<Text-field style="_pstyle256" layout="_pstyle256">Plotting Surfaces </Text-field>

Note: You may notice differences between this Maple worksheet and the equivalent Mathematica notebook. These differences were introduced to preserve the content of these modules and were necessary because of major functional differences between Maple and Mathematica.

<Text-field style="_cstyle256" layout="Heading 1">Introduction</Text-field>

OBJECTIVE: To learn how to use Maple to plot surfaces, contours, and level curves.

In this module, you will learn how Maple can help plot functions of two variables. For the standard form, z = f(x,y), you will examine contours (z-level curves) and x-level and y-level curves. Can you come up with a function that will stump your classmates when they attempt to match your functions with surface plots, contours, and level curves? This is a challenge for you to explore non-trivial functions and their graphs. All the codes for creating the necessary graphs are in this Maple worksheets.

When functions of two variables are expressed in cylindrical or spherical coordinates or are expressed parametrically, Maple can help you plot these surfaces.

<Text-field style="_cstyle258" layout="Heading 1"><Font size="18">Technology Guidelines</Font></Text-field>

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<Text-field style="Heading 1" layout="Heading 1">Part I : Surfaces, Contours, and Level Curves</Text-field>

The following code gives a set of Maple commands that can be used in defining and plotting surfaces in three-dimensional space. You can get more information on these commands and their options in the Help menu. You can insert several new functions of your own, each of a different type, to develop a visualization of different surfaces. Try some new options for the plots as well.

Once the graph is plotted, click anywhere on the graph and drag the mouse to show the plot from different angles. You can do this for all 3D plots.

First, we load the plots package.

restart; with(plots):

f:=exp(-(x^2+y^2)/8)*((cos(x)^2+sin(y)^2)): xmin:=-5: xmax:=5: ymin:=-Pi: ymax:=Pi: pict:=plot3d(f(x,y), x=xmin..xmax, y=ymin..ymax, numpoints=800, labels=["x","y","z"], axes=BOXED): print(pict);

Now, look at the curves along which the z-coordinate is a constant. What do the different colors signify?

contourplot(f, x=xmin..xmax, y=ymin..ymax, numpoints=1000, axes=FRAME, labels=["x","y"], filled=true, coloring=[blue, red]); contourplot(f(x,y), x=xmin..xmax, y=ymin..ymax, numpoints=1000, axes=FRAME, labels=["x","y"]);

Next, look at the curves formed when you hold x or y constant.

{seq(subs(x=xi,f),xi=xmin..xmax)}: plot(%, y=ymin..ymax);

{seq(subs(y=yi/2,f),yi=-6..6)}: plot(%, x=xmin..xmax);

<Text-field style="Heading 1" layout="Heading 1">You Try It : Part I : Stump Your Classmates</Text-field>

You are to submit a printout showing a function, its 3D Plot, its contourplot, and its x-level and y-level plots. Print each on a different sheet. Try to select a function that will stump your classmates. Your teacher may want to include all the students' results as a set of plots to match up on the next exam, so send your worksheets to the teacher electronically. I usually have my students work in pairs to come up with a good function. Then I include their problems on the next exam and ask students match the functions-surfaces-contours-level curves, giving points to the pair who finds a function that stumps the most classmates. Just put in your own functions and bounds. Be sure to use correct notation.

f:=exp(-(x^2+y^2)/8)*(cos(x)^2+sin(y)^2); xmin:=-5: xmax:=5: ymin:=-Pi: ymax:=Pi: pict:=plot3d(f(x,y), x=xmin..xmax, y=ymin..ymax, numpoints=800, labels=["x","y","z"], axes=BOXED): print(pict); display(pict, orientation=[90,90]);

contourplot(f, x=xmin..xmax, y=ymin..ymax, numpoints=1000, axes=FRAME, labels=["x","y"], filled=true, coloring=[blue, red]); contourplot(f(x,y), x=xmin..xmax, y=ymin..ymax, numpoints=1000, axes=FRAME, labels=["x","y"]);

{seq(subs(x=xi,f),xi=xmin..xmax)}: plot(%, y=ymin..ymax);

{seq(subs(y=yi/2,f),yi=-6..6)}: plot(%, x=xmin..xmax);

<Text-field style="Heading 1" layout="Heading 1">Part II : Implicit Plots in Three Dimensions </Text-field>

In the last chapter, you used implicitplot3d and plot3d commands to plot cylinders and quadric surfaces. Now you understand better what contours are. Now, imagine an object in 4 dimensions. Its contours would be 3-dimensional surfaces. That is how we interpret such surfaces when plotting them in Maple. Recall that the default assumes that the function is equal to 0.

f1:=z^3+2*x^2-3*y^2: implicitplot3d(f1, x=-3..3, y=-2..2, z=-4..4, axes=FRAME, labels=["x","y","z"]);

Let's see another level surfaces.

f2:=9*x^2+16*y^2-4*z^2+1=16; implicitplot3d(f2, x=-2..2, y=-2..2, z=-2..2, axes=FRAME, labels=["x","y","z"], grid=[15,15,15]);

<Text-field style="Heading 1" layout="Heading 1">You Try It : Part II</Text-field>

Select any function of three variables that you wish. Be certain that the function is defined over the domain (x, y, and z bounds) you specify.

f1:=z^3+2*x^2-3*y^2: plots[implicitplot3d](f1, x=-3..3, y=-2..2, z=-4..4, axes=FRAME, labels=["x","y","z"]);

<Text-field style="Heading 1" layout="Heading 1">Part III : Cylindrical, Spherical, and Parametric Plots</Text-field>

Certain surfaces are easier than others to express in cylindrical or spherical coordinates. For cylindrical plots, it is assumed that z is given as a function of r and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvRjNRJ2l0YWxpY0Yn. For spherical plots, it is assumed that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEmJnJobztGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0YzUSdpdGFsaWNGJw== is given as a function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEmJnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnL0YzUSdpdGFsaWNGJw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==.

Here is the graph of 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 .

cylinderplot(r^2/(1+cos(theta)), r=0..1, theta=0..2*Pi, labels=["x","y","z"],axes=BOXED, numpoints=111);

Next is a graph of 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 .

sphereplot(phi-theta, theta=0..2*Pi, phi=0..Pi, labels=["x","y","z"], axes=BOXED);

As long as we have this package opened, let's plot a parametric surface in three dimensions. The input is of the form {x, y, z}. All set equal to functions of parameters u and v.

plot3d([u+v, u-v, u*v], u=-2..2, v=-2..2, labels=["x","y","z"], axes=FRAME);

<Text-field style="Heading 1" layout="Heading 1">You Try It : Part III</Text-field>

Go back to any of the plots and alter the functions as you wish. Be sure to use correct notation. Noted that many spherical plots take quite a bit of time to plot. If you find you are waiting too long, you can always click the "STOP" button on the toolbar. Replace the function in the cylindrical plot with a function of r and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==.

restart: with(plots):

cylinderplot(r^2/(1+cos(theta)), r=0..1, theta=0..2*Pi, labels=["x","y","z"],axes=BOXED, numpoints=111);

Replace function in spherical plot with a function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEmJnBoaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjI= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg==.

sphereplot(phi-theta, theta=0..2*Pi, phi=0..Pi, labels=["x","y","z"], axes=BOXED);