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At the moment, I'm implementing an algorithm which defines a spiral using the method outlined in the following paper:

https://www.ams.org/journals/mcom/1992-59-199/S0025-5718-1992-1134736-8/S0025-5718-1992-1134736-8.pdf

One of the challenges I've yet to sort out is how to calculate a point on the spiral given a point that is offset from it. Essentially, it's calculating the orthogonal vector from a point near the spiral and the corresponding distance. This appears to require an iterative process to zero in on the location with a desired level of precision, as suggested here:

Distance between point and a spiral

I haven't taken the time to sort out how to adapt that for my case just yet, as I was hoping I could get some direction on whether or not this is the sort of approach I need to implement - or is there something more efficient / direct?

Joel Graff
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  • A problem and a question. The problem : the first pointer doesn't point on an AMS document but to the same MathStackExchange Question/Answer(s). The question : which kind of spiral are you mentionning ? There are so many of them... you mentionned a clothoid, whereas the MathSE document mentions a Fermat spiral. A remark : archimedean spirals are ideally fit for offset issues... – Jean Marie Jun 15 '19 at 18:20
  • The link is fixed in the first case.
    In the second, I referenced the post if only because it was one of the few references I could find relevant to what I want to do - despite the type of spiral. I suspect the approach would be similar in my case, although I'm not working in polar coordinates. I also imagined an approach similar to this - basically an iterative, binary search method. I'm curious to know if there's a more direct way of finding that coordinate, even if it isn't a clothoid. It may give me a clue as to where to go looking... or what question to ask next. :)     – Joel Graff Jun 15 '19 at 21:41
  • I should also note that the reason I'm focused on clothoids is because this is a project related to transportation engineering, which specifically uses clothoid spirals for the fact that they maintain a constant curvature. – Joel Graff Jun 16 '19 at 02:55
  • binary/adaptive step/ search on $s$ (distance along the clothoid) getting its corresponding $(x,y),\theta$ , while calculating the angle from the $(x,y)$ point on the clothoid to your given point, until this angle is $\theta+\pi/2$. – Will Ness May 01 '21 at 08:03

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