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There's a theorem that states that a tangent line to any point on the cycloid passes through a point, which can be defined in terms of t, located at the top of its corresponding rolling circle. I was told that this is Descartes' theorem, but it doesn't appear to be. Thus far, I've been unable to find any record of the theorem online. Has anyone heard of it? -- Ketsuekigata 19:06, 8 May 2007 (UTC)
Don't know whose theorem it is but it follows directly from that the tangent to the cycloid must be perpendicular to the point about which the circle is instantaneously turning - therefore you get a right angled triangle in the circle with one point at the bottom and the other at the top. This can be used via Visual Calculus to get the area of the cycloid easily. Dmcq 10:31, 24 June 2007 (UTC)
Are there any excamples of cycloids in nature or engineering?
- Every time something round rolls in a straight line, there's a cycloid. -- Ketsuekigata 20:54, 16 June 2007 (UTC)
When a charged particle is subject under orthogonal electric and magnetic field, its trajectory becomes a cycloid. HotBallah ( talk) 18:49, 14 February 2023 (UTC)
Quoting Article: The cycloid through the origin, generated by a circle of radius r, consists of the points (x, y) with
x = r(t - \sin t)\,
y = r(1 - \cos t)\,
where t is a real parameter, corresponding to the angle through which the rolling circle has rotated, measured in radians (not degrees). For given t, the circle's centre lies at x = rt, y = r.
--- It would seem that you could use degrees, as long as you made 0 < t < 360. The statement that t is measured in degrees is dubious, unless there is a statement saying 0 < t < 2*pi.
Am I missing something? jheiv ( talk) 16:37, 5 May 2009 (UTC)
- I edited the article to correct what I thought was wrong / confusing. jheiv ( talk) 16:41, 5 May 2009 (UTC)
- Yes, you're missing something. t has to be in radians, irrespective of any bounds. The bounds you've given just mean that you get one arch of the cycloid, which is not unreasonable I guess. —Preceding unsigned comment added by 81.152.169.12 ( talk) 20:58, 11 May 2009 (UTC)
--Im sorry if im not writing this right, its new to me writing in wiki. the equation that written there for x by not using t isn't right, the right form of the equation is: -sqrt((2 y)/a-y^2/a^2)+cos^(-1)(1-y/a) while a=r. fix it please, I almost got wrong in work for my proffesor. — Preceding unsigned comment added by 85.250.64.229 ( talk) 16:06, 7 June 2012 (UTC)
An editor put in an edit comment saying it was differentiable everywhere, see any textbook. Could they find a citation first please for things like that which manifestly contradict what one can see. Dmcq ( talk)
- That the curves for some parameterization are differentiable is not as far as I know a common documented property of curves. Differentiable is dy/dx and what you're saying will confuse people. And the bit about the derivative being 0 is only for th parameterization and is why the dy/dx is infinite, it would have been perfectly possible for both derivatives in the parameterizations to be zero. Also it is not standard in Wikipedia to give long derivations, only the major necessary points. You might be interested in Wikipedia:Reference_desk/Mathematics for asking general maths questions and Wikipedia talk:WikiProject Mathematics where people interested in improving maths articles discuss issues. Dmcq ( talk) 17:56, 21 January 2010 (UTC)
Oh, sorry, I didn't see this comment. Cycloids are only ever mentioned in math books in relation to parametrized curves. They are in a sense a "classic" parametrized curve. Textbooks never even mention the Cartesian equation because it's unnecessarily complicated. "Regular" curves and "singular points" are elementary properties of differential geometry. See, e.g., Manfredo do Carmo, "Differential Geometry of Curves and Surfaces." (Cycloids are mentioned on page 7, where it says they have singular points at 2pi.)
