WEEK 1

 Monday 5/16/05: Joe wanted us to study the curve created by glueing 2 pieces of race track together. The suggested approach is that if we can find the curvature and torsion of the curve then we can get its equation. We thought that the part where straight line meets straight line would have no curvature and 1/r otherwise. Tuesday 5/17/05: It seemed that our assumption about the curvature is wrong. But we still thought that the straight-straight part would have no curvature and straight-curve would have 1/r but the curve-curve should be something else. Stephanie picked out a book to read on how a curvature of a circle changed when it turned into a helix to see if we can get anything related to the curve we are studying. We also started working on the knight's visor curve today. At first we thought that the function for the curve when the angle of the card is 0 is 1-cox(x) (polar coordinate) and the shape of the curve would change uniformly as we close or open the card. We calculated the parametric equations of that curve then compare it to our guess and we realized we were wrong. Wednesday 5/18/05: Joe suggested us to find the intersection of 2 planes and a sphere to find the coordinate of a point on the curve. We used Mathematica and got the parametric equations of the curve with respect to the angle of the card and the distance from the center of the circle. We also got a very neat animation of it. Thursday 5/19/05: We are supposed to document all the derivation in Mathematica. Since we started out with a point (-a,0) the day before, which seems strange, we wanted to redo all the equations with the point (a,0). Somehow the new equations we got turned out to be a totally different curve. The problem was that we forgot that y had to be nonnegative so we add an expression of absolute value into the equation of y and we got what we wanted. Friday 5/20/05: We converted the parametric equations of the curve when card angle was 0 to cartesian and polar coordinates. We got very complicated results so we concluded that this curve could not be any classic curve that had been studied. And we learnt Dreamweaver today which was totally exciting :).

WEEK 2

 Monday 5/23/05: We figured out today that the tangent lines to the flat knight's visor at the endpoints of the curve are horizontal. We also showed that each rib of the knight's visor is actually perpendicular to the curve (in the case of flat knight's visor). In other word, every tangent line to that curve is a reflection of the x axis across the unit circle. We were still stuck on D-forms. Stephanie had some conjectures about some properties of the D-forms curves but we were not sure. She noticed that the sum of the maximum angle and the minimum angle betweent the 2 race tracks was 180 degrees. Tuesday 5/24/05: We made 2 illustrative figures today in Mathematica to show the idea of how we got the equation for the knight's visor curve. Then Joe showed us how to use Adobe Illustrator to polish the graphs. Adobe Illustrator is fun! I did some more reading on the curvature and torsion of the hellix. Wednesday 5/25/05: Joe gave us his Latex files on Knight's visor and we put in some explanation for how we got the equations of the curves. We also labeled the figures we made the day before. And we did some more reading on curvature and torsion. We thought that for the special case when the 2 race tracks are glued together almost like a square, the curvature is 0 with straight-straight part and 1/r for the rest with torsion 0. But we can actually draw a flat curve with the same torsion and curvature and we don't know for sure what makes the difference btw that curve and the one we are studying. Thursday 5/26/05: More Latex and figures for 3D Knight's visor. We spent most of the day playing around with Adobe Illustrator and working on the parametric equations of the "square" D-form. Friday 5/27/05: More Adobe Illustrator. We got the piecewise parametric equations for the "square" D-form.

WEEK 3

 Monday 5/30/05: Memorial Day. Tuesday 5/31/05: We used the parametric equations for the "tennis ball" D-form to make a picture of the object in Mma. We had to change all of our equations so that it would have a similar scale to Joe's model (length of the rectangular part and radius of the circle both 1). Joe decided that we should focus on the Gaussian curvature of the D-forms for now so we could understand some more about them. Wednesday 6/1/05: Joe explained to us the idea of angle deficit and we wanted to use Gauss-Bonnet theorem along with Bertrand-Puiseux formula to calculate the Gaussian curvature of D-form. We agreed that the curvature would be 0 everywhere on the surface except for the circular part. We knew that the sum of curvature of each of the 4 circular part would be Pi. But when we calculated the curvature at a point on the circular part using the formula with perimeter, we got an infinite curvature! Even when we use that formula for very simple objects like cube or sphere, the formula still gave the wrong answers. Thursday 6/2/05: We tried the Bertrand-Puiseux formula again for sphere and plane and it worked but it only held true for smooth surfaces. We started on a new problem: gluing 2 pieces of paper together at an angle, if it is a part of a convex object then the sides glued together must be a straight line in space. Friday 6/3/05: We knew that the Gaussian curvature at everypoint inside a piece of paper was 0 and K=Kmin.Kmax. Let p be a point on the glued side. If we can prove that any cross section curve through p that is not the glued side itself has curvature infinity then point p on the glued side has curvature 0.

WEEK 4

 Monday 6/6/05: 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! Tuesday 6/7/05: 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. Then if e is creased, then e is straight, where creased at a point= dihedral angle neq Pi, creased curve= if all points are creased. Wednesday 6/8/05: 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. Thursday 6/9/05: 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. Friday 6/10/05: 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.

WEEK 5

 Monday 6/13/05: We wrote down the complete proof for the gluing paper problem today. Joe was worrying that Huffman origami would be a counterexample to our proof but Huffman's curves were not geodesics. We worked on the ribbon curve problem and Joe thought that the idea should be at every point on the curve, build a plane that has the second derivative as normal and link all these planes together to form a ribbon curve. But the problem was that we could not prove if we unroll this newly formed surface, the two edges of it would be geodesics. Tuesday 6/14/05: I wrote the description for Women and Science today. Stephanie had an idea of exploring the ribbon problem using some formulas for the rule surfaces and the First Fundamental Form but i had to read more on the topic to understand it. Wednesday 6/15/05: I made some pictures and modification to the women and science description before Joe left. Thursday 6/16/05: We thought that bending a piece of paper in any way would result in an isometry of the flat piece of paper. Therefore, the first fundamental form of the surface would remain the same. But this would lead to the conclusion that each rule of the surface will be perpendicular to the tangent at the edge of the piece of paper, which i believed to be false. Somehow we could not find an error with Stephanie's calculation. Friday 6/17/05: Today is my last day. We are still stuck with our problem and we had a goodbye party for me.