Research Log: Week 3
 
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Monday, 5/30/05
Memorial day.
Tuesday, 5/31/05
On Friday, we found a parameterization for the tennis ball curve and graphed it. Joe, meanwhile, found the curvature and torsion step functions and integrated to obtain another graph of the tennis ball curve. We each made different assumptions about the lengths of the straightaway to simplify our respective calculations. Therefore, we spent the day today finding a new parameterization with Joe's prettier assumption of a straightaway with length one, graphing it, and drawing rules on the surface to indicate that we are considering a surface and not just a space curve. Joe will include the picture in his book.
Wednesday, 6/1/05
Since we know that any point off the intersection curve on our deform has Gaussian curvature zero, the Gauss-Bonnett theorem gives us that the total curvature integrating along the intersection curve must be four pi. We believe that the Gaussian curvature along the curve to straight parts integrates to pi over an arc-length of pi, and so is the constant function one, and that the curvature on straight to straight parts is everywhere zero. We tried, however, to compute the Gaussian curvature using the limit formula involving the area of a geodesic circle, and the one involving perimeter and found that the Gaussian curvature at each point of the straight to curved line was infinite. This seems to make sense with the osculating circle definition of curvature, but is incongruous with the angle deficit formulation.
Thursday, 6/2/05
Today, Joe explained the angle deficit definition of curvature to us. We spent much of the day trying to prove that when you join two straight edges together (edges of a developable surface that is) you cannot both preserve convexity and have curvature along the intersection curve. In other words, the straight to straight gluing has Gaussian curvature zero. We began by simplifying the problem to that of joining two rectangles together, and noticed that, regardless of the curvature of the edges of the rectangles after gluing, the four corners are fixed in space and the straight lines between the corners form "lines of convexity." Three corners span a plane. Since the two rectangles share two corners, there are two planes per rectangle. We deduced that the curve between the two vertices had to lie along or above the intersection of the planes. We wanted to show that it had to lie along. Then we realized that if there was any lateral convexity, it would produced concavity for one of the rectangles being glued. Therefore, we determined that the line between the two glued vertices had to lie in a vertical plane through the gluing. Then Joe came in and we started describing how convexity between any two points implied greater than or equal to zero curvature everywhere (we later realized that depending on how you orient your normal the curvature is all of one sign but not necessarily positive). (Note that greater than or equal to zero curvature is a necessary condition for convexity, but not sufficient.) Then we applied this to points along the curves cut by normal planes. It gave that the principle curvatures must always be greater than or equal to zero.
Friday, 6/3/05
We have been switching back and forth between considering a creased piece of paper and two rectangles glued together, but the two representations are equivalent. The creased or glued edge must have Gaussian curvature zero by applying the limit formula involving the perimeter of a geodesic disk over the crease. We use the formula that the Gaussian curvature is the product of the curvatures in the two principle directions to conclude that one of the principle directions must be flat. We observed that all possible planes cutting through a point along the crease have infinite curvature except the plane that cuts along the crease. Therefore, we concluded that the plane which cuts through the crease gives the minimum curvature, and that minimum value must be zero in order to have Gaussian curvature zero. We also calculated the Gaussian curvature along a curved to straight section using the limit formula and found that it was infinite. If you look at normal cross sections along the curve to straight part, the sectional curvatures seem to be infinite everywhere except in one direction where it is a positive constant. The product of the principle curvatures, therefore, would be positive infinity. Therefore, the argument from classical differential geometry is at least consistent with itself. It is probably wrong, however, since we are applying techniques of smooth surfaces to singular points. The Gauss-Bonnet formula gives that the Gaussian curvature along the curved to straight parts should be the constant value of one, not infinity.