# SMC Wizard Documentation File # Copyright: SageMathCloud Authors, 2015 # License: Creative Commons: Attribution-ShareAlike 4.0 International (CC BY-SA 4.0) language: sage --- category: Plotting / 2D --- title: Basic Plot code: | print "Basic plot: you need a variable, and a function of the variable here x*sin( x^2 )" # this defaults many many things you can control # here in 2 steps using plot() and show() (which can be combined into one line): pt = plot( x*sin(x^2), x ) # create plot show( pt ) descr: Basic plot of a function, letting most options default. --- title: Upper and lower limit code: | print "Basic plot extended with upper lower, limit on x" pt = plot( x*sin(x^2), x, 0, 6*pi ) # args in order: function, variable, lower limit, upper limit, show( pt ) descr: > Basic plot, but using plot arguments to control upper and lower limits on x. --- title: Many more options/arguments code: | print "Plot with many more options/arguments, used by name that are somewhat self explanatory -- tweak values to see what they do, or google them" # show also offers named arguments for control. pt = plot( x*sin(x^2), x, 0, 6*pi, rgbcolor="red", linestyle = "-", fill=False, thickness=1, legend_label ="a legend label" ) show( pt, aspect_ratio = 1, axes=true, frame=True, gridlines=false, figsize=4, xmin=-0, xmax=8*pi, ymin=-6, ymax=6 ) descr: > Plot showing arguments to both plot and show that allow a lot of control over various aspects of the graph. --- title: Multiple lines/function code: | print "Plot Multiple lines/function on plots" # also note that functions may be computed and put into variables outside of plot() or show() function1 = x*sin(x^2) function2 = x^.5*sin(x) pt1 = plot( function1, x, 0, 6*pi, rgbcolor="red", linestyle = "-", fill=False, thickness=1, legend_label ="functionl" ) pt2 = plot( function2, x, 0, 6*pi, rgbcolor="blue", linestyle = "--", fill=False, thickness=1, legend_label ="function2" ) show( pt1 + pt2, aspect_ratio = 1, axes=true, frame=True, gridlines=false, figsize=6, xmin=-0, xmax=8*pi, ymin=-8, ymax=8 ) descr: > Multiple plots can be plotted on the same graph, optionally using different line styles. --- title: Custom tickmarks on the axes code: | print "Plot with ticks, LaTeX formatting at specific locations" fx = 1 * (x^2)+1 pt = plot( fx ,x, 0,5, ticks=[[0,1,e,pi,sqrt(20)], 5 ], tick_formatter="latex") show( pt, figsize=[3,3] ) descr: > Tickmarks on the axes can be controlled both in position, spacing, special values, and using LaTeX for values such as pi. --- title: Using Python functions code: | print "Plot using Python functions defined using 'def:'" # see http://www.sagemath.org/doc/tutorial/tour_functions.html def f(z): # this and next line define a Python function return z^2 print "Define plot and show it in one line of code" show( plot(f, 0, 2), figsize=[6,2] ) print "Alternate syntax " var( "z" ) show( plot(f(z), z, 0, 2), figsize=[6,2], ) descr: > Plot using Python functions. Python functions are defined using the keyword def. --- category: plotting / 2D Advanced --- title: Implicit Plot code: | print "Implicit plot ( here function of x, y not solved for y )" var ('x y') imp = exp(x)==y^2 implicit_plot(imp, (x,-2,2), (y,-3,3), linestyle="--", figsize=4, axes="true", aspect_ratio=1) descr: > Implicit plots are used where the relationship is implicit in an equation rather than explicitly solved. --- title: Parametric Plot code: | print "Parametric Plot ( here 3 functions)" # for graphing parametric equations ( 2 functions with a common variable implying a relation between the 2 functions ) # 3 plots all controlled by the parameter t var( "t" ) # our parameter ll = -1. * pi # lower limit of graph ul = 1. * pi # upper limit of graph # parametric_plot( x axis function, y axis function ) , ( parametric variable, min value, max value ), other plot args ) p1 = parametric_plot((cos(t),sin(t)), (t, ll, ul), rgbcolor=hue(0.2)) p2 = parametric_plot((cos(t),sin(t)^4), (t, ll, ul), rgbcolor=hue(0.4)) p3 = parametric_plot((cos(t),sin(t)^8), (t, ll, ul), rgbcolor=hue(0.6)) show( p1+p2+p3, axes=false) descr: > Parametric plots are used where the relationship between variables is implied by their relationship to a third variable, the parameter. --- title: Polar Plot code: | print "Polar Plot" # Plot using polar coordinates: note that many of the options for plot() work here. var( "theta" ) f = 1-2*sin(2*theta); show( f ) pp = polar_plot( f, theta, 0, 2*pi, rgbcolor="red", linestyle = "--" ) show( pp ) descr: > Polar plots are used where the relationship between x and y, in Cartesian coordinates, is expressed in terms of r an theta, in polar coordinates. --- title: Slope Field code: | print "Plot Slope Field" var( "x" ) var( "y" ) slope = y ^ 2 # define the slope to be the value of the function y^2 pt = plot_slope_field( slope, ( x, -1, 