Consider the unit circle centered at the origin minus the north pole in $\mathbb{R}^{2}$, denoted by $C$, and the $x$-axis, denoted by $\mathbb{R}$.
In this question, the accepted answer shows an isotopy $E$ from the inclusion $i$ to the stereographic projection $f$. This isotopy can be described visually by starting with $C$ at $t=0$ and then as soon as $t>0$ it immediately explodes out to infinity left and right of the North pole. At the same time it is being compressed, and completely flattened onto the $x$-axis as we arrive at $t=1$. Note the two fixed points $(\pm 1,0)$ and the horizontal asymptote at $(0,1-t)$ for $t>0$. Also note how the isotopy produces unbounded arcs but for $t=0$ (see the picture below for $t=2/5$).
We can define a different isotopy from $i$ to $f$ by $E'(x,y,t)=((1-t)x+\tan(t\arctan(x/1-y)),(1-t)y)$. In this case, we are opening out $C$ left and right of the North pole while simultaneously stretching it and flattening it onto the $x$-axis, where we arrive at $t=1$. However, this time our isotopy does not have fixed points and produces bounded arcs for all $t<1$ (see the picture below for $t=4/5$).
In summary, we have two isotopies from $i$ to $f$. On the one hand $E$ produces all unbounded arcs but one and has fixed points, while $E'$ produces all bounded arcs but one and has no fixed points. Do features like these may in any way allow us to group/classify the set of isotopies between two given embeddings? Is there a general theory for this in the different categories (TOP, DIFF, PL)?
Following on this, in the related question, the embedding $i$ remains the inclusion, but then an embedding $f$ is defined, which I will call $f_{1}$ here. The corresponding isotopy from $i$ to $f_{1}$ is given by $E_{1}(x,y,t)=(1-t)i(x,y)+tf_{1}(x,y)$. Visually here we are opening up $C$ left and right of the North pole, without much stretching, while simultaneously compressing it until it lies flattened onto the interval $(-\pi/2,\pi/2)\times\{0\}$ when we arrive at $t=1$. Note how the process produces bounded arcs for all $t$ this time. Moreover, the isotopy has no fixed points (see the picture below for $t=1/2$).
Now, one feature that appears distinct for this $f_{1}$ is that when working with the one-point compactification of $\mathbb{R}^{2}$, denoted by $S^{2}$, $i$ and $f$ appear to be ambient isotopic embeddings (apply the appropriate rotations of $S^{2}$), but not so $i$ and $f_{1}$. That is, and $i$ and $f_{1}$ are not ambient isotopic embeddings in $S^{2}$.
Thus, we have two embeddings $f$ and $f_{1}$ both isotopic to $i$ in $S^{2}$, but only $f$ is ambient isotopic to $i$ in $S^{2}$. Do properties such as this type of arguably desirable attribute of $f$ of becoming ambient isotopic to $i$ serve as a way to classify isotopies between embeddings? For example, $E$ has two fixed points, and in $S^{2}$ there are precisely two points (the ones missing from $C$ and $\mathbb{R}$) which any ambient homeomorphism of $S^{2}$ carrying $C$ to $\mathbb{R}$ must map to each other. Is this simply a coincidence? What I am trying to understand from the examples above is if there is a theory to classify isotopies between embeddings, and what properties to look for that would help select a particularly appropriate representative of an isotopy class.
*Graphs made with the parametric equation plotter https://www.geogebra.org/m/cAsHbXEU