# The prolate cycloid

A cycloid is given by the parametric equations: $$x = 2 - \pi \cos(t)$$ and $$y = 2t - \pi \sin(t)$$.

The problem asks for the slope of the tangents on the cycloid at a point where the cycloid intersects itself. That point is not given, but it lies on the x - axis.

I wanted to find that point by cancelling the parameter, $$t$$, but I couldn't come up with important elimination. Is there such way of solving the problem ?

At $$t$$ and $$t'$$ the coordinates repeat: $$x(t)=x(t')\land y(t)=y(t')\implies2t-\pi\sin t=2t'-\pi\sin t'\land\cos t=\cos t'$$

$$\cos t=\cos t'\implies t'=t+k2\pi;\in\mathbb Z-\{0\}\lor (t'=-t+k'2\pi,t\neq n\pi);k,n\in\mathbb Z$$

a) $$2t-\pi\sin t=2(t+k2\pi)-\pi\sin (t+k2\pi)$$

$$\sin(t+k2\pi)-\sin t=4k$$, no solutions.

b) $$2t-\pi\sin t=2(-t+k'2\pi)-\pi\sin (-t+k'2\pi)$$

$$4t=\pi(4k'+\sin t-\sin(-t+k'2\pi))$$

$$4t=\pi(4k'+\sin t+\sin t)$$

$$t/\pi=k'+(1/2)\sin t$$

Having as a valid solution (Wolfram Alpha) $$t=\pi/2,t'=-\pi/2$$ with $$k'=0$$ Only this value is needed as the function is periodic and th tangents have the same slope at the other intersection points.

Now the slope at $$t$$ is $$\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}=\dfrac{2-\pi\cos t}{\pi\sin t}$$

And at the intersection $$\left.\dfrac{dy}{dx}\right|_{t=\pi/2}=2/\pi;\left.\dfrac{dy}{dx}\right|_{t=-\pi/2}=-2/\pi$$

• Thanks,but If we end up with an equation with structure $t = a + bsin(t)$, just like you did before the link to 'wolfram alpha' what about just $2t - \pi sin(t) = 0$ and hence $2t = \pi sin (t)$, at the intersection point... I mean why all those steps, if we used Wolfram to solve the equation. Appreciate your answer by the way. May 5, 2019 at 4:40
• With those steps we know why $y=0$. If you don't need this, you can use WA directly for that transcendental equation, of course. May 5, 2019 at 5:32
• Thanks that was of a great help. But Is it possible to solve it without wolfram alpha ? May 5, 2019 at 6:29
• Yes, by approximate methods as the equation is a trascendental one (e.g. here). Nevertheless, considering the nice the solution is, maybe it is meant you have to get the solution by guessing. May 5, 2019 at 7:12