Research Log: Week 4
 
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Monday, 6/6/05
Today Joe concluded that the Gaussian curvature of the straight to straight part is 0, of the straight to curved part is 1, and of the curved to curved part is 2. One justification for this is that Gaussian curvature is intrinsic so the Gaussian curvature should be constant throughout similar regions (i.e. curve to straight). Our other logic was to observe that straight to straight has Gaussian curvature zero, Gauss-Bonnet gives that the Gaussian curvature of curved to straight segments is one, and curved to curved segments overlap to give twice Gaussian curvature one, or Gaussian curvature two. This theory is consistent with the Gauss-Bonnet theorem. We attempted to use the theorems and definitions from smooth differential geometry (namely principle curvatures and the limit of circle circumference) to prove our conjecture, but failed. It seems that smooth differential geometry and polygonal differential geometry do not mix... at least the results of one cannot be applied so blithely to the conjectures of the other.
Tuesday, 6/7/05
We spent much of the day looking for ways to extend the angle deficit definition of Gaussian curvature to make a statement about straight to curved segments. Unfortunately, the straight to curved segments are neither completely polygonal nor completely smooth. One idea to overcome this difficulty would be to polygonize the straight to curved segments and take the limit of the resulting figure as the length of the edges along the straight to curved part go to zero. We thought of calculating the Gaussian curvature by polygonizing the surface first and then using the limit circumference formula. The result of this approach, however, is the same as that obtained from using the limit circumference formula without polygonizing. We still don't know how to compute the Gaussian curvature of the straight to curved section (or the curved to curved). If we can prove that joining two straight segments always yields a straight curve, we will know that the angle deficit formulation of Gaussian curvature may be applied to straight to straight segments. (The limit circumference formula also gives the correct Gaussian curvature for straight to straight segments.)
Wednesday, 6/8/05
Joe came up with a proof that the straight to straight gluing gives a straight curve. The argument was that the curve must be a geodesic on both peices of paper since bending is an isometry and geodessy is invariant under isometries. Since the curve must be a geodesic, its second derivative is the normal of both planes. I had claimed earlier that the curve must lie in a vertical plane since lateral curvature would lead to concavity for one of the planes. Joe deduced that, if the curve lay in a plane, and was a geodesic, it had to be straight. Then we realized that we couldn't prove that the curve lay in a plane.
Thursday, 6/9/05 and Friday 6/10/05
Joe came up with an idea that used the same argument as the first proof to show that straight to straight gives a straight curve without assuming that the curve lies in a plane. We spent most of the day testing it, and formalizing it. It seems to work.