Objectives
- Understanding the shapes of quadric surfaces including the intercepts and traces in each coordinate direction
- Understand how to manipulate graphs and interactive elements in the CalcVR app
Section 3.7 Quadric Surface Explorations
The following exploration (done as three activities) are meant to be done sequentially and consists of a scaffolded set of tasks that guides the user through an examination of the intercepts and traces of various quadric surfaces. The questions and tasks get progressively more open ended are are designed to show the user how to explore using the tools of the Quadric Surfaces Playground.
Activity 3.7.1. Hyperboloid of One Sheet.
To get started, make sure your headset and controller is set up with the CalcVR app. Start the CalcVR app go to the Graphs in 3D set of lessons. Start the Quadric Surfaces Playground and put your phone in your headset. Once the scene loads, you can select the surface you would like to examine using the buttons displayed. There should be an animation that shows some of the manipulations you can do to the quadric surfaces in this demo.
You should select the hyperboloid of one sheet surface but clicking on the appropriate button. You should turn off the level curves for x, y, and z by unselecting the boxes on the panel to the right of the graph. Before you go any further, take a few minutes to rotate and zoom in on the graph so that you get a better understanding for the shape of this surface. Be sure to pay attention to which axes are which and try examining the surface by rotating to look down each coordinate axis.
(a)
- For how many points does the surface intersect the x-axis?
- For how many points does the surface intersect the y-axis?
- For how many points does the surface intersect the z-axis?
(b)
Copy the equation from the panel on the right onto your paper and label this your Base Equation for the Hyperboloid of One Sheet.
(c)
- Plug y=0 and z=0 into your Base Equation. How many solutions for the variable x will you be able to solve for?
- Plug x=0 and z=0 into your Base Equation. How many solutions for the variable y will you be able to solve for?
- Plug x=0 and y=0 into your Base Equation. How many solutions for the variable z will you be able to solve for?
- Compare the number of intercepts from question 1 and the results of the previous parts of this question. Be sure to explain what connections you see.
(d)
We want to look at the traces of the Hyperboloid of One Sheet now. The traces or level curves of the surface are the set of points on the surface that are parallel to one of the coordinate planes. We will look at the y-level curves, which will be parallel to the xz-plane.
Turn on the y-level curve by selecting the appropriate box on the panel to the right with the Base Equation. You may want to turn off the x-and z-level curves.
Click and hold the red sphere to adjust the y-value of the level curve to a value close to y=1. You should see a plane at the appropriate y-value that will highlight the trace curve once you let go of the red sphere. What shape is the trace or level curve around y=1? Describe any relevant characteristics of this shape like orientation or ….
(e)
Move the red sphere slider close to y=2. What shape is the trace at y=2? Be sure to describe what changed as compared to the trace with y=1.
(f)
You should take a few minutes now to look at the traces for different values of y. You should write a few sentences summarizing your observations of the traces, including addressing questions like “Are all of the traces the same shape?” and “How does the trace change as the value of y is varied?” You need to consider all of the possibilities for the shape and orientation of the traces (even if you can’t quite settle the slider on a particular value).
(g)
We now want to look at the traces that will be parallel to the xy-plane. On the panel to the right with the Base Equation, uncheck the box for x and check the box for the z-level curve. Move the trace close to z=0 using the green sphere. What shape is the trace? Be specific and note details involving orientation and scale.
(h)
Move the green sphere on the slider close to z=-2. What is the shape of the trace? Be sure to note any changes between the traces at z=0 and z=-2.
(i)
You should take a few minutes now to look at the traces for different values of z. You should write a few sentences summarizing your observations of the traces, including addressing questions like “Are all of the traces the same shape?” and “How does the trace change as the value of z is varied?” Be sure to consider ALL possibilities for the traces, not just the ones described by the values in the previous parts.
(j)
Turn off all the level curves using the panel to the right. Take a minute to look at the surface again and be sure to note how the shapes of the traces are incorporated into the makeup of the surface.
Would you expect the x-traces to be like the y-traces, z-traces, or different than both? You should use both algebraic and geometric descriptions in your explanation.
Activity 3.7.2. Hyperbolic Paraboloid (Saddle).
To get started, make sure your headset and controller is set up with the CalcVR app. Start the CalcVR app go to the Graphs in 3D set of lessons. Start the Quadric Surfaces Playground and put your phone in your headset. Once the scene loads, you can select the surface you would like to examine using the buttons displayed. There should be an animation that shows some of the manipulations you can do to the quadric surfaces in this demo.
We are now going to examine the hyperbolic paraboloid, or saddle surface. If needed, you can go back to the surface selection by clicking on the button labeled “Select Another Quadric Surface” below the panel used for rotation (which should be to the left of the surface plot). In the surface selection, click on the button for the Hyperbolic Paraboloid. You should turn off all level curves using the panel to the right.
(a)
Rotate and zoom the saddle surface to look for all places that the surface intersects the x-, y-, and z-axes. Write down all the coordinates for all the intercepts.
(b)
Write down the Base Equation for the saddle surface. Use this equation to explain your results from the previous part.
(c)
Turn on the x-level curves (also called the x-traces) using the panel on the right. Use the blue sphere to look at the shape of the traces for x values close to -1.5, 0, and 1.5. Note the shapes and how the traces change as the x value is changed. Give as complete of a description as possible including the orientation of the shapes that you observe and where changes in the shape or orientation of the traces occur.
(d)
Turn off the x-level curve and turn on the z-level curve using the panel on the right. Use the green sphere to look at the shape of the traces for z values close to -2, -1, 0, 1, and 2. Note the shapes and how the traces change as the z value is changed. Give as complete of a description as possible including the orientation of the shapes that you observe and where changes in the shape or orientation of the traces occur.
Turn off the z-level curve using the panel on the right. Would you expect the y-traces to be like the x-traces, z-traces, or different than both? You may want to use both algebraic and geometric descriptions in your explanation. Your explanation should not be a description of the y-traces that the program shows but a description based on the equation for the surface or the shape of the surface.
Activity 3.7.3. Elliptic Cone.
To get started, make sure your headset and controller is set up with the CalcVR app. Start the CalcVR app go to the Graphs in 3D set of lessons. Start the Quadric Surfaces Playground and put your phone in your headset. Once the scene loads, you can select the surface you would like to examine using the buttons displayed. There should be an animation that shows some of the manipulations you can do to the quadric surfaces in this demo.
For the last exploration, go back to the surface selection and select the Elliptic Cone surface.
(a)
Use this demo to give a complete description of the intercepts and traces (x-, y-, and z-) for the Elliptic Cone surface. You should use both geometric and algebraic descriptions to make sure you have considered all possibilities.
Hint.
Be sure to look at what kind of trace would correspond to a zero value (even if you can’t get the slider to be exactly on zero).
