particle moving on the astroid at constant frequency

In parametric Cartesian equations, the astroid can be represented by

$$x=a{\cos }^3\omega t,y=a{\sin }^3\omega t,$$

where $a>0$ is a known constant, $\omega >0$ is the constant angular frequency, and $t\in [0,\mathrm{\infty })$ is the time parameter. Thus the position vector of a particle, moving over the astroid, is

$$𝐫=a{\cos }^3\omega t𝐢+a{\sin }^3\omega t𝐣,$$

and its velocity

$$𝐯=-3a\omega \sin \omega t{\cos }^2\omega t𝐢+3a\omega {\sin }^2\omega t\cos \omega t𝐣,$$

where $\{𝐢,𝐣\}$ is a reference basis. Hence for the particle speed we have

$$v=3a\omega \sin \omega t\cos \omega t.$$

From the last two equations we get the tangent vector

$$𝐓=-\sin \omega t𝐢+\cos \omega t𝐣,$$

and by using the well known formula

$$\parallel \frac{d𝐓}{dt}\parallel =\frac{v}{\rho },$$

$\rho >0$ being the radius of curvature at any instant $t$, we arrive to the useful equation

$$v=\omega \rho .$$