I notice that your lengthOfBezier() function does an approximation
similar to what we'd get in postscript by executing flattenpath and
then adding up the lengths of the resulting line segments.
I generally find that Quartz doesn't flatten by that much in practice
(though of course you can set this value), so the spacing is quite
coarse between each point especially for already quite "flat" curves,
making it not that suitable for calculating the length - you still end
up having to interpolate the last segment.
In my code it actually does a binary subdivision of each curve segment
until the point on the path is within some epsilon of the wanted point
- I typically use 0.1 (points) as certainly on screen this is very
adequate and takes very little time to execute to this degree of
accuracy. Even for printed output that's only one pixel error at
720dpi, so it's pretty close. Most of the time you're interested in
something that looks right, so a mathematically perfect solution is
not really warranted.
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.
Yup, that seems similar to my experience too, all talk and no trousers!
G.
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