As for derivations, I did only include the important points. No normal reader of wikipedia is going to know the trig identity that gets you to sin (t/2)--most probably won't even know the substitution rule. And clearly most are not advanced enough to know differential geometry! —Preceding unsigned comment added by 71.185.2.144 ( talk) 01:11, 22 January 2010 (UTC)
But why am I discussing this with you? Go ahead and leave this page as it was, with the claim that cycloids are not differentiable at 2n*pi. I'm moving the discussion to vandalpedia.org, where they appreciate the absurdity of wikipedia. —Preceding unsigned comment added by 71.185.2.144 ( talk) 11:02, 22 January 2010 (UTC)
- I see now what you are up to and I agree with you to some extent. But I'll deal with this thing of edit warring and using Wikipedia properly first. The relevant procedure about this discussion is Wikipedia:BOLD, revert, discuss cycle. You put in an edit. I reverted it and put in something in the talk page. You reverted again, I reverted again and put more on the talk page and edited your talk page. You reverted again and I reverted again and finally you followed what I said on your talk page. It was you who kept reverting without discussion. The whole point of the bold, revert, discuss cycle is to get people to go to the discussion page when they disagree. Calling a person a vandal is considered a grave insult on wikipedia and WP:personal attacks are very much frowned upon. Editors shuld concentrate on the material not the other editors. I'm not annoyed because I see you are new to wikipedia, please just put it down as a learning experience about working with others here rather than thinking you are entitled to put in your edits and calling a person who disagree with you vandal. However I can warn you that many editors here take a very dim view indeed of such remarks and they can get you banned. Civility and cooperation is considered more important than having editors who know about their subject but who go around insulting others. See WP:Five Pillars for a quick summary of the principles.
- If you do wish after to continue to contribute to wikipedia on mathematics can I point you also at Wikipedia talk:WikiProject Mathematics where editors discss general issues about maths articles and Wikipedia:Reference desk/Mathematics where people in general ask maths questions. You might find people more to your liking there than me but there's many editors a whole lot worse around and you'd have to cope with them too I'm afraid. I believe my dealing with some of them has probably made me a bit more abrupt and dismissive than I should be whereas I should try harder to find something good in edits.
- As to the particular edits you've done. I agree that the standard parameterization gives a differentiable curve and I misread the edits when I went through them originally. I think one would have to be a bit careful about that as I'm not sure a person would have come across the concept of a differentiable curve when they look up cycloids and even if so mny would have just read it as the differentiation of y to x. I'm happy with it in now though I'm wondering how to phrase it for people who've just done straightforward calculus so they're not misled. I think I'll add a link for differentiable curve to curve and try and point to the definition there, that a 'curve' here means a map won't be a concept even many first year undergraduates will have in their armory.
- You seem to assert that one only comes across cycloids as an example of parameterized curves. It has been around a long time before that and in fact has been parameterized a number of different ways. Roberval and Toricelli and Newton and Johann Bernoulli all used horizontal strips when investigating it, Wren used involutes of another cycloid to get equal time parameterization for the tautochrone, the Euler-Lagrange equations and the Beltrami identity just use x and y when using the calculus of variations, and recently I've seen Mamikon in visual calculus use tangent sweeps like the parameterization but that required no formal calculus or differentiation to find the area.
- As to your putting in a large number of steps in a derivation and then removing every single thing except the final result. That is I consider WP:POINTy. See Wikipedia talk:WikiProject Mathematics/Proofs for the current thinking about proofs. Wikipedia:Manual of Style (mathematics) is a more general guide about the style of maths articles. Dmcq ( talk) 13:46, 22 January 2010 (UTC)
hey I've seen the cycloid pendulum somewhere and i would like to give an illustration. http://commons.wikimedia.org/wiki/File:Cycloid_pendulum.png But I don't know how to use svg editors.
I hope someone can help to create an svg image or make a thumbnail of my image and then add it to the related part. —Preceding unsigned comment added by Kimkim0513 ( talk • contribs) 12:19, 20 February 2010 (UTC)
- That would be nice but that picture is completely wrong I'm afraid. The cycloidal jaws are of two cycloids side by side and the support point is at the cusp. +the length should be half the length of the cycloid curve. The path of the pendulum is another cycloid of the same size from the lowest point of the two cycloids. There should be some picture accessible from google somewhere. Dmcq ( talk) 13:30, 20 February 2010 (UTC)
- Oh...You pointed out the key facts! thanks for teaching me~ Kimkim0513 ( talk) 16:42, 22 February 2010 (UTC)
The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th-century mathematicians.
This is a teaser and the reference to a text book doesn't enlighten. What quarrels? Who called it Helen? And who, for that matter, is Helen (for those who don't know their Greek mythology)? — Preceding unsigned comment added by 92.25.9.36 ( talk) 07:31, 30 April 2013 (UTC)
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It seems like we could profitably cut several sentences about speculative attributions that were later shown to be wrong. This might be in scope for some deep dive or historiography study, but doesn't really seem that helpful to readers of this encyclopedia article. – jacobolus (t) 07:32, 1 April 2023 (UTC)
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There's a theorem that states that a tangent line to any point on the cycloid passes through a point, which can be defined in terms of t, located at the top of its corresponding rolling circle. I was told that this is Descartes' theorem, but it doesn't appear to be. Thus far, I've been unable to find any record of the theorem online. Has anyone heard of it? -- Ketsuekigata 19:06, 8 May 2007 (UTC)
Don't know whose theorem it is but it follows directly from that the tangent to the cycloid must be perpendicular to the point about which the circle is instantaneously turning - therefore you get a right angled triangle in the circle with one point at the bottom and the other at the top. This can be used via Visual Calculus to get the area of the cycloid easily. Dmcq 10:31, 24 June 2007 (UTC)
Are there any excamples of cycloids in nature or engineering?
