Talk:Euler spiral

Request a diagram

The article lacks a graphic, i.e. a portion of the Cornu spiral in the first quadrant and begins at the origin illustrating:

• the initial tangent
• the ending circular curve
• the spiral
• radius of curvature R and Radius Rc
• an arbitrary point at coordinate (x,y)
• angles θ and θs
• length measurement L, Ls

Ling Kah Jai (talk) 12:56, 5 February 2009 (UTC)

Diagram added--Ling Kah Jai (talk) 12:51, 15 August 2009 (UTC)

A question

So how do you graph this (The Cornu Spiral) on paper? Evey other spiral can be graphed, but I don't see how to do this one.

What do you mean by "Every other spiral can be graphed"? —Tamfang (talk) 17:38, 31 May 2010 (UTC)

Faulty example

The linked interactive graph is grossly inaccurate (try increasing curve length): removing the link might be an improvement. — Preceding unsigned comment added by 82.185.105.200 (talk) 14:36, 19 July 2013 (UTC)

Indeed, it crosses itself when len > 1.76, which ought never to happen. —Tamfang (talk) 18:18, 19 July 2013 (UTC)

recursive integration

$C(L)=\int _{0}^{L}\cos L^{2}\,dL$

$S(L)=\int _{0}^{L}\sin L^{2}\,dL$

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)

It may be not preferred in style but I believe there is nothing wrong. However, If you wish, you may amend to the variable to 's'. --Ling Kah Jai (talk) 09:33, 8 October 2010 (UTC)

vocabulary

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)

   This is a very general class without a unique name (which would just be jargon anyhow). 162.246.218.28 (talk)

Also, is there a word for the first derivative of curvature (second derivative of tangent angle)? —Tamfang (talk) 19:41, 12 November 2010 (UTC)

  This is related to jerk, which is related quantity. 162.246.218.28 (talk)

Road transition curve

Would it not be logical if transition curve would be an article in it's own right just as track transition curve is now? or at least be a seperate section within Euler spiral. p Peter Horn User talk 02:29, 10 June 2011 (UTC)

Well, the term "transition curve" is awfully broad; the first senses it suggests to my mind are nonspatial. —Tamfang (talk) 13:16, 12 June 2011 (UTC)

Mathematica syntax q

Incidentally, in the expression

   FresnelC[Sqrt[2/\[Pi]] t]/Sqrt[2/\[Pi]],

is the \[...] necessary? I thought Pi was a valid symbol on its own. —Tamfang (talk) 20:55, 18 June 2011 (UTC)

Simplify, using complexity!

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):

${\begin{aligned}x+iy&=\int _{0}^{L}e^{is^{2}}\,ds\\&=\int _{0}^{L}\left(1+is^{2}+{\frac {i^{2}s^{4}}{2!}}+{\frac {i^{3}s^{6}}{3!}}+{\frac {i^{4}s^{8}}{4!}}\cdots \right)\,ds\\&=L+{\frac {iL^{3}}{3}}+{\frac {-L^{5}}{5\times 2!}}+{\frac {-iL^{7}}{7\times 3!}}+{\frac {L^{9}}{9\times 4!}}\cdots \end{aligned}}$

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)

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)

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)

I assume you meant "dθ is independent". —Tamfang (talk) 02:24, 21 August 2023 (UTC)

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)

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;

Pierre-Dufresne (talk) 00:35, 7 July 2019 (UTC)

Strange math in the illustration of scaling

https://en.wiki.x.io/wiki/Euler_spiral#Illustration

Here it's stated that

${\begin{aligned}\theta _{s}&={\frac {L_{s}}{2R_{c}}}\\&={\frac {100}{2\times 300}}\\&=0.1667\ {\mbox{radian}}\\\end{aligned}}$

This can't be right? Theta is the integral of curve length, so proportional to length^2? Maybe I'm misunderstanding... — Preceding unsigned comment added by 176.10.216.197 (talk) 02:52, 10 June 2020 (UTC)

Since L ~ k ~ 1/R, we have $\theta$ ~ L^2 ~ L/R . Also, $\theta$ is the integral of the curvature k over the arc length. WillNess (talk) 15:27, 27 July 2023 (UTC)

This article needs work

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)

Radius and center

The first animation File:CornuSpiralAnimation.gif is very interesting but it begs the questions "how do we find the center of the circle" and "how do we find the radius of that circle."--Skintigh (talk) 16:29, 29 October 2021 (UTC)

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)

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)

Please find sources to reference for these fine points and include them in the article! Johnjbarton (talk) 14:07, 21 August 2023 (UTC)

Fresnel spiral

The Cornu spiral is often called instead the Fresnel spiral, but virtually never the "Euler spiral", as this article erroneously believes. 2601:200:C082:2EA0:4463:B040:E4D2:6386 (talk) 00:59, 4 October 2023 (UTC)

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.

There is no Fresnel spiral page, but Fresnel integral links this page with discussions of spirals related to Fresnel. Johnjbarton (talk) 02:26, 4 October 2023 (UTC)

I created Fresnel spiral as a redirect. Judging from a Google scholar search it does seem to be a (rarely used) name for this. –jacobolus (t) 02:52, 4 October 2023 (UTC)

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)

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)

Claims like

• the Cornu spiral is often called instead the Fresnel spiral, but virtually never the "Euler spiral"

are meaningless unless they are backed up by evidence, including evidence of "often" and "virtually never". Johnjbarton (talk) 16:23, 23 November 2023 (UTC)

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)

"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."

von Seggern, David H.. PHB practical handbook of curve design and generation. United Kingdom, CRC-Press, 1994. Johnjbarton (talk) 02:30, 4 October 2023 (UTC)