1 ), ( y, .001, 10 ), plot_points = 10 ) # plot_points: higher numbers plot more "slopes" show( pt, aspect_ratio='automatic' ) # aspect_ratio: try .1, 1, 10 'automatic' or leave out and let it default descr: > Slope fields show the slope of 2 variables defined over a "field", that is a relationship between x, an y that extends over an area of the plane ( 2 D). The slope is indicated by the slope of a short line segment. --- title: Vector Field code: | print "Plot vector field" var( "x" ) var( "y" ) A = x - sin(y) B = -y * cos(x) field = (A,B) H = plot_vector_field( field, (x,0,2*pi), (y,0,2*pi), plot_points=15, color='green' ) # plot_points: higher numbers plot more vectors show( H ) descr: > Plot of a vector field shows an arrow of variable length indicating the direction an magnitude of the vector at a large number of points over the area of the graph. --- title: List of points code: | print "Plot of a list of points" # either type of list works dictionary_list = {22: 3365, 27: 3295, 37: 3135, 42: 3020, 47: 2880, 52: 2735, 57: 2550} tuple_list = [ ( 22, 3365) , ( 27, 3295 ), ( 37, 3135 ), ( 42, 3020 ), ( 47, 2880 ), ( 52, 2735 ), (57, 2550 ) ] listPlot = list_plot( dictionary_list ) show( listPlot, xmin=0, xmax =50, ymin=2000, ymax=4000, figsize=[4,4] ) descr: > Points, contained int Python lists, are all plotted. The lists may be dictionary lists, or lists of 2D tuples. --- title: Scatter Plot with Line code: | print "Scatter plot and line" # http://www.packtpub.com/article/plotting-data-sage # python function def noisy_line(m, b, x): ret = m * x + b + 0.5 * (random() - 0.5) return ret slope = 1.0 intercept = -0.5 x_coords = [random() for t in range(50)] y_coords = [noisy_line(slope, intercept, x) for x in x_coords] sp = scatter_plot(zip(x_coords, y_coords)) sp += line([(0.0, intercept), (1.0, slope+intercept)], color='red') sp.show( figsize=[6,2]) descr: > Plot points, x coords in one list, y coords in another. --- title: Contour Plot code: | print "Contour plot" # this might be improved, some explanation f = lambda x,y: cos(x*y) show ( contour_plot(f, (-4, 4), (-4, 4)) ) descr: > This plot indicates the contour of a function using 50 shades of gray. --- category: plotting / Basic Elements --- title: Points as points (plain and fancy) code: | print "Plotting points as points ( plain and fancy )" plainPoints = point( [1, 1] ) + point( [2, 1] ) plot1 = plot( plainPoints ) show( plot1, figsize=4 ) fancyPoints = point((0.5, 0.5), rgbcolor=(1, 0, 0), size=30) fancyPoints = fancyPoints + point((0.7, 0.7), rgbcolor=(0, 1, 0), size=50) show( fancyPoints , figsize=4 ) descr: > The graphic object point may be plotted using plot() Plotting this way may combine various elements, both those in this category but also other 2D plots. --- title: Line with point at the end code: | print "Plot a line with point ( dot ) at end" # computed in polar form, not required r = 5 t = pi/4. x = r*cos( t ) y = r*sin( t ) aPoint = point( ( x, y ), rgbcolor="red", size=30 ) aLine = line([(0.,0.), ( x, y ) ] , rgbcolor="blue" , linestyle = "dashed" ) plot = aPoint + aLine show( plot, xmin=-1.2, xmax=5.2, ymin=-1.2, ymax=5.2 , figsize=[3,3] ) descr: > A line is plotted with a point at the end. --- title: Lines and points with Python function code: | Print "Plot lines and points with Python function" def unitRotator( theta ): x = cos( theta ) y = sin( theta ) aPoint = point2d( ( x, y ), rgbcolor="red", size=30 ) #point2d? aLine = line([(0.,0.), ( x, y ) ] ) myRotator = aPoint +aLine return myRotator graphic = point( (0, 0 ) ) for theta in srange( .0, 2*pi, 2*pi/10): graphic = graphic + unitRotator( theta ) show( graphic , xmin=-1.2, xmax=1.2, ymin=-1.2, ymax=1.2 , figsize=[3,3] ) descr: > A Python function defines many lines and points all of which are plotted. --- title: Arc code: | print "Plot an Arc" graphic = ( arc((.1,.2), 1, sector=(0, 1*pi /4) ) ) show( graphic , xmin=-1.2, xmax=1.2, ymin=-1.2, ymax=1.2, figsize=[3,3] ) descr: > Plot an a arc, a segment of a circle. Center, radius and section of arc are all parameters. --- title: Circle code: | print "Plot a Circle" anGraphic = circle((0,0), .5, rgbcolor=(1,0,0), fill=False ) # fill controls filling of circle with color show( anGraphic, aspect_ratio=1, figsize = 4 ,xmax = 1, ymax = 1) descr: Plot a circle. Center, radius, color are parameters. --- title: Arrow code: | print "Plot an Arrow" anGraphic = arrow((0,1), (2,3) ) # start cord, end cord as tuples show( anGraphic, figsize = 2 ) descr: Plot an Arrow, describe by starting and ending coordinates. --- title: Text code: | print "Plot Text" # text is followed by tuple for position aGraphic = text("Sample Text", (5,4), rgbcolor=(1,0,0)) show( aGraphic, aspect_ratio=1, figsize = 3 ) descr: > Plot Text, arguments for text, position, and color.