- Every time something round rolls in a straight line, there's a cycloid. -- Ketsuekigata 20:54, 16 June 2007 (UTC)
When a charged particle is subject under orthogonal electric and magnetic field, its trajectory becomes a cycloid. HotBallah ( talk) 18:49, 14 February 2023 (UTC)
Quoting Article: The cycloid through the origin, generated by a circle of radius r, consists of the points (x, y) with
x = r(t - \sin t)\,
y = r(1 - \cos t)\,
where t is a real parameter, corresponding to the angle through which the rolling circle has rotated, measured in radians (not degrees). For given t, the circle's centre lies at x = rt, y = r.
--- It would seem that you could use degrees, as long as you made 0 < t < 360. The statement that t is measured in degrees is dubious, unless there is a statement saying 0 < t < 2*pi.
Am I missing something? jheiv ( talk) 16:37, 5 May 2009 (UTC)
- I edited the article to correct what I thought was wrong / confusing. jheiv ( talk) 16:41, 5 May 2009 (UTC)
- Yes, you're missing something. t has to be in radians, irrespective of any bounds. The bounds you've given just mean that you get one arch of the cycloid, which is not unreasonable I guess. —Preceding unsigned comment added by 81.152.169.12 ( talk) 20:58, 11 May 2009 (UTC)
--Im sorry if im not writing this right, its new to me writing in wiki. the equation that written there for x by not using t isn't right, the right form of the equation is: -sqrt((2 y)/a-y^2/a^2)+cos^(-1)(1-y/a) while a=r. fix it please, I almost got wrong in work for my proffesor. — Preceding unsigned comment added by 85.250.64.229 ( talk) 16:06, 7 June 2012 (UTC)
An editor put in an edit comment saying it was differentiable everywhere, see any textbook. Could they find a citation first please for things like that which manifestly contradict what one can see. Dmcq ( talk)
- That the curves for some parameterization are differentiable is not as far as I know a common documented property of curves. Differentiable is dy/dx and what you're saying will confuse people. And the bit about the derivative being 0 is only for th parameterization and is why the dy/dx is infinite, it would have been perfectly possible for both derivatives in the parameterizations to be zero. Also it is not standard in Wikipedia to give long derivations, only the major necessary points. You might be interested in Wikipedia:Reference_desk/Mathematics for asking general maths questions and Wikipedia talk:WikiProject Mathematics where people interested in improving maths articles discuss issues. Dmcq ( talk) 17:56, 21 January 2010 (UTC)
Oh, sorry, I didn't see this comment. Cycloids are only ever mentioned in math books in relation to parametrized curves. They are in a sense a "classic" parametrized curve. Textbooks never even mention the Cartesian equation because it's unnecessarily complicated. "Regular" curves and "singular points" are elementary properties of differential geometry. See, e.g., Manfredo do Carmo, "Differential Geometry of Curves and Surfaces." (Cycloids are mentioned on page 7, where it says they have singular points at 2pi.)
As for derivations, I did only include the important points. No normal reader of wikipedia is going to know the trig identity that gets you to sin (t/2)--most probably won't even know the substitution rule. And clearly most are not advanced enough to know differential geometry! —Preceding unsigned comment added by 71.185.2.144 ( talk) 01:11, 22 January 2010 (UTC)
But why am I discussing this with you? Go ahead and leave this page as it was, with the claim that cycloids are not differentiable at 2n*pi. I'm moving the discussion to vandalpedia.org, where they appreciate the absurdity of wikipedia. —Preceding unsigned comment added by 71.185.2.144 ( talk) 11:02, 22 January 2010 (UTC)
- I see now what you are up to and I agree with you to some extent. But I'll deal with this thing of edit warring and using Wikipedia properly first. The relevant procedure about this discussion is Wikipedia:BOLD, revert, discuss cycle. You put in an edit. I reverted it and put in something in the talk page. You reverted again, I reverted again and put more on the talk page and edited your talk page. You reverted again and I reverted again and finally you followed what I said on your talk page. It was you who kept reverting without discussion. The whole point of the bold, revert, discuss cycle is to get people to go to the discussion page when they disagree. Calling a person a vandal is considered a grave insult on wikipedia and WP:personal attacks are very much frowned upon. Editors shuld concentrate on the material not the other editors. I'm not annoyed because I see you are new to wikipedia, please just put it down as a learning experience about working with others here rather than thinking you are entitled to put in your edits and calling a person who disagree with you vandal. However I can warn you that many editors here take a very dim view indeed of such remarks and they can get you banned. Civility and cooperation is considered more important than having editors who know about their subject but who go around insulting others. See WP:Five Pillars for a quick summary of the principles.
