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Isn't it bad style to use the same symbol for the integration variable and the integration limit? Or does this have some significance that I miss? —
Tamfang (
talk)
04:53, 27 September 2010 (UTC)reply
Is there a word for a curve whose curvature is a continuously differentiable function of arc length, but not necessarily linear? —
Tamfang (
talk)
06:18, 6 November 2010 (UTC)reply
This is a very general class without a unique name (which would just be jargon anyhow).
162.246.218.28 (
talk)
There's no mention of
imaginary numbers in the article. Is a reader likely to be frightened off by imaginaries and not by integrals and Taylor series? With the
complex plane view, we can use one Taylor series rather than two (each of which has more
Kolmogorov complexity than the complex exponential form):
To me it's "natural" to think of integrating exp(i·f(s)), a vector of constant magnitude and transparent direction. What's your opinion? —
Tamfang (
talk)
22:45, 28 August 2015 (UTC)reply
Handwaving derivation
To my understanding, the derivation starts from a misleading premise when it omits the definition of a coordinate system and instead presumes curvature as dθ/ds. This expression for curvature assumes a local coordinate system for θ corresponding to the osculating circle. That circle, however, is not fixed in the global coordinate system, but θ is later used in the global, fixed coordinate system to translate to x and y. A better (or only correct) way to derive the formula would be to first define the individual symbols, including the associate coordinate system(s). That, indeed, is a bit tricky since there is no uniquely distinguished "center" of any circle. — Preceding
unsigned comment added by
193.106.140.9 (
talk)
09:35, 3 July 2019 (UTC)reply
θ is independent of the coordinate system, it is the change in tangent angle, the difference between ending and starting tangent direction. ds = r dθ is intrinsic; r the radius curvature is also independent of the coordinate system.
WillNess (
talk)
15:35, 27 July 2023 (UTC)reply
no. θ is already a delta, a difference between the tangent direction at a given point, and the tangent direction at the starting point of the clothoid. it does not depend on any coordinate system. put it differently, it is the tangent direction in the clothoid's own coordinate system, where the X axis coincides with the starting direction, making the starting direction = 0. whatever the coordinate system, the difference between directions for the same two points on the clothoid does not change.
WillNess (
talk)
09:52, 21 August 2023 (UTC)reply
Suggestion for JavaScript drawEulerSpiral
Initializing the variable 't' to dt/2 instead of zero gives a MUCH better approximation, even with coarse dt. It's equivalent to accessing the curvature in the middle of the interval instead of at the end. I.e.,
Change:
var dx, dy, t=0, prev = {x:0, y:0}, current;
var dt = T/N;
To:
var dx, dy, prev = {x:0, y:0}, current;
var dt = T/N;
var t = dt/2;
In the section "Expansion of Fresnel integral" the article talks about normalization which is not defined properly anywhere, certainly not before this point.
Badly written article
This article contains a lot of irrelevant nonsense, like about rat vibrissae, and the great deal of the article that is wasted in discussion of how to "normalize" the curve. As well as the very poorly chosen symbols for variables whose definitions are not explained, and worse, there are multiple symbols for the same (unexplained) variables.
As well as the striking fact that nowhere in the article is it clearly stated that the curvature of the curve at point P is proportional to the arclength along the curve from the origin to P. This is in fact the geometric defining characteristic of this curve.
2601:200:C000:1A0:11C7:4EA3:9235:58D6 (
talk)
23:47, 30 June 2021 (UTC)reply
As you see in that gif, the center decreases perpendicular to the tangent of the curve at the given point, I wish I know the ratio but may be it's constant.
179.26.39.79 (
talk)
18:13, 21 March 2022 (UTC)reply
The radius is the reciprocal of the curvature, which (in this spiral) is a linear function of path length. The line to the center of the circle is perpendicular to the tangent line. —
Tamfang (
talk)
02:29, 21 August 2023 (UTC)reply
Please find a source and include this information in the article.
I would discourage you from "often" and "virtually never" as these imply some kind of census or poll. Instead find some good sources for the definitions of each of these terms. Sometimes sources conflict so we just add both points of view with references.
The link
Cornu spiral redirects to
Euler spiral. If you find references to show that is wrong we can work to fix it.
