I was thinking about chasing-the-beam-style raster-generation software in low memory, and how to achieve smooth curves.
Suppose you want to rasterize some kind of shape. You can approximate the border of the shape in a number of different ways: a polygon, a Bézier spline, etc. One particularly interesting way, to me, is as a tilewise polynomial.
Suppose you have divided your screen up into tiles of 256×256 pixels. The screen I'm looking at at the moment is 1024×600, so it would need four such tiles horizontally and three vertically, for a total of twelve. But coordinates within each tile are only 8 bits per coordinate, which means that you can do parallel computations over 16 tiles using SSE instructions.
Suppose you decide to represent the shape's border within a given tile as a polynomial, either for the horizontal axis in terms of the vertical or vice versa, along with limits. What degree of polynomial do you need?
Clearly if the shape border is of arbitrary complexity you need a degree-256 polynomial. But if you just want to hit three points in the tile, you only need a degree-2 polynomial. And if you need two points with specific derivatives — say, the top and bottom of the tile — you can do that with a degree-3 polynomial.