Joe showed us that Gaussian curvature is intrinsic so we all agreed that on a D-form, the curve-straight part had K=1, straight-straight part had K=0 and curve-curve part had K=2. We can also explain this idea by poligonizing the circle part and approximating angle deficit. The Bertrand-Puiseux formula still didn't work even when we made the surface a smooth one and approximated the limit. Also, the logic for the straigh + straigh = straight must be wrong since we could not apply it to the curve to straight part. Or maybe we could but we still haven't figured out how yet!
We tried unsuccessfully to reconcile the definitions of Gaussian curvature and Stephanie thought that they just made up the definition of angle deficit for fun! And then we restated our problem: Glue two rectangles along a common edge e, so that
(a) e is smooth, (
b) the resulting object is convex.
if e is creased, then e is straight, where creased at a point= dihedral angle neq Pi,
creased curve= if all points are creased.
The question was: if we had a curve on a plane, and we glued an edge of the rectangle piece of paper to it, can that piece of paper not be perpendicular to the plane? The answer was yes only when that curve is a straight line. We proved it using the definition of geodesics that the normal of a geodesic is perpendicular to the surface that contained that geodesic.
We figured that we could not use the same proof with geodesic to our gluing 2 pieces of paper problem since it would be hard to prove that the intersection of 2 pieces of paper was planar, given that the object was convex. But then Joe figured out that convexity was not even a problem and he came up with a new statement: Glue 2 rectangles along a common edge forming a space curve gamma(t). Then tangent planes of R1 and R2 coincide at all points of gamma(t) at which gamma dot dot (t) neq 0. We could prove this using the same logic of geodesic as the day before without assuming gamma(t) is planar.
We started thinking about the ribbon curve problem today. Can we glue the edge of one rectangular piece of paper (supposedly thin) to any space curve? I thought that it wouldn't be true if we had a planar curve segment then at the end of the curve we lifted it off the plane then it would not be a ribbon curve. But then Joe showed us that that could be possible.