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Section 5.4 Intersections of General Surfaces

Although CalcPlot3D does not determine the intersection of two surfaces for you, it does make a great tool for visually checking that the parameterization you have worked out as the intersection of two surfaces is indeed correct.

Below you will find two examples to explore this concept. Hopefully this visualization process will help increase your confidence in and your appreciation for the results you have obtained.

Exploration 5.4.1

Determine the vector-valued function that traces out the intersection of the surfaces defined by the equations below using the parameter \(t\text{.}\)

\begin{equation*} z = x^2 + y^2\quad\text{and}\quad y=x^2 \end{equation*}

State the answer as a vector-valued function. Then visually verify this result, plotting the two surfaces in CalcPlot3D, and then plotting the curve to check that it really does represent the intersection of these two surfaces.

Answer

Here we can let \(x=t\text{.}\) Then \(y=t^2\text{.}\) What does \(z\) equal (as a function of \(t\))?

Then the vector-valued function we obtain that traces out the intersection of these surfaces is:

\begin{equation*} \vec{\textbf{r}}(t) = t \hat{\textbf{i}} + t^2 \hat{\textbf{j}} + (t^2 + t^4) \hat{\textbf{k}} \end{equation*}

This represents the same curve as specified by the parametric functions:

\begin{align*} x &= t\\ y &=t^2\\ z &=t^2 + t^4 \end{align*}

Now let’s verify this in CalcPlot3D visually!

a. Open the CalcPlot3D app.

b. Once the app is loaded and active, enter the first function listed above (\(z =\) x^2 + y^2) in the default function object on the left and press Enter (or click on the Graph button). The surface plot of this paraboloid should appear in the plot window.

c. Now, to enter the second function, go to the Add to graph dropdown menu (just above the default function), and select Function: y = f (x, z). This will set the function to y = 1 by default. Enter x^2 in the textbox and press Enter (or use the Graph button).

d. To extend the second surface farther up the paraboloid, change the range of \(z\) (just below y = x^2) to go from -2 to 4. You may want to use the scroll-wheel on the mouse to zoom-out a little. [Alternatively you could click on the Format Axes button located just to the right of the 3D Mode button. Then set \(z\)-max to 4. This will automatically change the upper \(z\)-clip value to 8. Let’s change this value to 4 also.]

e. Make the surfaces semi-transparent using the button or by typing the T key to get a clearer view of the intersection of the surfaces. Press the E key to turn off the edges on the surfaces.

f. Next we need to graph the space curve to see how well it fits the intersection of the surfaces. Select Space Curve: r(t) from the Add to graph dropdown menu. A space curve object should appear just below this menu.

Enter the three parametric equations we obtained (each in terms of \(t\)). Then enter a range of -2 to 2. If you press Enter on the second value, it should produce the curve on the plot. If it does not appear, click the Graph button.

g. I like the look of this one better with a constant color, so select the checkbox titled Use Constant Primary Color, at the bottom of this object.

h. Finally rotate the graph to see if it looks like we found the correct intersection curve. Except for different coloring, this should look like the image in Figure 5.4.1 below.

Intersection of two surfaces
Figure 5.4.1 Intersection of the surfaces \(z = x^2 + y^2\quad\text{and}\quad y=x^2\text{.}\)
Exploration 5.4.2

Determine the vector-valued function that traces out the intersection of the surfaces defined below, assuming that \(y = 2\sin t\text{.}\)

\begin{equation*} 2x^2 + y^2=4\quad\text{and}\quad z = x^2 + y^2 \end{equation*}

State the answer as a vector-valued function. Then visually verify the result, plotting the two surfaces in CalcPlot3D, and then plotting the curve to check that it really does represent the intersection of these two surfaces.

Answer

Note that we can parameterize the ellipse given by the first equation with

\begin{equation*} x = \sqrt{2}\cos t \quad\text{and}\quad y = 2\sin t\text{.} \end{equation*}

Now that we've parameterized \(x\) and \(y\text{,}\) the second surface equation makes it easy to determine \(z\) in terms of \(t\text{.}\)

After simplifying the expression, we obtain, \(z = 2 + 2\sin^2 t\text{.}\)

Then the vector-valued function we obtain that traces out the intersection of these surfaces is:

\begin{equation*} \vec{\textbf{r}}(t) = \sqrt{2}\cos t \hat{\textbf{i}} + 2\sin t \hat{\textbf{j}} + (2 + 2\sin^2 t) \hat{\textbf{k}} \end{equation*}

This represents the same curve as specified by the parametric functions:

\begin{align*} x &= \sqrt{2}\cos t\\ y &= 2\sin t\\ z &= 2 + 2\sin^2 t \end{align*}

Now let’s verify this in CalcPlot3D visually!

a. Open the CalcPlot3D app.

b. Once the app is loaded and active, enter the second function listed above (z = x2 + y2) in the default function object on the left and press Enter (or click on the Graph button). The surface plot of this paraboloid should appear in the plot window.

c. Now, to enter the first function, which is stated implicitly, go to the Add to graph dropdown menu (just above the default function), and select Implicit Surface. Once this new object definition appears on the left, enter the implicit equation defining the first surface in the textbox and press Enter (or use the Graph button). This surface should look like an elliptic cylinder.

d. To make the intersection more clear, let's first zoom out once using the Zoom-out button, so the \(x\)- and \(y\)-axes run from -4 to 4 instead of from -2 to 2. Now click on the Format Axes button located just to the right of the 3D Mode button, and set the upper \(z\)-clip to 4.

e. Next make the surfaces semi-transparent using the transparency button or by typing the T key to get a clearer view of the intersection of the surfaces. Press the E key to turn off the edges on the surfaces.

f. Next we need to graph the space curve to see how well it fits the intersection of the surfaces. Select Space Curve: r(t) from the Add to graph dropdown menu. A space curve object should appear just below this menu.

Enter the three parametric equations we obtained (each in terms of \(t\)).

You will enter them like this:

\begin{align*} x &= \textbf{sqrt(2)cos(t)}\\ y &= \textbf{2sin(t)}\\ z &= \textbf{2 + 2(sin(t))\^{}2} \end{align*}

g. Now enter a range of 0 to \(2\pi\text{.}\) If you press Enter on the second value, it should produce the curve on the plot. If it does not appear, click the Graph button.

I also like the look of this one better with a constant color, so select the checkbox titled Use Constant Primary Color, at the bottom of this object.

h. Finally rotate the graph to see if it looks like we found the correct intersection curve. This should look like the image in Figure 5.4.2 below.

Notice that the intersection of these two surfaces may appear a little polygonal rather than smooth like the curve we just graphed. This is because we used an implicit surface with the default resolution. To improve this resolution, you can adjust the # Cubes/axis to a higher number, like 25. See the result in Figure 5.4.3.

Intersection of two surfaces
Intersection of two surfaces
Figure 5.4.2 Intersection of the surfaces \(2x^2 + y^2=4\quad\text{and}\quad z = x^2 + y^2\text{.}\)
Figure 5.4.3 Intersection of the surfaces \(2x^2 + y^2=4\quad\text{and}\quad z = x^2 + y^2\text{,}\) using a higher resulution on the implicit surface.