HW #4

1. We have seen that a hyperboloid of one sheet is a ruled surface. Make an animation that show the hyperboloid's shape for various twist angles. Here is what I have:
   

   Hint: Think of the bottom rim as a fixed circle and the top rim as a rotated version of that circle. The surface is formed by lines that connect the two circles' corresponding points.

2. Consider a surface $S_1$ parameterized as $x(u,v)$, and let $N(u,v)$ be its unit normal vector. Then the surface $S_2$ defined through $y(u,v) = x(u,v) + \delta N(u,v)$ for some constant $\delta$ is “parallel” to $S_1$ in the sense that every point of $S_2$ is obtained by moving off of $S_1$ by a distance $\delta$ in the normal direction.

   If $S_1$ and $S_2$ are colored differently but $\delta$ is very small, then we obtain the appearance of a single surface painted with different colors on its two sides as in the example below. Pick your favorite surface and plot a two-colored version of it.

   
3. The Möbius strip is a surface that can be formed by gluing the ends of a strip of paper together after giving it a half-twist. See the Wikipedia page for a lot of information.

   The animations below show a graphical construction of a Möbius strip as a ruled surface. The surface is constructed as $x(u,v) = \alpha(u) + v \beta(u)$ where $\alpha$ is a parameterized circle, say $\alpha = \langle \cos u, \sin u, 0 \rangle$ (the directrix), and $\beta$ is a parameterized vector (the ruling) that rotates by 180 degrees as it goes once around the circle.

   Find $\beta$ with that property and use it to plot the Möbius strip. It will be a plus if you can do an animation of some sort as an enhanced demo.

   Hint: Constructing the Frenet triad of $\alpha$ can help you with constructing $\beta$.

   
   Remark: Interestingly, twisting a strip of paper into a Möbius strip is more complex that it appears. A strip of paper cannot be twisted into the Möbius strip without stretching it, in the same way that it cannot be pasted onto a sphere without stretching it. If the strip is long and narrow, then the amount of stretching is negligible and does not noticeably harm the paper. But try twisting a not-so-narrow “strip”, such as a whole page from your notebook, into a Möbius strip to see the impossibility of it.
4. Make an animation that shows a line segment being wrapped around a circle without stretching. The animations below show two (out of infinitely many) possibilities. The one on the right is prettier and is actually easier to implement.

5. [You need to do the previous exercise before attempting this.] A torus may be constructed by (a) rolling a rectangle into a cylinder, and then (b) bending the cylinder as shown in this animation. Make an animation that shows the step (a) of this construction. You will do step (b) in a later exercise.