So one thing I've wondered is why more GIS systems can't/don't use B-splines, opting instead for simple lines. The OGC Simple Features standard defines a Curve, which must specify its method of interpolation between points, but then only defines one method – straight linear interpolation.
Obviously there are disadvantages to using b-splines, such as the assumption of accuracy which permeates so many people's interactions with GIS data. At least when you using simple linear features, as you move to a large scale map, you can see the clumsy straight lines that make up your data. But it sure would look nicer when outputting the same data to use B-splines.
Not being hugely math skilled anymore (woe for math atrophy), I'm not even sure how to go about answering the question. But some of the more interesting issues that bounce around in my mind are:
1) Do B-splines/bezíer curves even work in non-euclidean (in particular spherical) geometry?
1a) If they do, then how would one go about projecting them into a euclidean space – if one can meaningfully do so at all?
2) Assuming a "bezíer polygon" were defined as closed series of curves just like linear polygons, how would one calculate the area of said polygon?
Any math über-geeks out there in my vast (hah!) readership have any insights?