Research Log: Week 1
 
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Monday, 5/16/05
We had our first meeting today. Joe explained the construction of the ruled racetrack-surfaces, which are examples of D-forms, and of their predecessors, the pita-forms. He explained how discontinuous curvature and torsion can integrate to give a smooth curve. We suppose that the curvature of sections where the straightaway meets the half-circle is 1/r where r is the radius of the circle, and that sections where two straightaways meet have zero curvature. It was formerly thought that when half-circle met half-circle the curvature was still 1/r. There doesn't seem to be much justification for this, and perhaps even some experimental data against the hypothesis. Is this curvature even a constant value? I have no intuition wrt this question. If, however, the curvature and torsion were simple step functions, we would be able to calculate the curve quiet easily since we would simply measure the arc-length (in the plane, since we do not stretch the surfaces when we glue them, and we have parameterization for the racetracks in the plane) of the sections where one of the above three combinations of curve to curve, curve to straightaway, and straightaway to straightaway met to determine the curvature and torsion functions, and then apply the integration algorithm from Gray's differential geometry/Mathematica book to obtain the curve. We have been focusing on thinking of methods to prove our first two conjectures about the curvature, have been ignoring the curve to curve case, and haven't even touched the seemingly more complicated subject of torsion.

It seems that there are two different approaches to calculating the curvature and torsion of the racetrack D-forms: one, use the fact that the surface is an intersection of two ruled surfaces with Gaussian curvature zero at all interior points, or two, somehow use distances and arc lengths in the plane to calculate the curvature and torsion of the intersection curve itself and then form the surface by taking the convex hull of the intersection curve. The first approach seems to be a bit of a dead end since Gaussian curvature of zero tells us nothing about the principle curvatures, only that, since we're looking at a ruled surface the minimal curvature and, therefore, one of the principle curvatures is zero and so the product is zero. The other principle direction is in the direction of the tangent line to the intersection curve where the "ruled reflection line" hits the intersection curve. This is just a result of the surface being a ruled surface and is not particularly useful.

The second approach is difficult because the curvature of the space curve, which I've been calling the intersection curve, will be different from the curvature of the plane curve before the two racetracks are glued together. The curvatures should change in a predictable way since we know the shapes of the two surfaces being glued together, but it does not seem like a simple task to find the way. Joe has already proven that the d-form is the convex hull of the curve so if we could find the intersection curve we would have the entire object.

Tuesday, 5/17/05
Today, we started work on the Knight's visor. We started by calculating the curve traced out by the Knight's visor when the paper is completely

flat. I assumed that since the maximum height of the folded curve is 2r and the minimum height is zero that the curve must be the cartiod. This turned out to be an over simplification. As Joe pointed out, the curve is traced out by reflecting points on the x-axis across the tangent line to the circle at the point in question. We computed the true equation (in parametric form) of the curve when the Knight's visor is completely flat and found that it is not the cartiod. We also began searching through a book which listed smooth plane curves to see if the curve we found might have a name.
Wednesday, 5/18/05
Noticing that the family of curves traced out by the Knight's visor card as it is folded and unfolded does not depend uniformly on the flat case, we needed to find a new method for describing the family. Joe pointed out that we could do this by taking the intersection of a plane through folding angle, a plane along a given folded rib of the card, and the sphere traced out by the end point of the same rib. We computed the general parameterization and began cleaning up our results in a new Mathematica file.
Thursday, 5/19/05
Though we found the equation of the flat Knight's visor curve on Tuesday, we did it in a somewhat akward form using a parameterization with the variable -a. Today, we attempted to edit our equation to get a more intuitive parameterization, but ran into several Mathematica glitches and made some computational errors so that our calculations took up most of the day. In the end, however, we obtained a nice set of parametric equations. We also completed our ongoing attempt to find the polar and implicit forms of the curve. We used Mathematica to find the polar form because the calculations were beyond our comprehension. The result was messy. We attempted to use Mathematica to find the implicit form as well. It produced messy results for the implicit form as well, but this time we knew how to find a simpler form. We substituted the parametric form into the equation of the original circle from which we cut the ribs of the card. While this technique gave us a simple expression, it was not simple enough to be a classic curve, and the parameterize form was still the most simple.
Friday, 5/20/05
Today we finished up writing neat versions of our work in Mathematica. After lunch we learned the beginnings of Dreamweaver.