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Raph Levien

A week ago I figured out that the evolute of an Euler spiral (Cesàro equation κ(s) = s) is another simple power law spiral, the Cesàro equation being κ(s) = -1/s³. The writeup is at: linebender.org/wiki/curves/eul

None of the math for this was very hard, in fact I can see the result being obtained hundreds of years ago if anyone had bothered to look. A challenge for math enthusiasts: does the result generalize to other spirals with a simple power law?
16 March at 14:13 | Open on mastodon.online
4 comments
Evan Martin

@raph I think there's some omitted words(?) in the post in the sentence "In this derivation, we'll fluidly mix complex numbers and 2D vectors, writing." At least I wasn't sure how to parse it.

16 March at 18:18 | Open on inuh.net
Raph Levien

@evmar Good catch, thanks. I have a fix up (rolling in some other changes) here: github.com/linebender/linebend

Also, I should link to discussion on the Zulip: xi.zulipchat.com/#narrow/strea. That's got some more discussion, including other examples that suggest a power law relation, and a link to a paper by Gravesen which may contain a clue.

16 March at 19:02 | Open on mastodon.online
Evan Martin

@raph TeX inline with your markdown seems like a lovely way to publish!

16 March at 19:04 | Open on inuh.net
Raph Levien

@evmar Update on this thread: I did some deep searching in the literature and did indeed find the result. It's in github.com/linebender/linebend which will shortly be merged to linebender.org/wiki/curves/eul (the gentle reader is invited to contribute a review).

I also updated the Wikipedia articles on Cesàro equation and Evolute to include this result. I was originally worried about "no original research" but I'm on firm ground now, as I cite published papers.

@evmar Update on this thread: I did some deep searching in the literature and did indeed find the result. It's in github.com/linebender/linebend which will shortly be merged to linebender.org/wiki/curves/eul (the gentle reader is invited to contribute a review).

17 March at 16:31 | Open on mastodon.online
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