Let $n \ge 3$ and $x(t) \in \mathbb{R}^n$ be a vector-valued function of $t \in I = [0,M]$. Suppose $x_1(t) = [t,\cdots,t]^T = \textbf{1}t$ which is a line, a 1-dimensional manifold. Then $$x_1(I) = \{x(t)|t \in I\}$$ can be viewed as a subset in 1-dimension $\mathbb{R}$. Similarly, if $$x_2(t) = [1,1,1,0,\cdots,0]^T\sqrt{1-(t/M)^2} + [0,0,0,1,\cdots,1]^T(t/M),$$ $x_2(I)$ can be viewed as a half circle, as a subset in 2-dimenision $\mathbb{R}^2$.
Now let consider $$ x_3(t) = [\cos(t-\theta_1), \cos(t-\theta_2), \cdots, \cos(t-\theta_{n})]^T. $$ In this case, it is unclear that what dimensional object $x_3(I)$ is.
I am wondering if there is a mathematical notion or terminology which describing the dimensionality of $x(I)$ in general (e.g. $x_3(I)$). But I don't want to (even continuously) deform $x(I)$ to identify the dimensionality. If I allow any deformation on $x(I)$, I believe $x_2(I)$ and $x_3(I)$ become a 1-dimensional "manifold" as they only depend on a single variable $t$.
I can see that $x_1(I)$ is isomorphic to $I$ (as a set) and $x_2(I)$ is isomorphic to the half circle $\{(x,y) | x^2 + y^2 = 1, 0 \le x, y\}$ (as a set). It seems that these might be related to some isomorphic properties, however, I am not sure.
Any comments/suggestions/answers will be very appreciated.