Talk:Fresnel integral

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[edit] Question regarding acceleration

At the Cornu spiral: a vehicle with constant speed will not have a constant rotational acceleration, but a constant _change_ of r.a., won't it?

In this image of the [[Image:Cornu_Spiral.svg|spiral]], the radius of curvature is zero at the origin, and tends to a constant towards the two attractors ("holes"). So a vehicle travelling along this yellow-brick road would experience zero acceleration as it passes the origin, then increasing acceleration as it spirals in towards either attractor, then in the limit constant acceleration, that is, it would experience a circular orbit (within diminishing error). That isn't exactly what you said, but does it answer the question ok? Pete St.John (talk) 16:03, 10 March 2008 (UTC)

short answer, someone corrected that in the text already; now it says "constant change" :-) So good catch Pete St.John (talk) 16:10, 10 March 2008 (UTC)

[edit] Cornu spiral images

The article says that clothoid is another name for a Cornu spiral. But the clothoid loop image is quite different to that of the spiral, and I think it must confuse readers of the article. Also, why does the caption say "f(r) = r − π"? Is the image actually a clothoid loop or is it an Archimedean spiral? JonH (talk) 12:36, 10 March 2008 (UTC)

One image is the graph in polar coordinates, the other in cartesian coordinates. Pete St.John (talk) 16:12, 10 March 2008 (UTC)

Plotting S(t) and C(t) as cartesian or polar coordinates would give different shapes, but the article talks about a clothoid as being the same a Cornu spiral and there is no mention of polar coordinates. Also I think that r = θ − π is a formula for the polar coordinates of an Archimedean spiral, not for a Cornu spiral. JonH (talk) 18:27, 10 March 2008 (UTC)

The diagram caption at that section labels f as a function of r, shows a background of polar coordinates, and calls it a parameterization, so it would seem that representing the function from Cornu Spiral in polar coordinates is supposed to give the shape "Clothoid". From Archimedean spiral it would seem the Clothoid is a special case of Archimedean (the constant equals pi). I admit the section is not very clear, we might check a reference. Pete St.John (talk) 19:29, 10 March 2008 (UTC)

I think I've been blithe, I have to think about this some. What happens if we express r and theta in the archimedean spiral parameterized by arclength? Pete St.John (talk) 20:39, 10 March 2008 (UTC)

"Cornu spiral" and "clothoid" are synonyms, and refer to the curve with an S-like shape with two spiraling ends. Apparently there is such a thing as a "clothoid loop", but the article does not give a definition, nor does it explain how it is related to the clothoid curve. From some web pages I get the impression that the loop consists of two segments of the clothoid glued together (e.g. here), but I don't know of a reliable source for this definition. The "Parametric graph" sure looks like an Archimedean spiral if you let the angle run across 0. I don't get the meaning of the equation f(r) = r − π, which looks very wrong. Possibly r = θ − π is meant.  --Lambiam 17:35, 11 March 2008 (UTC)

In the Archemedean spiral case where the constant is -pi, the equation would be theta = r - pi (so f(r) is theta). I think that's what was meant. I haven't had time yet to answer my own question, expressing r and theta as functions of arclength, I want to. And sorry for the primitivist typography, I never typed much math myself, it would have been nroff in my day :-( Pete St.John (talk) 02:58, 12 March 2008 (UTC)

You mean Eqn? What will you do with your expressions of r and theta as functions of arclength, once you've got them? I think arclength can easily be expressed as a function of r or theta, but you can't invert the transcendental equation into a closed form for r and theta as functions of arclength. It is easy, though, to study the asymptotics for small or large arclengths.  --Lambiam 08:31, 12 March 2008 (UTC)

I meant from the Archimedean spiral / vartheta = r - / pi, write r and theta as functions of arclength. I'm wondering how to interpret C and S for this "parameterized" graph. But I'm rusty (at best) and would need some uninterupted time with pencil and paper. (And yeah, Eqn would have been available by then but I wasn't aware of it. I paid a professional who had the greek and math balls for her Selectrix :-) Pete St.John (talk) 12:46, 12 March 2008 (UTC)

I've removed the "clothoid loop" image: until someone can describe exactly what this is, and how it relates to the ordinary clothoid, it does not belong in the article. Not only was it of questionable relevance, it also drew attention away from the correct image. -- The Anome (talk) 10:51, 13 March 2008 (UTC)

Update: I've now found a cite for the use of the term "clothoid loop" as used in rollercoaster design: I've put it in the vertical loop article and made the term redirect there, and adjusted this article accordingly. The image stays out: I'm not sure what it illustrates, but since it seems neither to be a clothoid curve or a rollercoaster "clothoid loop", and does not have any idenfiable mathematical significance, it does not seem relevant to either article. -- The Anome (talk) 12:02, 13 March 2008 (UTC)