Some Context:

I'm working with some data given in the form of Bezier curves. I need to sort these (partially ordered) Bezier curves by "outwardness" (described below) and have come across an interesting problem. Such an interesting problem, that I'm confident others have worked on the same problem. Due to the limited complexity of the problem (only considering simple cubics) I'm guessing a solution is known too.

Terminology/Notation/Assumptions:

Let $c:(0,1)\to\mathbb{C}$ be a cubic polynomial with complex coefficients. Further, assume this curve is simple, oriented and let the outward (unit) normal vector to $c$ at the point $t$ be denoted by $n_c:(0,1)\to\mathbb{C}$.

For my purposes, we can also assume that $c$ has a weak form of convexity in that it is not "outward of" itself (according to the following definition).

Below I'll refer to a point, $z\in\mathbb{C}$, as being *outward of the curve* $c$ if there is some $s>0$ and $t\in (0,1)$ such that $z=sn_c(t)+c(t)$. For simplicity (and with some abuse of common notation), I'll let $\mathcal{N}^+_c$ denote the region of all points in $\mathbb{C}$ that are outward of $c$.

The Goal:

I want a simple way to check whether one such curve (like that described by $c$ above) is outward of another such curve. In other words, a curve $c_1$ and outward of another curve $c_2$ if and only if $\text{Image}(c_1)\cap \mathcal{N}^+_{c_2}\neq\emptyset$.

Note that in the data set I'm working with, it can be assumed that $\text{Image}(c_1)\cap\mathcal{N}^+_{c_2}\neq\emptyset \:\Rightarrow\: \text{Image}(c_2)\cap \mathcal{N}^+_{c_1}\neq\emptyset$ (which is equivalent to the condition that a set of such curves admits a partial ordering w.r.t. "outwardness").

Current Progress (and **the interesting problem**):

Currently I'm using generating $N$ outward normal lines and testing for intersections, but this a very slow. Trying to speed it up, I thought maybe I could explicitly solve for the curves that define the boundary of $\mathcal{N}^+_c$. This is the interesting part... what are these curves?

I've included some pictures below (the curve, $c$, is in black). You can see sometimes these boundary curves of $\mathcal{N}^+_c$ are (piecewise) linear, and other times they are curved (depending on the curvature of $c$).

Ideas:

1)I could do something really adhoc (like doing a floodfill of the outer white space in the example images and fitting a cubic spline to the boundary), but that seems messy.

2)I could look at the family of curves offset from (parallel to) $c$ at a distance of $s$ and find values of $s$ and $t$ at which self-intersections occur... which sounds like fun, but a little outside my wheelhouse so I thought I'd ask on here before investigating that path further.

Anyone know how to do this? The ideal solution would be a function that takes in the coefficients of $c$ and outputs (some explicit parameterization of) the two boundary curves that $\partial\mathcal{N}^+_c$ is the union of.

Note: In the examples below, $c$ is black (the red curve is parallel curve $sn_c(t)+c(t)$ at some fixed $s$ I chose for display purposes).