Quick Maple worksheet to produce graphs of ellipse,  and elliptical spiral,  

for solutions for Homework No. 1 

First, for simplicity, let us simply look at the 2-dimensional projection of this motion  

into the x,y-plane. 

         We define the 2-dimensional portion of the  

curve as a parametric equation; below is the syntax to do this: 

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(1)

 

The equation above has an arbitrary parameter in it, the ratio of the semiaxes in the two directions. 

 Since we must insert  a value for this parameter,  

the syntax below creates a collection of ellipses  

with 4 different, specific values for this ratio. 

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(2)

 

The command below then produces a plot with each of these distinct, specific ellipses. 

  Note the use of the "character" \n  (in the title command),  

                   which gives a carriage return inside the title name. 

  Also note one may use Greek characters for labelling, by using the SYMBOL font. 

  Also note the use of the command scaling=constrained,  

                  which insists on the same length scale for both axes, thereby  

                                                  making the "eye" see it more naturally. 

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However, now let's go ahead and produce something with the full 3-dimensional. 

This will in fact be rather more difficult to visualize; however, inside a true  

Maple file one may use the cursor to actually rotate the displayed graph around  

any axis desired, to achieve a better perspective on it. 

Here, of course, we will now need two parameters, the new one specifying the  

speed of vertical rise, as described in the textual material. 

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(3)

 

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(4)

 

The object above is a "list" of 3-dimensional curve commands. 

The command below shows how to extract a single element from that list: 

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(5)

 

Next we want to plot one or more of them.   

They are a little complicated to plot all together on the same set of axes,  

so I will display (just) two of them, separately. 

The command to plot a 3-dimensional curve lives in a Maple "package" called plots. 

It must be "loaded" before it can be used, as is done just below. 

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(6)

 

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The command above is NOT part of this document, exactly, but, rather, shows a way to acquire more informtation concerning plot options, for 2-dimensional plots.  Use plot3d  

for 3-dimensional plots. 

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