tangent

The unit tangent vector can be computed from the first derivative of the parametric equation: $\vec{T}=P'(t)/\vert\vert P'(t)\vert\vert$, which means its coordinates are:

$$\vec{T} \left\{\begin{aligned}\frac{x'}{\sqrt{x'^2+y'^2}}\\ \frac{y'}{\sqrt{x'^2+y'^2}} \end{aligned}\right.$$

This unit tangent vector is an intrinsic property of the curve, it is independant of the parametric equations used as long as they define the same orientation. If two different orientations are used, they will define opposite unit tangent vectors.