Google scholar turns up >900 results for "Euler spiral", so I would say you are incorrect / are in a bit of a bubble w/r/t this topic. Note that it is common for the same object/idea to have different names in different disciplines or regions. With that said, I think it would be fair to move this page to
Cornu spiral, which is apparently a (moderately) more popular name. Or perhaps
clothoid would be an even better name than either of those; it is the most popular and doesn't involve someone's name. ("Fresnel spiral" is a much rarer name, which I would recommend against using as a primary title or even putting in bold in the page, but could be a redirect.)–
jacobolus(t)02:28, 4 October 2023 (UTC)reply
Claims like "Google scholar turns up >900 results for "Euler spiral" are mostly meaningless, since a very large number of people, and hence websites, take what they find in Wikipedia as the gospel truth and then use the same language. So the reasoning is circular. — Preceding
unsigned comment added by
2601:200:C082:2EA0:D14D:7183:D20A:AB89 (
talk)
16:06, 23 November 2023 (UTC)reply
Claims like
the Cornu spiral is often called instead the Fresnel spiral, but virtually never the "Euler spiral"
Google scholar doesn't show results from "websites", but instead from the academic literature, predominantly published materials (but also some preprints etc.). Most results there are not going to have anything to do with Wikipedia. –
jacobolus(t)16:36, 23 November 2023 (UTC)reply
"7.3.1. Spiral of Cornu A particularly beautiful spiral is based on the integrals of Fresnel, which arise physically in the diffraction of electromagnetic and acoustic waves. This spiral has several names: spiral of Cornu, Euler's spiral, and clothoid."
The name "Fresnel spiral" for this is quite rare in the academic literature (like 10x less than the names bolded in the lead section here; from what I can tell from a brief skim the name "Fresnel spiral" is more often used for a kind of
Fresnel lens constructed with a spiral). It most often seems to appear in sentences such as "The Cornu spiral is also commonly referred to as clothoid, Euler spiral or Fresnel spiral" (
doi:
10.1007/978-3-030-05798-5_4), rather than as the primary name used. But I added a redirect from
Fresnel spiral, as discussed a month and a half ago. I don't anticipate any significant confusion if someone arrives expecting "Fresnel spiral" and then immediately reads the sentence "Euler spirals are based on Fresnel integrals". –
jacobolus(t)16:39, 23 November 2023 (UTC)reply
I chose "Euler spiral" as the primary name simply because Euler is the first person to characterize it in detail. It's been rediscovered a number of times, so "Cornu spiral" is also common, but I think less appropriate. I think most of the people who use it are simply aware of Euler's priority. "Clothoid" is a fine neutral name, but was proposed by Cesàro well after the other names were in widespread use. I've never seen "Fresnel spiral" in actual use, but of course "Fresnel integral" is standard.
Raph Levien (
talk)
13:52, 26 November 2023 (UTC)reply
Towal reference for whisker shapes
In the section on 'Whisker shapes" a reference has been added by @
Aachapp which I don't believe is necessary or correct:
Towal, R.B.; et al. (7 April 2011). "The Morphology of the Rat Vibrissal Array: A Model for Quantifying Spatiotemporal Patterns of Whisker-Object Contact". PLoS Computational Biology. 7 (4): e1001120. doi:10.1371/journal.pcbi.1001120. PMC 3072363. PMID 21490724
I didn't check the citation, but even if were discussed this way, the level of detail added about rat whiskers seems out of scope for this article. –
jacobolus(t)19:16, 20 January 2024 (UTC)reply
I understand the concern, particularly about the level of detail. However, as written, the Wikipedia article misrepresents the history of finding that whisker shapes can be fit with Euler spirals.
The Results section of Towal et al., 2011 uses the term "Cesaro Equation" to describe fitting the shape of rat whiskers to the curve K(s) = As + B (an Euler spiral). The peer-reviewed plot is
here, and demonstrates that rat whisker shape is described by an equation that linearly relates the curvature to the arclength (an Euler spiral).
Starostin et al, 2020 reference the results of Towal et al., 2011, when they indicate that they are performing the identical analysis: "Consequently, we fit the data to Euler spirals, given by the Cesàro equation κ(s) = As + B, where s, an element of [0,1], is the scaled arc length, κ is the curvature, and A and B are constants, called the Cesàro coefficients (Towal et al, 2011)."
I hope that we can work together to find a way to eliminate the great detail about rat whiskers, and yet also find a way to indicate precedence and contributions to this admittedly-niche field.
Aachapp (
talk)
19:54, 20 January 2024 (UTC)reply
I just listed both papers in the same footnote. Note that references on Wikipedia don't really work the same way as references in scientific publications. The purpose is primarily to validate the claims in the article rather than to give credit to the first (or every) author who wrote about something, per se. There's not really a problem with adding the first or most important sources from the literature, but in the event that only a later source is cited it does not "misrepresent the history"; no claim about the history of this idea was being made. –
jacobolus(t)20:08, 20 January 2024 (UTC)reply