<Text-field style="Heading 1" layout="Heading 1">Parametric Equations</Text-field>

Worksheet #11

Math 303, Fall 2008

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As you may remember, we can describe curves using parametric equations by

x=f(t)

y=g(t)

a \342\211\244 t \342\211\244 b

Here are some examples of curves defined by parametric equations. The coordinates x and y are entered as functions of t. The Maple format is:

plot(['equation for x', 'equation for y', t='t range']);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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Notice that I didn't give a range of values for x and y, but Maple just chose reasonable values. You can choose these values yourself if you like. Here's the graph of the same curve from the previous example, where I've changed the x and y ranges:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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Here's an example of a parametric plot of a linear function:

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Looking at the graph, what is the slope of the function?

What are the x and y coordinates when t=0?

The general form for a line in parametric equations is

plot([mt+c,nt+d, t=a..b])

What is the slope or such a line? What do the numbers c and d tell us about the line?

We can also draw 3d curves using parametric equations. The format is:

plot3d(['function for x','function for y', 'function for z'], t='t range', u='u range');

Note: even though the variable u is never used, it is part of the format, and Maple will squawk if you don't include it.

As always, it is easier to make sense of 3d graphs if you put axes on your graphs. Here are some examples:

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JSFH

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

Notice that sometimes the graph looks a little choppy. This is because Maple didn't use enough points when plotting. To fix this you can increase the number of points as in:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JC1GLDYlUSdwbG90M2RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Ny1GOzYmLUYjNiktRiw2JVEkY29zRicvRjVRJmZhbHNlRicvRjhRJ25vcm1hbEYnLUY7NiQtRiM2Iy1JJW1zdXBHRiQ2JS1GLDYlUSJ0RidGNEY3LUYjNiMtSSNtbkdGJDYkUSIyRidGSC8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGSC1JI21vR0YkNi1RIixGJ0ZILyUmZmVuY2VHRkcvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGRy8lKnN5bW1ldHJpY0dGRy8lKGxhcmdlb3BHRkcvJS5tb3ZhYmxlbGltaXRzR0ZHLyUnYWNjZW50R0ZHLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEkc2luRidGRkZILUY7NiQtRiM2I0ZRRkhGZ25GUUZILyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmduRlEtRmhuNi1RIj1GJ0ZIRltvL0Zeb0ZHRl9vRmFvRmNvRmVvRmdvL0Zqb1EsMC4yNzc3Nzc4ZW1GJy9GXXBGYXEtRlc2JFEiMEYnRkgtRmhuNi1RIy4uRidGSEZbb0ZfcUZfb0Zhb0Zjb0Zlb0Znby9Gam9RLDAuMjIyMjIyMmVtRicvRl1wRltwRlYtRmhuNi1RIn5GJ0ZIRltvRl9xRl9vRmFvRmNvRmVvRmdvRmlvRltyLUYsNiVRJSZwaTtGJ0ZGRkhGZ25GXHItRiw2JVEidUYnRjRGN0ZccUZjcUZmcS1GVzYkUSIxRidGSEZnbkZcci1GLDYlUSpudW1wb2ludHNGJ0Y0RjdGXHEtRlc2JFElOTAwMEYnRkhGSEYr

Here's another unusual example:

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Polar Coordinates</Text-field>

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Here are some examples of parametrized curves using polar coordinates. You can enter the radius, r, as a function of t, and the angle, 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, as a function of t. The format is:

plot(['function for r', 'function for 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', t='t range'], 'x range', 'y range', coords=polar);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

If r is a function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YyLyUwZm9udF9zdHlsZV9uYW1lR1EwUGFyYWdyYXBoU3R5bGUzRidGNQ==, we graph the curve in either of the following two formats:

1. plot(['function for r', 'function for theta', theta= 'LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEnJiM5NTI7RicvJSdpdGFsaWNHUSV0cnVlRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi9GMi8lMGZvbnRfc3R5bGVfbmFtZUdRMFBhcmFncmFwaFN0eWxlMkYnRjU= range'], coords=polar);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

2. with(plots):polarplot('function for r',theta='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 range');

Note: The with(plots): at the beginning tells Maple that you want to use the plots package. You only need to enter this once per worksheet.

