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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 S1 parameterized as x(u,v), and let N(u,v) be its unit normal vector. Then the surface S2 defined through y(u,v)=x(u,v)+δN(u,v) for some constant δ is “parallel” to S1 in the sense that every point of S2 is obtained by moving off of S1 by a distance δ in the normal direction.

    If S1 and S2 are colored differently but δ 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)=α(u)+vβ(u) where α is a parameterized circle, say α=cosu,sinu,0 (the directrix), and β is a parameterized vector (the ruling) that rotates by 180 degrees as it goes once around the circle.

    Find β 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 α can help you with constructing β.

    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.