A line segment in $\mathbb{R}^2$ is defined as $\mathbb{L} = Conv\{\textbf{p}_1,\textbf{p}_2\}$. Obviously, its area is 0. Given a time-parameterized continuous trajectory $\textbf{T}(t)$ in $\mathbb{R}^2 \times SO1$, which contains position trajectory $\textbf{P}(t)$ and attitude trajectory $\psi(t)$. $\mathbb{L} (t) = \textbf{P}(t) + \psi(t)\mathbb{L}$. Define a circle $\textbf{C} = \{ \textbf{p} \ | \ ||\textbf{p} - \textbf{o}||^2 \leq r , \textbf{p}, \textbf{o} \in \mathbb{R}^2 \}$. How to calculate the area of the following set: $$ \{ (\cup \mathbb{L} (t) \ | \ t \in [t_{a} , t_{b}] ) \cap \textbf{C} \}$$
Here is an example figure. The green line segment $\mathbb{L}$ move along a trajectory and the orange part is the target set.
