Graphing

  ?plottingguide 

Rectangular Coordinate Plots, Multiple Plots

# STANDARD COORDINATES
# Maple can construct many kinds of graphs, a feature that you can use to
# visualize mathematical objects and processes. The command

 plot( sin(3*x), x = -Pi..Pi );

# produces a plot containing the curve y = sin(3 x) for x in the
# interval from -Pi to Pi. 

# No space is allowed between the double-dots in a plot command. Scales
# on the x-axis and y-axis are chosen in the maple engine.  To change
# this behavior:

 plot( sin(3*x), x = -Pi..Pi, scaling = constrained );

# Sometimes it is useful to restrict the range over which y varies.  We
# get a misleading graph from

 plot( tan(x), x = -5..5 );

# It does not accurately represent the vertical asymptotes of y = tan(x).
# Better results are obtained with

 plot( tan(x), x = -5..5, y = -2..2 );
 plot(tan,-5..5,-2..2); # Same result with less typing

# MULTIPLE CURVES ON ONE PLOT
# To plot y = sin(x) and y = sin(3 x) one a single axis, code
 
 plot( { sin(x), sin(3*x) }, x = -Pi..Pi ); 

# The curve equations are inside set-delimiters of curly braces
# You can also use square brackets, helpful for assigned colors.

 plot([sin(x),sin(3*x)],x = -Pi..Pi,color=[red,blue] ); 

# Decimal points can conflict with double-dot range parsing 

  plot(1+x^2,x=1 .. 0.8); # Compare syntax x=1...8

# THE PLOTS PACKAGE
# The DISPLAY command is found in the PLOTS package.
# It allows you to assemble several plots onto one set of axes.

 with(plots):
 plot1 := plot(x^2,x=0..4): # Colon to omit graphic display
 plot2 := plot(3*x^2-2,x=0..4):
 display([ plot1,plot2]);
 

Parametric, Polar and Raw Data Plots

# PARAMETRIC EQUATIONS
# A curve in the plane can be described as the graph of a function, as in
# the graph of 
#                           1     /     2\
#                       y = - sqrt\4 - x /
#                           2             
# for x in the interval from -1 to 1. It can be given parametrically as
#  
#               (x(t), y(t)) = (2 cos(t), sin(t))
#
# for t in the interval from 0 to Pi. Often we interpret such a curve as
# the path traced by a moving particle at time t . Use the plot command
# to draw a curve from this parametric description:

 plot( [2*cos(t), sin(t), t = 0..Pi] ); 
   # unconstrained, distorted graphic
 plot( [2*cos(t), sin(t), t = 0..Pi], scaling = constrained);  
   # constrained is not distorted

# POLAR COORDINATES
# Polar plots are a special kind of parametric plot.  The polar
# coordinates ( r, theta ) of points on a curve can be given as a
# function of some parameter t. In many cases the parameter is just the
# angle theta. Consider the ellipse defined in standard coordinates by
#
#                         x^2  + 4 y^2  = 4
#
# To find an equation relating the polar coordinates r and theta of a
# typical point on this ellipse, we make the substitution x = r cos(t)
# and y = r sin(t), where t = theta.

 subs( x = r*cos(t), y = r*sin(t), x^2 + 4*y^2 = 4);
 simplify( % );
 solve( %, r );

# We find that the ellipse is the collection of points whose polar
# coordinates ( r, theta ) satisfy
#
#                      2           4        
#                     r  = -----------------,
#                                          2
#                          4 - 3 cos(theta) 
#
# and the following commands draw the right half of the ellipse:

 r := 2/sqrt( 4 - 3*cos(t)^2 ):
 plot( [ r, t, t = -Pi/2..Pi/2 ], coords = polar );
 
# Here are some more of examples of polar plots that you can try:

 plot( [ 1, t, t = 0..2*Pi], coords=polar ):
 plot( [t, t, t = 0..2*Pi], coords=polar ):
 plot( [ sin(4*t), t, t = 0..2*Pi], coords=polar ): 
   # a more realistic picture is obtained with scaling = constrained.

# PLOTTING RAW DATA

# Maple can plot data consisting of pairs of x and y values.  For
# example, the double list

 data := [ [0, 0.53], [1, 1.14], [2, 1.84], [3, 4.12] ];

# defines data as a sequence of five data points.
# A double list is something enclosed in square brackets, with
# comma-separated elements in square brackets. Lists are used
# for collections of objects where the order matters.

 data[1]; data[2]; data[3]; data[4]; 
   # Individual items are accessed this way
 data[3][2]; 
   # Extract 2nd coordinate 1.84 from the 3rd data point [2,1.84]

# To plot the points in a double-list we use commands like

 plot(data); # The simplest interface to plotting points.

 plot(data,opts); # Plot with an option definition like
 opts:=style = point, symbol = circle, color = black;
 opts:=style = line, view = [0..4, 0..5];
 opts:=style = line, title = "Experiment 1";

# The double quote is used to specify a plot title string.  See
# ?plot[options] for more information, e.g., about symbols and line
# styles available. See ?plottingguide for ideas and examples.

# Display help ?readdata and ?writedata to find out how
# to read or write a file of data points into a Maple session.

Graphs in Three Dimensions

# EXAMPLE. Graphing functions of two variables.
 f:=(x,y)->x^2-y^2;
 plot1:=plot3d(f(x,y),x=-1..1,y=-1..1,color=blue):
       # Graph of z=x^2-y^2
 plot2:=plot3d([.5*cos(theta),.5*sin(theta),z],
          theta=0..2*Pi,z=0..1,color=pink):
       # Graph of a cylinder, defined parametrically                    
 plot3:=plot3d(.5,x=-1..1,y=-1..1,color=brown): 
       # A horizontal plane z=0.5
 with(plots):
 display([plot1,plot2,plot3],axes=boxed);  
       # Click on the plot to move it around in space.
       # A box in upper left of window will display
       # the spherical coordinates you're looking from!

# EXAMPLE. Implicit plots.
 with(plots): f:=(x,y)->x^2-y^2;
 implicitplot(f(x,y)=.5,x=-1..1,y=-1..1,color=black); 
       # The level curve where x^2-y^2=.5
 with(plots):g:=(x,y)->3*x^2-2*x*y+5*y^2:  
       # A quadratic function of two variables
 implicitplot(g(x,y)=1,x=-2..2,y=-2..2,color=blue,grid=[80,80]);
       # Default grid unappealing, needed better resolution.