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

I pulled the theta from the pull-down menu. If you don't like greek letters use whichever letter you like:

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Cylindrical Coordinates</Text-field>

Here are some examples of plotting in cylindrical coordinates. Here r is defined as a function of 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 and z, and the format is:

plot3d('function for r', theta='LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY2LUkjbWlHRiQ2NlEnJiM5NTI7RicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHRjcvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJSdvcGFxdWVHRjcvJStleGVjdXRhYmxlR0Y3LyUpcmVhZG9ubHlHRjcvJSljb21wb3NlZEdGNy8lKmNvbnZlcnRlZEdGNy8lK2ltc2VsZWN0ZWRHRjcvJSxwbGFjZWhvbGRlckdGNy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdGNy8lMGZvbnRfc3R5bGVfbmFtZUdRMFBhcmFncmFwaFN0eWxlMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0YvRjJGNUY4RjpGPEY+RkBGQ0ZGRkhGSkZMRk5GUEZSRlRGVkZZ range', z='z range', coords=cylindrical);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Spherical Coordinates</Text-field>

How about a 3D spherical plot? You can do this by entering \317\201 as function of 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 and 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 . (If you don't like using greek letters, you can replace 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 and 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 with the letters of your choice). The format is

plot3d('function for rho', theta='LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EnJiM5NTI7RicvJSVzaXplR1EjMTRGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjVGOA== range', phi='LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2J1EnJiM5NjY7RicvJSVzaXplR1EjMTRGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjVGOA== range', coords=spherical);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEncGxvdDNkRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNjUtSSNtbkdGJDYkUSIyRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEoJnRoZXRhO0YnL0YwRkZGPi1GQTYtUSI9RidGPkZEL0ZIRkZGSUZLRk1GT0ZRL0ZUUSwwLjI3Nzc3NzhlbUYnL0ZXRlxvLUY7NiRRIjBGJ0Y+LUZBNi1RIy4uRidGPkZERmpuRklGS0ZNRk9GUS9GVFEsMC4yMjIyMjIyZW1GJy9GV0ZVRjotRkE2LVEifkYnRj5GREZqbkZJRktGTUZPRlFGU0Zmby1GLDYlUSUmcGk7RidGZm5GPkZALUYsNiVRJiZwaGk7RidGZm5GPkZnbkZeb0Zhb0Zqb0ZALUYsNiVRJ2Nvb3Jkc0YnRi9GMkZnbi1GLDYlUSpzcGhlcmljYWxGJ0YvRjJGPg==

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

plot3d(2,theta=-1..2*Pi,phi=0..Pi,coords=spherical);

plot3d(phi/theta,theta=-1..2*Pi,phi=0..Pi,coords=spherical);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Exercises</Text-field>

1. Graph a cylinder with radius 3.

2. Graph a cone with height 4 and and radius 3

2. Graph a sphere of radius 4.

3. Graph a circle of radius 2 in the x,z - plane.

4. Graph each of the following parametrizations. What is the difference between the two?

curve 1:

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JVEiMEYnLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi5RJiZsZXE7RidGL0YyLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkotSSNtaUdGJDYmUSJ0RicvJSdpdGFsaWNHUSV0cnVlRidGLy9GM1EnaXRhbGljRidGNS1GTjYmUSUmcGk7RicvRlJGO0YvRjJGMg==

curve 2:

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5. Consider the following two parameterizations of the line through the points (-1,4) and (5,7):

line 1:

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIj1GJ0YyL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZPLUYsNiZRInRGJ0YvRjJGNS1GOTYuUScmcGx1cztGJ0YyRjxGPkZBRkNGRUZHRklGSy9GTlEsMC4yMjIyMjIyZW1GJy9GUUZZLUkjbW5HRiQ2JVEiNEYnRjJGPEY8

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

line 2:

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a. Check that each of these is in fact a parameterization of the line through the given points.

b. To approximate the velocity of a moving object, you must compute distance traveled divided by elapsed time. For each of these parameterizations, approximate the velocity of a partical as it moves along the parameterized line from the point (-1,4) to the point (5,7). [Hint: the distance traveled is the same in both cases. What is the elapsed time in each case?]

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JSFH

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