Comment by ux
I'm interested in the approach you're describing but it's hard to follow a comment in the margin. Is there a paper or an implementation example somewhere?
I'm interested in the approach you're describing but it's hard to follow a comment in the margin. Is there a paper or an implementation example somewhere?
What you described in your first message seemed similar to the approach used in the degree N root solving algorithm by Cem Yuksel; splitting the curve in simpler segments, then bisect into them. I'd be happy to explore what you suggested, but I'm not mathematically literate, so I'll be honest with you; what you're saying here is complete gibberish to me, and it's very hard to follow your point. It will take me weeks to figure out your suggestion and make a call as to whether it's actually simpler or more performant than what is proposed in the article.
I have written some gist to illustrate the approach I suggest. The code run but there may be bugs, and it don't use the appropriate optimizer. The purpose is to illustrate the optimisation approach.
https://gist.github.com/unrealwill/1ad0e50e8505fd191b617903b...
Point 33 "intersection between bezier curve with a circle" may be useful to find the feasible regions of the subproblems https://pomax.github.io/bezierinfo/#introduction
The approach I suggest will need more work, and there are probably problematic edge cases to consider and numerical stability issues. Proper proofs have not been done. It's typically some high work-low reward situation.
It's mostly interesting because it highlight the link between roots and local mimimum. And because it respect the structure of the problem more.
To find roots we can find a first root then divide the polynomial by (x-root). And find a root again.
If you are not mathematically literate, it'll probably be hard to do the details necessary to make it performant. But if you use a standard black-box optimizer with constraints it should be able to do it in few iterations.
You can simplify the problem by considering piece-wise segments instead of splines. The extension to chains of segment is roughly the same, and the spatial acceleration structure based on branch-and-bound are easier.
The general technique is not recent I was taught it in school in global optimisation class more than 15 years ago.
Here there is a small number of local minimum, the idea is to iterate over them in increasing order.
Can't remember the exact name but here is a more recent paper proposing "Sequential Gradient Descent" https://arxiv.org/abs/2011.04866 which features a similar idea.
Sequential convex programming : http://web.stanford.edu/class/ee364b/lectures/seq_notes.pdf
There is not really something special to it, it just standard local non linear minimization techniques with constraints Sequential Least Squares Quadratic Programming (SLSQP).
It's just about framing it as an optimization problem looking for "Points" with constraints and applying standard optimization toolbox, and recognizing which type of problem your specific problem is. You can write it as basic gradient descent if you don't care about performance.
The problem of finding a minimum of a quadratic function inside a disk is commonly known as the "Trust Region SubProblem" https://cran.r-project.org/web/packages/trust/vignettes/trus... but in this specific case of distances to curve we are on the easy case of Positive Definite.