Rounding out flat paper

Some observations on problem surfaces

One of the biggest problems with paper models is the fact that you can't really curve paper in two different directions at one time. I should actually qualify that statement that you can't do that severely or easily in two directions at one time. The other problem is that many 3 dimensional surfaces are not directly mappable to a 2 dimensional space. This second problem is easily illustrated by taking a large section of orange peel and flattening out on a surface. The peel will inevitably crack and split.

Getting paper to go around in more than one direction

The reason for the former has to do with the material itself. Paper is a mesh of microscopic fibers (typically wood pulp) that are woven together under pressure. There is often a general alignment of these fibers that result in the paper being easier to roll in one direction rather than another. Some model designers actually recommend taking advantage of this property when fitting pieces for printing to take advantage of this natural roll which in most factory milled paper would be around its long axis. A crease or fold in the paper actually breaks some of these fibers creating a hard corner, but a roll will simply bend and stretch the fibers into a curve much like the way you would cup your fingers to give someone a "leg up" somewhere. While in theory it should be possible to stretch and bend those fibers around in more than one direction, like the way fabric will stretch and wrap, in practice paper fibers are too static.
     A limited amount of orthogonal roll (that is rolling around 2 axis) can be done by rubbing the paper over a curved surface such as the outside of a large spoon bowl. This technique is unfortunately limited to small areas and curvatures that are relatively shallow. If the paper is damp, it can be molded to some extent, of course if the paper is printed in a water soluble ink, this is not an option. Don't expect very dramatic curves, but it might be OK for miniature dishes and the like.

A sheet of paper is Euclidean

If you try to wrap paper around an apple to match the curve you will quickly come to the reason for the second problem. When one is first introduced to geometry, one of the first tasks is to draw a typical flat X/Y (Cartesian) coordinate system. Parallel lines will always remain equidistant to each other. This is a geometry whose properties were initially described by Euclid, and so hence Euclidean. On a non-Euclidean surface, by definition this is not true. So two parallel lines, being lines who can share a normal (a right angle intersection), do not actually remain equidistant. The classic example are longitude lines which while parallel at any particular latitude, in that they meet the common latitude at right angles, they do actually meet at the poles. This critically means that the area covered by any section between these two lines is not constant. This is why the orange peel splits, since there isn't enough area of peel to compensate for the increase in space as you try to map non-euclidean parallel lines to a flat euclidean space. The inverse with paper is the appearance of folds that try to take up the extra area.
     Surprisingly, not all 3D surfaces result in this type of area difficulties. Consider the rolling of a sheet of paper into a cylinder. This is a 3D surface, and it is quite trivially mapped onto a planar sheet of paper. Similarly you can bunch it up in a cone shape and again, this is not a problem. It turns out that a cylinder is basically a subset of a cone section, that is a segment of a cone whose vertex lies at an infinite distance from this segment. For a trickier shape, consider rolling over two corners like a cone and hold the other corners flat on the table. This shape is still related to the cone, but contains a more complicated set of curvatures.

A standard flat sheet on far left, curled up into a cylinder, then roughly into a cone, and finally the pseudo hood like shape that is flat at one end and curled at the far end

     The ability to create relatively complex curves that are based on shapes that can be mapped onto a plane is discussed in an article downloadable from papermodelers.com (alas membership required) called "Cardboard Models Design Principles" by Mad44ms. The paper can actually be a tough read if one is not well experienced in 3D geometry and CAD software, but it does provide some insight into what can be done with flat paper and how to minimize the number of parts to make a curve. To create the examples, Mad44ms takes advantage of the software to maintain the plane-mapping qualities of the surface that is being worked. This is unfortunately not always an option since the ability to do this may not be part of your software or at least a feature that is not easily understood - 3D software is often very complex and the learning curve can be very steep.
     Still, the basic cylinder and cone can provide a variety of different surfaces to work with, even in cases that the cylinder or cone are sectioned and somewhat asymmetrical.

Cylinders and cones. Note that a tapered cylinder is just a cone cut off horizontally. Similarly a cone section can be cut at any angle and the resulting shape can be mapped to a plane. Even a cone where the vertex is offset can be mapped to a plane (and a section of that type of cone as well). See next figure

On left, a simple cone. Center, a cone with an off center vertex,. Right, tapered cylinder.

