Parametric equation for a cycloid

[yuiop]

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 :)

 

[HallsofIvy]

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?

 

[yuiop]

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?"

 

[yuiop]

Can the Lambert W function help here?