Traces of a Quadric Surface

The graph of the hyperboloid of one sheet


is shown below along with a vertical plane x = n. The trace formed by the intersection of the plane and the surface is graphed separately. Launch the animation and complete the following exercises.

click image to launch animation in Viewer

Exercises

1. Change the position of the vertical plane by using the slider to change the value of n. Describe the family of traces in the planes x = n. In particular, identify any transitional values of n where the shape of the trace changes.

2. Determine equations for the family of traces in x = n. Do the equations confirm your observations from Exercise 1?


Next investigate the traces formed by the intersection of the vertical plane y = n with the surface. Launch the animation and answer the following questions.

click image to launch animation in Viewer

Exercises

3. Change the position of the vertical plane by using the slider to change the value of n. Describe the family of traces in the planes y = n. In particular, identify any transitional values of n where the shape of the trace changes.

4. Determine equations for the family of traces in y = n and explain the similarities to the traces in x = n.


Finally, investigate the traces formed by the intersection of the horizontal plane z = n with the surface. Launch the animation and complete the following exercises.

click image to launch animation in Viewer

Exercises

5. Change the position of the horizontal plane by using the slider to change the value of n. Describe the family of traces in the planes z = n.

6. Determine equations for the family of traces in z = n. Do the equations confirm your observations from Exercise 5?



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