Project 8 is due Thursday 10 Apr 2003

Unstapled assignments will not be accepted.
The first line of the worksheet should include your name.

Plots must be rotated to a "nice" position and "look good (smooth)".
Plots must include axes and titles. Titles must include your name.


To plot the surface <x(u, v), y(u, v), z(u, v)> a <= u <= b, c <= v <= d
with Maple use plot3d([x(u, v), y(u, v), z(u, v)], u=a..b, v=c..d)

1 Plot the following surfaces from u=0..2, v=0..4 (from Test 3 Fall 98)
Each plot is separate.
  1a. <cos v, sin v, u>
  1b. <(2+u)cos 2v, (2+u)sin 2v, v>
  1c. <u cos v, u sin v, u>
  1d. <sin u cos v, sin u sin v, cos u>

2. Find parameteric equations for torus with radii one and two.
This is surface generated when the circle in the xz-plane
given by (x-2)^2+z^2=1 is rotated about the z-axis. Plot your
parametric equations. (Problem 38 in HH 17.5) [Note that a tubeplot
is NOT acceptable, you must have and plot the parametric equations
of the torus.]

3. Add a `knot' (This knot is called a trefoil knot). Do a `tubeplot' of
the curve below for 0 <= theta <= 6*Pi (3 times around) of r = r(theta) =
 <(2+cos(2*theta/3))*cos(theta),(2+cos(2*theta/3))*sin(theta),sin(2*theta/3)>
and display it twice, once alone and once with a copy of your torus.
(The trefoil is called a Torus Knot and now you know why.) [tubepoint has
a default option `radius=1' which is too big. Use a smaller radius like
1/10 or 1/5.]

4. This surface is related to a Mobius strip and the trefoil knot above.
It is a triangular torus with a 1/3 twist. It is made up of 3
surfaces, which are in turn based on 3 curves. If r(t) is the curve in
part 3. The three curves are
r1(t) = r(t - 0*Pi) 0 <= t <= 2*Pi
r2(t) = r(t + 2*Pi) 0 <= t <= 2*Pi
r3(t) = r(t + 4*Pi) 0 <= t <= 2*Pi
The three surfaces are
s12 = s*r1(t)+(1-s)*r2(t) 0 <= s <= 1 0 <= t <= 2*Pi
s23 = s*r2(t)+(1-s)*r3(t) 0 <= s <= 1 0 <= t <= 2*Pi
s31 = s*r3(t)+(1-s)*r1(t) 0 <= s <= 1 0 <= t <= 2*Pi
Plot this beauty twice, once alone and once with your trefoil tubeplot
(without torus).

http://www.math.fsu.edu/~bellenot/class/s03/cal3/trefoiltorus.pdf
has the combined picture, each of the surfaces sij is a different
color.

The with(linalg); package contains a matadd command and
s12 can be written matadd(r1, r2, s, 1-s) if r1&r2 are lists like [x, y, z].

5.  Plot a Klein bottle. The following should help. The handle and
bulb `fit together' to make one smooth surface.
c:=.6;a:=.2;
plot1 the handle:
Range u=Pi/2..5*Pi/2,v=Pi/4..5*Pi/4
A:=c*(cos(v)*sin(v) -0.5 + a*sin(u)*sin(v)/sqrt(sin(v)^2+cos(2*v)^2));
B:=a*c*cos(u);
C:=cos(v)+a*c*sin(u)*cos(2*v)/sqrt(sin(v)^2+cos(2*v)^2);
parametric equations of this part of the surface [A, B, C]

plot2 the bulb: v=5*Pi/4..9*Pi/4
Range u=Pi/2..5*Pi/2,v=5*Pi/4..9*Pi/4
r:=sin(v)*cos(v) - (a+1/2);
A:=c* sin(u) * r;
B:= -c * cos(u) * r;
C:=cos(v);
parametric equations of this part of the surface [A, B, C]

There is a picture of a Klein Bottle at
http://www.math.fsu.edu/~bellenot/class/s03/cal3/klein.pdf