a is a constant that affects the shape of the surfaces.

Helicat is a function that interpolates between the helicoid (x,y) -> [x*cos(a*y), x*sin(a*y), y]

and the catenoid (x,y) -> [(1/a)*(sqrt(1 + (a*x)^2)*cos(a*y),

(1/a)*(sqrt(1 + (a*x)^2)*sin(a*y), (1/a)*arcsinh(a*x)].

Helicat(x,y,0) is the helicoid, Helicat(x,y,1) is the catenoid, and Helicat(x,y,t) wheret is between

0 and 1 is an intermediate surface. 

The statement below will give you a catenoid. 

Try changing the "Helicat(x,y,1)" part to "Helicat(x,y, 0.5)"

to see an intermediate surface. Try other values 

(between 0 and 1) in the third coordinate as well.

Are the intermediate surfaces minimal?

First, input a radius between 0 and 2:

Create a cylinder with that  radius: 

Create a strake with inner radius r and outer radius 2:

Display them together:

Now we create the surface of revolution, defined by (u,v) -> (u, f(u)*cos(v), f(u)*sin(v)) 

Notice that when the surface first appears, we are looking down the y axis. The upper boundary of the surface is f(x) and the lower boundary is -f(x). Try rotating the surface to see how it looks from various angles.

Experiment with other plane curves by changing the definition of f on your worksheet. 
Note that you may want to vary the range of u in the "plot3d" line.

Can you make a cylinder? a cone? a sphere? a paraboloid?
Keep these surfaces in mind as you do Problem 1.7.

Now define the vector:

Now define the surface and the domain of s (from b1 to b2). s multiplies the vector r and
determines the width of the surface.

Now choose the number of divisions (div) of the curve domain. We'll draw the rulings (calculated by Rule) at the endpoints (points) of each division. 

Now we calculate the coordinates we need to put everything on the screen.

Finally, we display everything.