I would replace the last two operations by another multiplication and one reciprocal square root. There is extensive SIMD support for very fast approximations, and getting better in newer chips.
back
1 comments
Yes of course. I just mean, conceptually that is what is required. You can replace a bunch of the multiplications and additions by FMA instructions as well.
By the way Raph, I think you might be interested in this draft paper I have been working on (well, not working on for the past two months, but anyway...)
Very cool! Some of the later sections are broken. The fact that this is interactive is great, maybe I'll get around to putting my stuff in this form.
The broken parts are intentionally commented out, either now redundant or not finished yet. (Observable isn’t the best for version control / etc. yet.)
There are still a bunch of diagrams to make but I got a bit stalled on the project after going on a trip (and taking care of a toddler full time).
There are still a couple of research problems to figure out. In particular how to best set the tangent and curvature at the knots. Just fitting circles through triples of points isn’t the best method.
But I think this thing should compare favorably to Spiro curves for some use cases: in particular it is pretty local, a bit more robust to pathological inputs, and a lot simpler to compute (and explain). (But of course isn’t going to be globally optimizing for some smoothness metric, and isn’t extensional.)
Edit: sorry to bystanders for a completely off-topic conversation.
Agreed. And (as was largely the point of my thesis) these things are tradeoffs, a spline that's perfect in all these ways is not possible. I played around a bit and am not feeling that they'd be great for font design (they play pretty similarly to Séquin's circle splines, which of course you cite, though I think better).
We're now pretty far afield from the subject of the superiority of Rust over C++^W^W^W tau over pi^W^W^W geometric algebra over quaternions. I'd be more than happy to continue the discussion somewhere else.