Timeline for Intrinsic definition of arc length [closed]
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| May 22, 2015 at 10:03 | history | closed |
Alex Degtyarev Stefan Waldmann András Bátkai Dima Pasechnik alvarezpaiva |
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| May 19, 2015 at 11:09 | answer | added | Duchamp Gérard H. E. | timeline score: 1 | |
| May 19, 2015 at 0:26 | comment | added | Qiaochu Yuan | @Felix: you do not need a parameterization. Provided that you know where the two endpoints are, you can take the limit (in the sense of nets) over all PL approximations with the same endpoints. That just requires that you know where the points on the curve are. | |
| May 18, 2015 at 21:18 | history | edited | Ricardo Andrade |
replaced inadequate tags
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| May 18, 2015 at 20:52 | answer | added | Liviu Nicolaescu | timeline score: 4 | |
| May 18, 2015 at 16:45 | comment | added | Ryan Budney | There's the Cauchy-Crofton formula for the arc length of a curve. It does not use a parametrization. It also generalizes to formulas for surface areas, etc. en.wikipedia.org/wiki/Crofton_formula The Crofton formulas also have versions that work for parametrized curves (arc length) vs. just the image/trace of the curve. | |
| May 18, 2015 at 12:11 | comment | added | Joonas Ilmavirta | @DavidRoberts, yes, by trace I mean image. A finite set of self intersection is ok, but the OP didn't seem to assume it. | |
| May 18, 2015 at 8:45 | comment | added | Felix Goldberg | @DavidRoberts I've considered this option, but I think there's a catch: to talk about line segments created by a set of points $S$ sampled from the curve, we would need to rely on a (sensible) ordering of the points in $S$ - which seems to throw us back to the need for a parametrization of sorts. Do you agree or am I missing something here? | |
| May 18, 2015 at 8:09 | comment | added | Włodzimierz Holsztyński | Indeed, one would like a statement of the possible (research) potential of an alternative definition. This would help to focus the discussion. | |
| May 18, 2015 at 8:05 | review | Close votes | |||
| May 18, 2015 at 23:09 | |||||
| May 18, 2015 at 8:03 | answer | added | Włodzimierz Holsztyński | timeline score: 12 | |
| May 18, 2015 at 7:48 | comment | added | Loïc Teyssier | Am I the only one doubting the research-level relevance of the question (at least in its present form)? | |
| May 18, 2015 at 7:40 | comment | added | David Roberts♦ | @Joonas by 'trace' do you mean the image of a defining function? Assuming a finite set of self-intersection points should sort that out (or perhaps measure zero!) | |
| May 18, 2015 at 7:17 | comment | added | Joonas Ilmavirta | If you don't want to use a parametrization, do you only want to work with the trace of the curve? In that case you cannot distinguish between a line traversed once and a line traversed several times if you assume no injectivity. Oh, and how about the one dimensional Hausdorff measure of the trace? That requires no parametrization but does measure length. | |
| May 18, 2015 at 7:03 | comment | added | David Roberts♦ | @Cusp probably not. Such an approximation only requires a bunch of points on the curve, and one can define the error of the approximation via the sum of the maximum and the minimum of the length of the line segments in the approximation. Presumably for rectifiable curves as this 'error' approaches zero the usual distance (L^1 metric, say) approaches zero. | |
| May 18, 2015 at 6:31 | answer | added | user73805 | timeline score: 0 | |
| May 18, 2015 at 5:59 | comment | added | Cusp | @QiaochuYuan Isn't approximation by piecewise linear curve also requires parametrization? | |
| May 18, 2015 at 5:48 | comment | added | Qiaochu Yuan | Yes, approximate it by a piecewise linear curve and take the limit (if it exists). | |
| May 18, 2015 at 5:37 | history | asked | Felix Goldberg | CC BY-SA 3.0 |