Definition: A real smooth analytic function $f$ defined on some neighborhood of $t_0$ is said to have an $A_k$ singularity at $t_0$ if and only if $$ f'(t_0) = f''(t_0) = \dots = f^{(k)}(t_0) = 0$$ and $f^{(k+1)}(t_0)\neq0$
Definition: A smooth plane curve $\gamma : I \to \mathbb{R}^2$ is said to have a contact of order $k$ with the implicit curve $F(x, y)=0$ at $\gamma(t_0)$ if and only if $F(\gamma(t_0))=0$ (i.e. they intersect) and $F(\gamma(t))$ has an $A_{k-1}$ singularity at $t_0$
In particular, given $\gamma$ and $t_0 \in I$, the line (resp. circle) with the highest order of contact with $\gamma$ at $t_0$ is the tangent line (resp. osculating circle) at $\gamma(t_0)$.
The order of contact at the points of the curve with tangents lines/osculating circles at that point can give insights about the local geometrical behavior at that point. For instance, at ordinary vertexes of the curve (i.e. $\kappa'=0$ and $\kappa'' \neq 0$), the order of contact with the osculating circle is $4$ (whilst at points with $\kappa' \neq 0$, it is $3$). Also, the contact of a line and $\gamma$ at an ordinary inflexion (i.e. the point $(0,0)$ of $\gamma(t)=(t,t^3)$) is $3$, whilst at a higher inflexion, it could be $5$ (i.e. the same point of $\gamma(t)=(t, t^5)$). It could also be $4$ at the same point of $\gamma(t)=(t, t^4)$, even though these curves have distinct local geometries.
So the order of contact with lines and circles reveals geometric information about the local behavior of the curve at that point.
We also know that some interesting geometric phenomenon occur at some singularities. For instance, if we consider the parallel curves of an ellipse, its four ordinary vertexes "are responsible" for giving rise to swallowtail singularities. Also, if we think of the evolute, the vertexes of the ellipse "are responsible" for giving rise to the four cuspids of the evolute.
In conclusion, singularities at smooth plane curves give information about its local geometry and are clearly visible at some curves originated from the original curve, like swallowtails at parallels and cuspids at evolutes.
With all that said, my question is the following:
Is it possible to "do something" with the original curve $\gamma$ (i.e. create a curve originated from the original curve) which gives a intuitive and clear geometric reason for the curve having an order of contact $k$ with lines or circles for all $k$? Namely, how do we know the difference between, say, the inflexions (local behavior at $t=0$) of $\gamma(t)=(t, t^5)$ and $\gamma(t)=(t, t^7)$ just by "looking" at $\gamma$ itself near $t=0$ or some related curve? Can curves created from these be geometrically different (visually) because of the different behavior at $t=0$? You could say that for this case, when we consider the "proximity" to the line $y=0$, $t^7$ is closer to it than $t^5$ (near $t=0$). But that is generally not immediately clear just by looking at curves the same way as swallowtails at parallels and cuspids at evolutes clearly indicate the existence of a point in which $\kappa'=0$ at the original curve.
