Reparameterizing a curve in path length parameter

[Marioweee]

Homework Statement

See below

Relevant Equations

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Given the parameterization of an inverted cycloid:
$$x(t)=r(t-\sin t)$$
$$y(t)=r(1+\cos t)$$
where $$t \in [0, 2\pi]$$.
I am asked to parameterize the curve in its natural parameter. To do it:
$$s=\int_{t_0}^{t} ||\vec{x}'(t*)||dt*$$
The modulus of the squared velocity is:
$$||\vec{x}'(t*)||=2r\sin(t/2)$$
Therefore, the integral is:
$$s=\int_{t_0}^{t} 2r\sin(t/2)dt*=-4r\cos(t/2)|_{t_0}^{t}=-4r(cos(t/2)-cos(t_0/2))$$
My doubt is, I can take any arbitrary value for the parameter $$t_0$$ for example $$t_0=\pi$$ which would simplify the expression or does it have to be $$t_0=0$$ since the parameter $$t$$ starts on 0.
Thank you very much for all the help.

 

[Orodruin]

Both will be parametrizations of the curve in a parameter that is path length along the curve. You will get a different interval for the parameter depending on your choice though as they will be path lengths from different points on the curve.

The $t_0 = 0$ choice does not seem half bad to me either to be honest. Note that $\cos(0) - \cos(t/2) = 1 - \cos(t/2) = 2 \frac{1-\cos(t/2)}2 = 2 \sin^2(t/4)$.