Why can a piece of paper be folded along a straight line but not along a curved line? It seems that if something can be stretched then it can be folded along a curved line. Since paper can't be stretched I can't fold it along a curve. But it's just an observation not an answer to the question. Does anyone know the answer? Is it because of some kind of area conservation?
 A: Apparently, while you can't fold paper along a curve in 2 dimensions, you can do it in 3 dimensions.
This article from the journal Physics describes how a flat ring of paper, folded along its circular center, deforms into a saddle as the material buckles to absorb the stresses.
(Longer discussion at Physicsworld courtesy of an answer to a related question.)
A: No, paper can't be stretched because its structure contains no slippage mechanisms which would allow it. To some extent, you can provide slippage by wetting the paper into mush, which unlocks the entanglements between the paper fibers and allows the sheet to deform.
A: The short path between 2 points on the paper before folding is a straight line, all points of which belong to the paper.
Now suppose that we managed to completely fold it along a curve between these 2 points. All paths between them on the paper are now curves, with bigger lengths compared to the straight line, (which passes out of the paper).
So, it is not possible to fold it along curves keeping the same distance between all points. And  it requires streching to change distances.
A: Paper can be more-or-less folded along a curve. You can do it with your fingers.

It could be argued that this is really just a lot of little short straight lines.

But it could be argued that any curve is just a lot of little short straight lines.
A: The supposition that "something" must be conserved is correct. That something is the flat angle of the material. At every point on a flat sheet of paper (a plane), there is 360° of flat angle about the point. When you fold the paper, it necessarily turns into a 3D shape, but it still has 360° of flat angle at each point along the folded edge. If the fold is straight, then there is 180° on either side of the fold totaling 360 and everything is cool. Furthermore, the material on either side of the fold remains flat as in the sides of a box. However, if the fold follows a curve (which it certainly can), then the flat angle on the convex side of the fold is larger than the angle on the concave side. A consequence of this mismatch is that the paper arches up into two simple curved surfaces on either side of the fold.  Compound curves - arched surfaces where no straight line passes through a particular point as in a dome - cannot be generated from a flat sheet without cutting or stretching.
In a related thought experiment, consider a cube.  At each corner of the cube, 3 sides come together.  Each side forms a 90° angle at the corner. So there are $3 \times 90° = 270°$ of flat angle at the corner.  If you are folding this shape from a single flat sheet of paper, that means you had to remove 90° of flat angle from the sheet to form the corner. (It is also possible to add more flat angle by cutting the sheet and pasting in extra paper.) Since there are 8 corners on the cube, that means $8 \times 90° = 720°$ are removed from the entire cube. It turns out that no matter how many corners a solid object has or how irregular it might be, 720° of flat angle have been removed.
A: Paper represent a flat surface with no elasticity, view from the side (say Z-plane), a flat surface is basically a line.
The next question is what is mean by "folding". I think it mean the two surface "coincide" with no distance between them. From the side view, it mean it is the same meaning of no distance between two points at the point of folding, that is a "single" point (from Z-plane). Take a horizontal folding as an example, in 3D view, it is a straight-line (with same Z-plane co-ordination but different X, Y coordinate). From the point of view of Z-plane, a curve line represent more than one points, that mean the paper need to be torn apart so as to curve, which is impossible.
