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.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

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

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.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

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

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

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)

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

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

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