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by raphlinus 133 days ago
I don't agree that the clothoid is a math nightmare. One of the central problems you have to solve for roads is the offset curve. And a clothoid is extremely unusual in that its offset curve has a clean analytic solution. This won't be the case for the cubic parabola (which is really just a special case of the cubic Bézier).

Sure, you have to have some facility with math to use clothoids, but I think the only other curve that will actually be simpler is circular arcs.
1 comments
Ef996 133 days ago
I mean they are not a math nightmare per se if you’re comfortable with the theory. What I meant is that they become comparatively complex to integrate into a system like this. Think about arc length, compute intersections, reparametrization, etc., and with clothoids that usually means some complex numerical algorithms.

Using circular arcs or even simple third-degree polynomials (like cubic parabolas) reduces many of those operations to trivial O(1) function calls, which makes them much cheaper to evaluate and manipulate procedurally, especially when you're computing it 60 times per frame

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masfuerte 133 days ago
You might be familiar with these, but GP wrote a couple of excellent pieces on Euler spirals:

https://raphlinus.github.io/curves/2021/02/19/parallel-curve...

https://levien.com/phd/euler_hist.pdf

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