On 26 Apr '08, at 7:24 PM, John C. Randolph wrote:
I've asked several math majors over the years if they could give me an exact function for the length of a bezier curve. They've generally told me it was possible, but none of them actually produced it.
A math major's notion of "an exact function" probably allows integral signs. :-P
I looked into similar operations on cubic curves (though not specifically path length) years and years ago when I worked on a font- design program. There are a few easy operations, like getting the bounding-box, but otherwise the math tends to get really hairy. In the case of length, you're solving a path integral, and if I remember my college calculus*, the integrand will be a tangle of roots of sums of sixth powers.
Everything I've read since indicates that the universal solution is to flatten the curves and do the rest of the work on polygons, which is far easier. (One of the exceptions was Apple's old QuickDraw GX: like TrueType fonts, it was based on quadric curves, which are less flexible but mathematically more tractable. GX apparently had code for doing a lot of operations directly on curves without having to flatten them first.)
—Jens * I don't. o_O
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