphics primitive

Sage Live

u, v = var('u v')
f = exp(-(u^2 + v^2))
plot_vector_field(f.gradient(), (u,-2,2), (v,-2,2), color='blue')

Plot two orthogonal vector fields:

Sage

sage: x,y = var('x,y')
sage: a = plot_vector_field((x,y), (x,-3,3), (y,-3,3), color='blue')
sage: b = plot_vector_field((y,-x), (x,-3,3), (y,-3,3), color='red')
sage: show(a + b)

Python

>>> from sage.all import *
>>> x,y = var('x,y')
>>> a = plot_vector_field((x,y), (x,-Integer(3),Integer(3)), (y,-Integer(3),Integer(3)), color='blue')
>>> b = plot_vector_field((y,-x), (x,-Integer(3),Integer(3)), (y,-Integer(3),Integer(3)), color='red')
>>> show(a + b)

Sage Live

x,y = var('x,y')
a = plot_vector_field((x,y), (x,-3,3), (y,-3,3), color='blue')
b = plot_vector_field((y,-x), (x,-3,3), (y,-3,3), color='red')
show(a + b)

We ignore function values that are infinite or NaN:

Sage

sage: x,y = var('x,y')
sage: plot_vector_field((-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')
>>> plot_vector_field((-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')
plot_vector_field((-x/sqrt(x^2+y^2),-y/sqrt(x^2+y^2)), (x,-10,10), (y,-10,10))

Sage

sage: x,y = var('x,y')
sage: plot_vector_field((-x/sqrt(x+y),-y/sqrt(x+y)), (x,-10, 10), (y,-10,10))
Graphics object consisting of 1 graphics primitive

Python

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

Sage Live

x,y = var('x,y')
plot_vector_field((-x/sqrt(x+y),-y/sqrt(x+y)), (x,-10, 10), (y,-10,10))

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

Sage

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

Python

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

Sage Live

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