Parametric equation for a cycloid

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Discussion Overview

The discussion revolves around the challenges of reversing the parametric equations of a cycloid, specifically the equation x=R(θ - sin(θ)) to express θ in terms of x. Participants also explore the implications of this reversal in the context of finding intersections between the cycloid locus and the perimeter of an ellipse.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses difficulty in reversing the equation x=R(θ - sin(θ)) to solve for θ, questioning whether it is impossible or if they are missing something.
  • Another participant states that there is generally no formula for solving equations where the unknown appears both inside and outside a transcendental function, suggesting numerical solutions as a potential approach.
  • A clarification is made regarding the non-parametric equation for the cycloid, correcting it to x = ±cos⁻¹((R-y)/R) ± √(2Ry - y²), which is derived from substituting for θ in the original parametric equation.
  • A later reply introduces the possibility of using the Lambert W function to assist in the problem, though the effectiveness of this approach is not evaluated.

Areas of Agreement / Disagreement

Participants generally agree on the complexity of reversing the equations, with some expressing skepticism about the existence of a straightforward solution. However, there is no consensus on the applicability of the Lambert W function or other methods to resolve the issue.

Contextual Notes

Participants note the challenges posed by the transcendental nature of the equations involved, as well as the potential need for numerical solutions. The discussion highlights the complexity of the relationships between the cycloid and other geometric figures, such as ellipses.

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Hi, I am having trouble reversing the formula x=R(\theta - \sin(\theta)) to get \theta in terms of x. Am I missing something obvious or is it just impossible?

To put it into context this is part of the parametric equation for a cycloid. The other part of the parametric equation is y = R (1- \cos(\theta)). Setting R to 1 (the radius of the rolling wheel) does not seem to help. The non parametric equation for the cycloid is \pm \cos^{-1}((R-y)/R) \pm \sqrt{2 R y -y^2}. I would also like to reverse this full equation to get y in terms of x but I am having trouble with that too. The reason I am trying to reverse the equations is that I am trying to get the intersection of two loci (the cycloid locus and the perimeter of a ellipse).

Any help appreciated. Thanks :)
 
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In general, there is no "formula" for solving an equation in which the unknown number occurs both inside a transcendental function (such as cosine) and outside it. typically, the best that can be done is a numerical solution.

I ahve no idea what you mean by "The non parametric equation for the cycloid is \pm \cos^{-1}((R-y)/R) \pm \sqrt{2 R y -y^2}" because that is not an equation. Did you leave something out?
 
HallsofIvy said:
In general, there is no "formula" for solving an equation in which the unknown number occurs both inside a transcendental function (such as cosine) and outside it. typically, the best that can be done is a numerical solution.

I ahve no idea what you mean by "The non parametric equation for the cycloid is \pm \cos^{-1}((R-y)/R) \pm \sqrt{2 R y -y^2}" because that is not an equation. Did you leave something out?

Yes I did! That should read "The non parametric equation for the cycloid is x = \pm \cos^{-1}((R-y)/R) \pm \sqrt{2 R y -y^2}" which is obtained by substituting \pm \cos^{-1}((R-y)/R) for \theta in x=R(\theta - \sin(\theta)).


It seems ridiculous that there is no easy solution to the question "If the point on the perimeter of a wheel has advanced linearly by x then what angle has the wheel rotated through?"
 
Last edited:
Can the Lambert W function help here?
 

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