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:

QyQ+SSNubUc2IjclLUkkY29zR0YlNiNJJHRhdUdGJSomSSJCR0YlIiIiLUkkc2luR0YlRilGLS9GKjsiIiEsJEkjUGlHJSpwcm90ZWN0ZWRHIiIjRi0=

NyUtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kkdGF1R0YoKiZJIkJHRigiIiItSSRzaW5HRiVGKUYtL0YqOyIiISwkSSNQaUdGJiIiIw==

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.

QyQ+SSRubXNHNiItSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSVzdWJzR0YoNiQvSSJCR0YlLCRJImpHRiUjIiIiIiIjSSNubUdGJS9GMDtGMiIiJUYy

NiY3JS1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSR0YXVHRiksJC1JJHNpbkdGJkYqIyIiIiIiIy9GKzsiIiEsJEkjUGlHRidGMTclRiRGLUYyNyVGJCwkRi0jIiIkRjFGMjclRiQsJEYtRjFGMg==

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

Note the use of the "character" \134n (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.

QyQ+SSRubXNHNiI3JS1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSSJ0R0YlKiZJIkJHRiUiIiItSSRzaW5HRilGLEYwKiZJIldHRiVGMEYtRjBGMA==

NyUtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kidEdGKComSSJCR0YoIiIiLUkkc2luR0YlRilGLSomSSJXR0YoRi1GKkYt

QyQ+SSVubXNxRzYiNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSVzdWJzR0YpNiQ3JC9JIkJHRiUsJEkiakdGJSMiIiIiIiMvSSJXR0YlLCRGMiNGNCIjNUkkbm1zR0YlL0YyO0Y0IiIlRjQ=

NyY3JS1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ0R0YpLCQtSSRzaW5HRiZGKiMiIiIiIiMsJEYrI0YwIiM1NyVGJEYtLCRGKyNGMCIiJjclRiQsJEYtIyIiJEYxLCRGKyNGPEY0NyVGJCwkRi1GMSwkRisjRjFGOA==

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

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

QyQmSSVubXNxRzYiNiMiIiMiIiI=

NyUtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kidEdGKC1JJHNpbkdGJUYpLCRGKiMiIiIiIiY=

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.

QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlIiIi

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

QyQtSStzcGFjZWN1cnZlRzYiNiomSSVubXNxR0YlNiMiIiMvSSJ0R0YlOyIiISwkSSNQaUclKnByb3RlY3RlZEciIiUvSShzY2FsaW5nR0YlSSxjb25zdHJhaW5lZEdGJS9JKmxhYmVsZm9udEdGJTckSSdTWU1CT0xHNiRGMUkoX3N5c2xpYkdGJSIjNy9JJ2xhYmVsc0dGJTclUSJ4RiVRImhGJVEiekYlL0klYXhlc0dGJUkmYm94ZWRHRiUvSSZjb2xvckdGJUkkcmVkR0YlL0kmdGl0bGVHRiVJWkVsbGlwdGljYWx+U3BpcmFsflxufndpdGh+YXhpc35yYXRpb35vZn4xfmFuZH5yaXNlfm9mfjAuMkdGJSIiIg==

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

QyQtSStzcGFjZWN1cnZlRzYiNiomSSVubXNxR0YlNiMiIiUvSSJ0R0YlOyIiISwkSSNQaUclKnByb3RlY3RlZEdGKi9JKHNjYWxpbmdHRiVJLGNvbnN0cmFpbmVkR0YlL0kqbGFiZWxmb250R0YlNyRJJ1NZTUJPTEc2JEYxSShfc3lzbGliR0YlIiM3L0knbGFiZWxzR0YlNyVRInhGJVEiaEYlUSJ6RiUvSSVheGVzR0YlSSZib3hlZEdGJS9JJmNvbG9yR0YlSSRyZWRHRiUvSSZ0aXRsZUdGJUlaRWxsaXB0aWNhbH5TcGlyYWx+XG5+d2l0aH5heGlzfnJhdGlvfm9mfjJ+YW5kfnJpc2V+b2Z+MC40R0YlIiIi

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEiP0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSwwLjExMTExMTFlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEucGxvdFtvcHRpb25zXUYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHRjRGLw==

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.

JSFH