- Use Matlab to draw a picture of the surface described on page 16
of the text, using the parametrization given by equations (5) on that page.

- Determine which points on the surface correspond to parameter
values (
*t,u*) with*t = u.*(Give the coordinates (*x,y,z*) of such points.) Similarly, determine which points on the surface correspond to parameter values (*t,u*) with*t = -u.*Presumably, the answer will show that the surface somehow crosses itself along this set.

- Using the "hold on" command, add a curve to the sketch that traces the set that you found in the previous part.

**Exercise 5: some surfaces of rotation**

** General strategy:** Consider a plane in the

The surface obtained by rotating this curve about the z-axis is then given parametrically by:

(x,y,z) = (g(t)cos(), g(t)sin(), h(t)).

Usually, we'll try to work with curves that stay in the 1

- Draw the surface obtained by rotating the curve given
parametrically by:

(y,z) = (t^{3}, t^{2}), 0__<__t__<__1.

Experiment with varying the color, for instance by using commands such as set(S,'cdata',x).

[This makes sense if your plotting command was S = surf(x,y,z);.]

- Verify that one half of the hyperbola
z
^{2}= y^{2}- 1 is given parametrically by:

(y,z) = (cosh(t), sinh(t)).

(Recall that cosh(t) = (e^{t}+ e^{-t})/2 and sinh(t) = (e^{t}- e^{-t})/2 . )

Use this information to draw a picture of the*hyperboloid of one sheet*z^{2}= x^{2}+ y^{2}- 1.-1.5*Suggested parameter range:*__<__t__<__1.5 corresponds approximately to -2__<__z__<__2.

The command`axis square`may help to make the scale look more realistic.

**Exercise 6**

This is an ** advanced exercise.** It is about another way of
looking at the surface from exercise 5

- Does the picture that you drew for exercise 5
**b**make it appear that there are any straight lines on this surface?

- Recall that the tangent line to the unit circle
x
^{2}+ y^{2}= 1 at the point (cos(), sin()) is given parametrically as follows:

*u*---> (cos() - sin()*u,*sin() + cos()*u*).

Use this information to find the points*(x,y,z)*on the hyperboloid z^{2}= x^{2}+ y^{2}- 1 such that*(x,y)*is a point on this tangent line.substitute for*Suggestion:**x*and*y,*then solve for*z.*There should be 2 choices for the sign.

- Using one of your sign choices, find a new parametrization for
the hyperboloid of one sheet. {It will involve
and
*u.*} Use this parametrization to draw a new picture of the hyperboloid.

- Experiment with the commands set(S,'meshstyle','column')
set(S,'meshstyle','row') to exhibit a family of straight
lines on the hyperboloid.

- Repeat the same process as parts
**b**and**c**with the other sign choice in your solution from part**a.**

Comments and questions to:
`roberts@math.umn.edu`

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