### Fabric triangulations

 D dress from Continuum Design
What with cheap and easy 3D scanning and laser cutting, I see very interesting possibilities for bespoke clothing. But I think there are some technological issues to resolve, and in particular, there's a particular mathematical problem I'm not quite sure anyone has studied. Or at least, I'm not sure how to find the work on it (suggestions welcome!).

Actually, there are sort of three categories of clothing that are relevant here. Ready-to-wear is clothing that is made from standardized patterns to standardized sizes; the buyer chooses the least-poorly-fitting. Made-to-measure is clothing that is cut from a parametrized pattern with the user's measurements fed in. And truly bespoke clothing is designed from scratch to fit the wearer.

 Olympic athletes photographed by Howard Schartz and Beverly Ornstein
Perhaps this is a jaundiced view from someone who is a fairly unusual size, but ready-to-wear is kind of a disaster. I've heard many stories of people from places like India or Iraq coming to North America and being puzzled by the clothes everyone was wearing: such lovely fabrics, fitting so badly! The normal procedure, as I understand it, is to design one garment for one size of person, then turn it into a parametrized family by introducing darts of various sizes to scale it up or down. Of course, humans are not well-described by any one-parameter family. Personally I've had a much better time finding clothes that fit in army surplus shops, where Canadian Forces clothes are offered in nine sizes: all combinations of short, medium, and tall with small, medium, and large. (I'm usually a tall/medium.) A few manufacturers will introduce variants like "plus size" (wider), "petites" (proportioned for shorter people), or "tall", but (for example) Lands' End's Tall line offers no women's shirts with sleeves long enough for me, regardless of other size parameters.

Even very primitive made-to-measure is better. That same Lands' End offers jeans in various sizes, but you get to specify the inseam, and they will basically cut the jeans to any length you like no matter how bizarre your proportions happen to be. Not perfect, since to fit well you'd want the jeans to also match your waist, "hip", and thigh circumferences, but at least it's possible to fit some unusual sizes.

Full-on bespoke clothing suffers from the industrialization problem: mass production has made materials and standardized assembly cheap (albeit largely through the use of sweatshops). But the cost of paying skilled workers has increased, so while you can get clothes made out of all sorts of fabrics, paying a tailor to design them to fit you has become inconveniently expensive. The wisdom I've heard is that if you're going to be spending much time in India (say), you should get yourself a suit made, because the tailors there are working for Indian salaries and can make you a lovely well-fitted suit for a very modest price by North American standards.

The advent of technology promises to help bring bespoke clothing back into the realm of affordability. For one thing, with modern technology a 3D scanner is not at all a complicated device; a stereo pair of cameras will often do fine, or for human faces (because of all the face-recognition work that has gone on) often a single image is enough. But even structured-light  scanners are not expensive or difficult to use. So it's not difficult for a machine to obtain a decent model of the intended wearer of the clothing.

On the manufacturing side, sewing and assembly is not very easily automated (yet), but with laser cutters, making specific shapes and sizes is an automatable task. And sewing pre-cut and pre-marked fabric is not very difficult; even I can do it. So the only problem is going from the required 3D shape to the 2D fabric pieces.

Of course, the easiest way to do this is a sort of made-to-measure approach; designers come up with parametric families of clothing, and software selects the best-fit item from a modest multidimensional space of sizes; the parameters are then used to generate cut shapes. Such an approach can work, but it's not as flexible as truly bespoke clothing, and it's not simple for the designer.

For truly bespoke clothing entirely generated by computer, the mathematical problem is this: you have a three-dimensional shell specified in some way, and you have to approximate it with a modest number of pieces of fabric. This is non-trivial because the fabric is flat while the shell will be curved.

 Topology control in approximation
There's a classic problem in 3D graphics, of approximating some three-dimensional surface with a modest number of flat polygons (often triangles). Naive algorithms can make topological errors (no, that arm should not be connected to the hip no matter how little it adds to the mean squared error) but enough work has been done that there are algorithms to do this cleanly and efficiently.

That's not the problem here, though. While it's superficially similar - approximate a three-dimensional surface with a modest number of flat polygons - there is a key difference in what is meant by "flat" here. Without diverging into differential geometry, even the restricted part appropriate for understanding general relativity, the difference is between intrinsic and extrinsic curvature. To make a cylinder or a cone out of fabric, all you need is one piece, because the fabric itself can be bent. But to make a sphere, you need more pieces, because the sphere is intrinsically curved; triangles on a sphere, for example, all have angles adding up to more than 180 degrees. So for the same reason map projections all distort and/or cut the sphere's surface to put it on a flat plane, either you need to rely on the fabric stretching or you need to use some other means to approximate the shape. Multiple pieces, darts removed or stitched out, ruching, and other techniques allow tailors/seamstresses to make flat cloth fit curved bodies. But I don't know of any research into algorithms for approximating curved surfaces with flexible-but-intrinsically-flat pieces. And it's not at all obvious how to go about it.

 Schematic of mechanism for a baseball cover sewing machine
Think about approximating a sphere, for example. Triangulating it gives you something like an icosahedron or a geodesic dome; anyone who's used 3D modelling software is probably familiar with the sorts of shape. But making a sphere out of fabric is another question. You can imagine covering the sphere with a series of strips, each of which forms part of a cone, but baseballs (for example) do it with a pair of vaguely peanut-shaped pieces of fabric. I can't easily picture software that would come up with that design.

Of course, there are more considerations than simply approximating the curvature; fabric is distinctly anisotropic, stretching differently along the warp, weft, and diagonal directions. And of course if you have really stringent requirements, like the skin-tight permeable spacesuit designers, you may want to lay out every fibre in the garment based on the user's shape and movements. These present even more complex requirements for design and assembly systems. Nevertheless, we are already seeing clothing designers take advantage of 3D scanning to make clothes that fit better. Thank goodness.