Suppose I have the $x$ axis and some function $f(x)$. If I want to visualize them in $\mathbb{R}^2$, for instance, it would be easy - I would set $y=f(x)$. But what if I want to visualize them in $\mathbb{R}^3$? For that, of course, I would need to choose some plane for them to lay in. If if choose, for example, the $XZ$ plane, then suitable parametrizations would be:
$$\Gamma_x(t)=(t,0,0)\\ \Gamma_f(t)=(t,0,f(t))$$
Now, let's say that the $x$ axis goes through some transformation $T$ and becomes a new curve in the $3D$ space. I will denote it $T(\Gamma_x)$. Good thing - I have a parametrization for $T(\Gamma_x)$. Bad thing - I have no idea what $T$ is. My goal is to find $T(\Gamma_f)$ with a suitable plane.
I will demonstrate what I am trying to do with an example. Let's say that $f(x)=\sin x$ and that $T(\Gamma_x)$ is given by:
$$T(\Gamma_x)(t)=(t,0,t)$$
There is a bunch of transformations that could transform $\Gamma_x$ to $T(\Gamma_x)$. An example that I thought of is a $45^\circ$ clockwise rotation around the $y$ axis. This transformation is naturally represented with a rotation matrix. Applying it to $\Gamma_f(t)=(t,0,\sin t)$ one receives:
$$T(\Gamma_f)(t)=(t-\sin t,0,t+\sin t)$$
Notice that the choice of $T$ is also a choice of the plane - this specific $T$ comes with the plane $XZ$. I could easily rotate $T(\Gamma_f)$ $90^\circ$ degrees around $T(\Gamma_x)$ and I would get a different curve, now laying in some other plane, but it is still "wrapped" around $T(\Gamma_x)$ like a sine wave would. The choice of $90^\circ$ is also, of course, arbitrary.
But what if $T$ is more complicated? For example, what if $T(\Gamma_x)$ is the unit circle laying in the $XY$ plane? What would $T(\Gamma_f)$ be then? How do I "wrap" the sine function around a circle? And for the general question - What if $T(\Gamma_x)$ is some random $3D$ curve, and I want to find a parametrization for $T(\Gamma_f)$ such that it is "wrapped" around $T(\Gamma_x)$?
(In the picture - The function $f(x)=\sin(3x)+1$ wrapped around the unit circle in the $XY$ plane)
I'm sorry if I didn't explain myself well enough, but I hope some understand my intentions. Thanks!