Think of the frenet-serret formulas as a way of keeping track of your driving using the odometer, steering wheel angle, road bank angle (nonzero on turns), and uphill/downhill gradient. If you logged all 4 numbers precisely at all times, you'd know where you were without GPS
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This is a better approach to doing math on twisty-turny paths like roller coasters or spirals or... threads... than the usual way of extrinsic coordinate systems (either explicit like y=f(x) or implicit like f(x,y)=0) but it's not super intuitive.
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Heh, I just remembered. At Xerox, as a random side thing, I wrote some code, using frenet-serret frames, for modeling how paper path (paper flow through a printer) could take arbitrary twists and turns through modular printers that put multiple print engines in parallel/series
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This is like rack-mounted cloud-like print architecture (known as TIPP: tightly integrated parallel printing). Never very popular/practical, but fascinating to think about. Like distributed computing for printing. Paper flow being effectively a packet-switched network
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(basic idea is that instead of building a single giant extremely high-speed production printer, like old-school Cray supercomputers), you assemble a lot of smaller printers into an parallel array, with a shared paper path (kinda like a rack-scale high-speed bus in a datacenter)
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If paper weren't dying, this would open up fascinating possibilities for real-time variable printing at binding level. Variable information printing already allows every copy of a page to be somewhat different. TIPPed printer racks could effectively shuffle pages like card decks
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I am the king of pointless bunnytrails... still, to get back to original point, I'm trying to think about threaded longform content in a "frenet serret" way where book-like formats are more extrinsic coordinate based.
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Writing naturally has something like an arc-length parameter (the independent variable in frenet-serret)... word count. But tweets and blog posts are "scalars". To F-Size them, you'd need to think of them as vectors. Every chunk having an orientation and direction as well.
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In books pages/chapters are scalars, though we think in terms of quasi-vector qualities (direction of narrative arc, "twist" endings), but these can lean on inherited logic of a "plot" scaffolding conceived beforehand as a "plan". I want the dead-reckoning internal equivalent.
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Why do things in this complicated way? Well, we're moving from a Newtonian zeitgeist to a relativistic zeitgeist. Frenet-serret frames lead on to curve space times of general relativity. If the space you are living in is curved, intrinsic coordinates work better.
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Why do things in this complicated way? Well, we're moving from a Newtonian zeitgeist to a relativistic zeitgeist. Frenet-serret frames lead on to curve space times of general relativity. If the space you are living in is curved, intrinsic coordinates work better.
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It sounds like what I learned as "parametric curves" only more so, somehow.
Also path integrals would seem to be involved somehow.
Looking forward to digesting the reference.
Thanks!