---

It should not be left unmentioned in the article that this curve is also called a "Fresnel spiral". 2601:200:C082:2EA0:D14D:7183:D20A:AB89 (talk) 16:03, 23 November 2023 (UTC)

Please provide a reliable source. Johnjbarton (talk) 16:21, 23 November 2023 (UTC)

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)

In Levien, Raph. "The Euler spiral: a mathematical history." Rapp. tech (2008) the phrase "Fresnel spiral" never appears. Johnjbarton (talk) 17:04, 23 November 2023 (UTC)

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)

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 don't see anywhere in this work where the Euler spiral is discussed. Johnjbarton (talk) 19:13, 20 January 2024 (UTC)

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)

yes, the Towal is not needed at all. Johnjbarton (talk) 19:19, 20 January 2024 (UTC)

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)

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)

I reduced the paragraph to match the context of the Euler spiral. The material may fit in articles on whiskers IDK. Johnjbarton (talk) 19:30, 20 January 2024 (UTC)

Unsourced additions.

@Kaba3 recently added content with no references. Please add sources, thank you.

At present the new content seems unconnected to Euler spiral, but I can't repair this missing info without the source. Johnjbarton (talk) 20:10, 21 October 2024 (UTC)

@Johnjbarton I'll find references. Meanwhile, for intuition, it is because you get the error function by rotating the Fresnel integrals 45 degrees. The formula can be visualized either in Mathematica or Sagemath. Kaba3 (talk) 20:28, 21 October 2024 (UTC)

I added a simple proof sketch for the formula. Although it is simple to do by hand, here is Mathematica verifying the formulas:

Alpha = (1 - Sign[k]*I)*Sqrt[Pi/8];

L= Sqrt[(Pi/2)/Abs[k]];

f[t_]:=L*Erf[Alpha * t/L];

Assuming[k>0,Simplify[D[f[t],t]]]

Assuming[k < 0, Simplify[D[f[t], t]]]

Assuming[k>0,Limit[f[t],t-> Infinity]]

Assuming[k < 0, Limit[f[t], t -> Infinity]]

which returns

((1-I) E^(1/2 I k t^2))/Sqrt[2]

((1 + I) E^(1/2 I k t^2))/Sqrt[2]

Sqrt[\[Pi]/2]/Sqrt[k]

Sqrt[\[Pi]/2]/Sqrt[-k]

Noting that (1 - I) / Sqrt[2] = E^(-Pi/4) and (1 + I) / Sqrt[2] = E^(+Pi/4), the formulas in the article are obtained.

Mathematica can be used to visualize as follows:

kPlot =1;

ParametricPlot[{Re[f[t]]/.{k->kPlot}, Im[f[t]]}/.{k->kPlot},{t,-10,10}]

which plots the expected Euler spiral, with limit points symmetrically on the x-axis, and at (-L, 0) and (+L, 0).

I'll still look for a reference. Kaba3 (talk) 23:00, 21 October 2024 (UTC)

I mean, the proof is rather simple: apply Euler's formula to $e^{-ix^{2}}=i\sin(-x^{2})+\cos(-x^{2})=-i\sin(x^{2})+\cos(x^{2})$, then antidifferentiate both sides (constants can easily be dealt with). This doesn't strike me as particularly interesting nor worthy of inclusion in the article: if no sources describe it, then it's undue weight to put it in.--Jasper Deng (talk) 23:30, 21 October 2024 (UTC)

Seems like you reverted already. First of all, I demonstrated verifiability, because there is a simple way to prove it. I'm not seeing why there would need to be sources at all in this case. Second, currently there is no single explicit representation in the article for an arbitrary affine curvature function. My edit was the first attempt at providing one. Third, the connection to the error function is not trivial to notice, even if it can be verified simply in retrospect. Fourth, the error function is one of the best ways to evaluate the Euler spiral: there is an excellent algorithm to evaluate the Faddeeva function at https://epubs.siam.org/doi/abs/10.1137/20M1373037, which can then be used to evaluate the complex argument erf very accurately. Hence, that the connection to the error function is not interesting is quite a subjective opinion. Fifth, articles should preferably include a maximal amount of information, not minimal. I'd prefer you put it back. Kaba3 (talk) 01:17, 22 October 2024 (UTC)

A simple way to prove it is not good enough here: you need a source to demonstrate that this is something that exists in the literature (ideally some demonstration that other secondary sources about the Euler spiral specifically discuss this, to show that this is considered an important aspect of the topic). Otherwise it is "original research", which is not allowed here even in cases where the result is relatively obvious. This is, in my opinion, well beyond the scope of WP:CALC. –jacobolus (t) 02:33, 22 October 2024 (UTC)

For what it's worth though, if you think this is important and "not trivial to notice", you should pursue publishing it in some other venue. Ideally a peer-reviewed paper, but it could also be a preprint, blog post, or whatever. –jacobolus (t) 05:16, 22 October 2024 (UTC)

Levien 2008 has:

The Fresnel integrals are closely related to the error function [Abramowitz & Stegun, p.301], §7.3.22: $$C(z)+iS(z)={\frac {1+i}{2}}\operatorname {erf} \left({\frac {\sqrt {\pi }}{2}}(1-i)z\right)$$

Which is certainly something we could mention. doi:10.1109/HISTELCON56357.2023.10365736 also mentions the error function and points to:

• J. Weideman, “Compution of the complex error function,” Siam Journal of Numerical Analysis, vol. 31, no. 5, pp. 1497–1518, 1994.
• M. Alazah, S. Chandler-Wilde, and S. La Porte, “Computing Fresnel integrals via modified trapezium rules,” Numerische Mathematik, vol. 128, pp. 635–661, 2014.

arXiv:2011.10936 also looks like it might be relevant.

–jacobolus (t) 05:44, 22 October 2024 (UTC)