     Regarding exceptions, if you recall the rolled sheets, it can be possible to extend some shapes along a line which indicates a change in curvature. If the line is straight in 3D space between the faces they can be kept together. In practice, without software assistance to maintain the required constraints (matching slopes on both faces, etc), this isn't really possible.  A possible surface that can be done manually by visualizing denting a cone in, or adding a cone in from the opposite direction such that the slope of the intersecting line segments are exactly opposite. The easiest example is with centered cones in which case the common seam is centered and circular making a crater-like shape. There often isn't much call for this particular shape, but I've seen it used for creating fancy eyes on dinosaurs and dragons.

The particular crater-like surface on the right consists of two intersecting cones of exactly the same absolute slope (although one is negative the other positive). The flattened pattern can then be two concentric sections (right).  The inner seated inside the outer. In theory, when folded it automatically generates a perfect circular section (in practice it's somewhat hard to glue perfectly...oh well). 

The crater does not need to have equally long slopes on each side, it doesn't even really need to be centered. This last point is because it only requires for the absolute slopes to match at the point of intersection, but not that the slopes on the whole be equal (the flattened version of the such a shape would have the middle circular fold line off center). To see the range, make a paper cone and dent the apex in such that you get a smooth surface on both sides. Without software to help you get this right, only the centered one is easy: make a cone, then halfway up, subdivide, reset apex to same height as base, then set segment lengths to required lengths by subdivision or translating vertices along the line to keep the slopes unchanged..

Triangles...why I mentioned them

An unfolder script for SketchIt made use of the fact that any face can be flattened against another face along a common edge. This can be then be extended to the next face and so on. Some designs that use this algorithm can appear as long snakes or like some kind of angular octopus.
     While Blender does allow for polygons of pretty much any size, the paper folder software only likes faces that are co-planar, i.e. flat. If for some reason you stretch them unevenly or apply a change to any individual vertex, you may very well introduce a twist in the face. This is also true if you make new 4 vertex faces on your own. When this happens in Blender, it is easy to find non-flat faces as the unfolder script will highlight them (although the feature can be buggy). A better method to deal with this is converting suspect faces, or even whole suspect sections, to triangles. Converting triangles not only guarantees flatness (3 points define a plane), you can also control the type of twist the surface gets for your purpose, such as a more complementary curve to the shape itself.
     Now when looking at a set of adjacent triangular surfaces, you can conceive a part that consists of the entire path of adjacent sides. You can also continue the path outward on additional sides provided that they only meet the main part along only one adjacent side (Note: exceptions do exist). If the curvature reverses (say from concave to convex), you may have to separate the part the mapping might cause an overlap.

A nonsense pseudo vampire with flowing cape that can nonetheless be mapped onto a single sheet.  Simply follow the path of the triangular edges from left to right. The "head" is also attached to only one triangular edge and itself has appendages that are only hinged on one edge.

Concave and Convex and Overlaps

So some intricate shapes are possible, but there are limits. As seen, it is possible to accommodate surfaces that twist and present concave and convex surfaces. Objects with both concave and convex surfaces can create faces that when unfolded results in overlaps. Sometimes with careful planning, you can allow the parts to unfold in a manner that the parts will miss each other when unfolded. In the Blender unfolder script, this is not always caught and you end up with an unworkable part. When it is caught, the part might be cut off arbitrarily and placing it back where it belongs can be tricky.

This surface consist of a rolled surface with down angled sides. The colors refer to the particular parts that overlap when unfolded ; the overlap being shown as blended colors(bottom). In this case, this part would need to be cut into different parts (at least 3, center plus 2 outer panels)

    Unfortunately, I don't have a sufficiently mathematical background to address when this would happen based on the geometry involved. Some surfaces that go in and out can be resolved without a problem, others can't. On the other hand, you may be able to visualize the problem parts as you plan the path of the common seams. Unfold it mentally and you may see not only where the problem arises, but how you can unfold it differently and avoid the problem altogether. I'll talk about decisions regarding seams and patterns for parts in a future posting.

Nautilus ice breaking ram from ongoing project. The highlighted section is only made up of triangles with single common "hinge" segments. It should unfold as one part if so desired.