Section 6: More on Graphing

Parametric Equations

Maple's plot  command can also be used to graph curves described by parametric equations.

To graph the parametric curve corresponding to the pair of parametric equations:    and  on the parameter interval  use the command:

plot([f(t), g(t), t = a..b], x = xmin..xmax, y = ymin..ymax);  

There are two things to take careful note of here. First note that there are three entries in the square brackets : the two parametric expressions for x  and y   and  the parameter domain. Also note that the viewing window for the plot is separately specified by the x-  and y -ranges (i.e., x = xmin..xmax, y = ymin..ymax ).

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Example 1

Plot the parametric curve determined by     and     over the t -interval [-2, 2] .

>	plot([t^2-t,2*t-t^3,t=-2..2],x=-2..5,y=-5..5);

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Exercise 6.1

Plot the parametric curve defined  and  over the t -interval  .

For a viewing window let x  and y  range between -2 and 2 .

Student Workspace 6.1

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Answer 6.1

>	plot([sin(3*t),sin(4*t),t=0..2*Pi],x=-2..2,y=-2..2);

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Implicit Plots

Maple can plot curves that are implicitly defined by an equation in the variables x  and y .

Example 1

To plot the graph of the hyperbola given by the equation:  use the implicitplot  command. To use this command we must first load the "plots" package using the "with" command.

>	with(plots):

Warning, the name changecoords has been redefined

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Note the syntax for this command on the next line.

>	implicitplot(x^2/4-y^2/4=1,x=-5..5,y=-5..5);

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Example 2

Graph the equation    using the implicitplot  command.

Recall that this is the equation of an ellipse with the lengths of major and minor axes equal to 10 and 6 respectively.  

Our first attempt at getting the expected graph comes up short !  

>	implicitplot(x^2/25+y^2/9=1,x=-5..5,y=-5..5);

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Why did we get a circle instead of an ellipse ?

The problem here is that the x-  and y -scales are not equal. To force equal scaling add "scaling=constrained" or click on the graph to expose the graphing toolbar, and select the button marked 1:1.

The graph then appears as seen in the following figure.

>	implicitplot(x^2/25+y^2/9=1,x=-5..5,y=-5..5,scaling=constrained);

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Exercise 6.2

Graph the equation  

Student Workspace 6.2

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Answer 6.2

>	implicitplot(x^2+4*y^2=4,x=-3..3,y=-2..2,scaling=constrained);

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Polar Graphs (optional)

Graphs of polar equations  are handled by the polarplot  command, which is part of the plots package accessed using with(plots).

Here are some examples. Note that we include the option scaling = constrained  to get geometric perspective.

>	polarplot(1+cos(theta),theta=-Pi..Pi,scaling=constrained);

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>	polarplot(sin(3*theta),theta=-Pi..Pi,scaling=constrained);

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Another way of graph polar graphs is to use the plot option coords=polar  and graph the curve using parametric equations. The general form of the command is:

plot([r(s), theta(s), s=a..b], coords=polar);

If the parameter  is actually the angle , the command becomes

plot([r(theta), theta, theta=a..b], coords=polar);

For example, to graph 1+cos( ) in polar coordinates using the plot  command, type:

>	plot([1+cos(theta),theta,theta=-Pi..Pi],coords=polar);

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The coords=polar  option can be applied to implicitplot  command as well.

For example, to graph the lemniscate  over the  interval , type:

>	implicitplot(r^2=4*cos(2*theta), r=0..2, theta=0..2*Pi,coords=polar, scaling=constrained, grid=[50,50]);

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Plot Options

There are many options available when you use the plot  command.  To see a list, execute the next line to go directly to Maple's Help Page on this command.  Skip this if you wish.

>	?plot[options];

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