- If you do wish after to continue to contribute to wikipedia on mathematics can I point you also at Wikipedia talk:WikiProject Mathematics where editors discss general issues about maths articles and Wikipedia:Reference desk/Mathematics where people in general ask maths questions. You might find people more to your liking there than me but there's many editors a whole lot worse around and you'd have to cope with them too I'm afraid. I believe my dealing with some of them has probably made me a bit more abrupt and dismissive than I should be whereas I should try harder to find something good in edits.
- As to the particular edits you've done. I agree that the standard parameterization gives a differentiable curve and I misread the edits when I went through them originally. I think one would have to be a bit careful about that as I'm not sure a person would have come across the concept of a differentiable curve when they look up cycloids and even if so mny would have just read it as the differentiation of y to x. I'm happy with it in now though I'm wondering how to phrase it for people who've just done straightforward calculus so they're not misled. I think I'll add a link for differentiable curve to curve and try and point to the definition there, that a 'curve' here means a map won't be a concept even many first year undergraduates will have in their armory.
- You seem to assert that one only comes across cycloids as an example of parameterized curves. It has been around a long time before that and in fact has been parameterized a number of different ways. Roberval and Toricelli and Newton and Johann Bernoulli all used horizontal strips when investigating it, Wren used involutes of another cycloid to get equal time parameterization for the tautochrone, the Euler-Lagrange equations and the Beltrami identity just use x and y when using the calculus of variations, and recently I've seen Mamikon in visual calculus use tangent sweeps like the parameterization but that required no formal calculus or differentiation to find the area.
- As to your putting in a large number of steps in a derivation and then removing every single thing except the final result. That is I consider WP:POINTy. See Wikipedia talk:WikiProject Mathematics/Proofs for the current thinking about proofs. Wikipedia:Manual of Style (mathematics) is a more general guide about the style of maths articles. Dmcq ( talk) 13:46, 22 January 2010 (UTC)
hey I've seen the cycloid pendulum somewhere and i would like to give an illustration. http://commons.wikimedia.org/wiki/File:Cycloid_pendulum.png But I don't know how to use svg editors.
I hope someone can help to create an svg image or make a thumbnail of my image and then add it to the related part. —Preceding unsigned comment added by Kimkim0513 ( talk • contribs) 12:19, 20 February 2010 (UTC)
- That would be nice but that picture is completely wrong I'm afraid. The cycloidal jaws are of two cycloids side by side and the support point is at the cusp. +the length should be half the length of the cycloid curve. The path of the pendulum is another cycloid of the same size from the lowest point of the two cycloids. There should be some picture accessible from google somewhere. Dmcq ( talk) 13:30, 20 February 2010 (UTC)
- Oh...You pointed out the key facts! thanks for teaching me~ Kimkim0513 ( talk) 16:42, 22 February 2010 (UTC)
The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th-century mathematicians.
This is a teaser and the reference to a text book doesn't enlighten. What quarrels? Who called it Helen? And who, for that matter, is Helen (for those who don't know their Greek mythology)? — Preceding unsigned comment added by 92.25.9.36 ( talk) 07:31, 30 April 2013 (UTC)
Hello fellow Wikipedians,
I have just modified one external link on Cycloid. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
- Added archive https://web.archive.org/web/20060127232450/http://historical.library.cornell.edu/math/index.html to http://historical.library.cornell.edu/math/index.html
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Cheers.— InternetArchiveBot ( Report bug) 00:19, 16 August 2017 (UTC)
It seems like we could profitably cut several sentences about speculative attributions that were later shown to be wrong. This might be in scope for some deep dive or historiography study, but doesn't really seem that helpful to readers of this encyclopedia article. – jacobolus (t) 07:32, 1 April 2023 (UTC)