Math 5385    

Supplementary Matlab exercises #4, #5, and #6

 

Exercise 4

  1. 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.
     
  2. 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.
     
  3. 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 yz plane, given parametrically by (y,z) = (g(t),h(t)), with a < t< b.
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 1st and 4th quadrants, to avoid confusion about self-crossings of the surface.

  1. Draw the surface obtained by rotating the curve given parametrically by:
     
               (y,z) = (t3, t2),   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);.]
     
  2. Verify that one half of the hyperbola   z2 = y2 - 1   is given parametrically by:
     
               (y,z) = (cosh(t), sinh(t)).
     
    (Recall that   cosh(t) = (et + e-t)/2   and   sinh(t) = (et - e-t)/2 . )
    Use this information to draw a picture of the hyperboloid of one sheet   z2 = x2 + y2 - 1.
    Suggested parameter range: -1.5 < 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 5b.

  1. Does the picture that you drew for exercise 5b make it appear that there are any straight lines on this surface?
     
  2. Recall that the tangent line to the unit circle   x2 + y2 = 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 z2 = x2 + y2 -  1   such that (x,y) is a point on this tangent line.
    Suggestion: substitute for x and y, then solve for z. There should be 2 choices for the sign.
     
  3. 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.
     
  4. Experiment with the commands set(S,'meshstyle','column')   set(S,'meshstyle','row') to exhibit a family of straight lines on the hyperboloid.
     
  5. 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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