As a follow-up to my previous question about non-intersecting curves with the pleasantly simple answer "thickness times arc length", the next question is:
Given a curve $\gamma:\mathbb R\to\mathbb R^2,\ s\mapsto \vec\gamma(s)$ (parametrized in arc length such that $|\vec\gamma'(s)|\equiv 1$) of thickness $t$, what is the area covered by it for $s\in[a,b]$ if the thick curve has self-intersections, i.e. curvature $\kappa > \tfrac 2t$, at some points?
Is there a simpler way to calculate this than "manually" obtaining the intersection surfaces to subtract?
I do not believe thta you will find a general solution to the area of a thick line that crosses itself except under the most trivial conditions (see the zeroth-order solution below). In the first place, there may be multiple crossings, and in the second place, the thick lines themselves may overlap as in the first figure below.
I proffer two approximate solutions and a stochastic method that can approach an exact solution.
My first inclination is a zeroth-order solution wherein it is assumed that (a) the line is relatively thin, and (b) the crossings are at, or nearly, normal. In that case,
$$ a=s\cdot\delta-n\delta^2 $$
where $s$ is nominal line length, $\delta$ is the line thickness, and $n$ is the number of crossings.
As we relax those conditions, and allow that the crossings are skew, but the thickness is still sufficiently small that the crossings are nearly parallelograms, then we can say that area is
$$ a=s\cdot\delta-\delta^2\sum_{k=1}^n\frac{1}{\cos\big(\frac{\pi}{2}-\alpha_k\big)} $$
where $\alpha_k$ is the acute angle between the crossing lines.
Anything beyond that may require extensive analysis and/or computation. I've selected an epitrochoid consisting of five internal, but non-overlapping loops. The equation for the epitrochoid is
$$ z=me^{it}-he^{i\frac{m}{b}t},\quad t\in[-\pi,\pi] $$
The upper and lower boundaries of the thick line are parallel to the curve (i.e, equidistant on the normal) as follows,
$$ z_{u,l}=\pm i\frac{\dot z}{|\dot z|}\frac{\delta}{2} $$
I used a Monte Carlo simulation in which we throw random positions into a rectangle containing the thick line and keep count of those falling inside and outside of that line. The area of the line is then the area of the rectangle times the fraction of points falling within the line. The second figure below shows the first 10,000 of eight million throws. The epitrochoid parameters for this case of $m=12$, $h=7$, $b=2$, and the thickness $\delta=2$. (Incidentally, the first figure is the same epitrochoid, but $\delta=4$.)
The Monte Carlo simulation for this case gave an area $A\approx 514.72$ compared withe the first-order approximation $a\approx 515.01$. The difference here is less than 0.1%.
The Monte Carlo simulation (again, eight million throws) took about 370 seconds on a 16-cluster parallel processor. This is a 12-fold speed improvement over normal processing. In other words, you don't need a supercomputer to do these calculations.