Relationship between tangent normal of parametric curve and range of parameter

Question

It is intuitive that by the right-hand rule one can choose direction of motion along a parametric curve by selecting one of the unit normal vectors. However, the relationship to the parameter is less clear. Is the direction defined by the range of values of the parameter?

Answer 1

I'm not sure what you mean. For a parametric, regular (tangent vector is never $0$) and infinitely differentiable curve $\alpha: I \subset \mathbb{R} \to \mathbb{R}^2$, at each $t \in I$ one usually defines the unit tangent vector as $T(t) = \dfrac{\alpha'(t)}{\vert \vert \alpha'(t) \vert \vert} = (f'(t), g'(t))$. Out of convenience, the unit normal vector is $N(t) = (-g'(t), f'(t))$ (the choice is made like this so that $T$ and $N$ form a positively oriented orthonormal basis - also notice that this just means to take $T$ and rotate it $\dfrac{\pi}{2}$ counter clockwise). The direction of the normal vector at each $t \in I$ is indeed dependent on the parameter $t$.