Checking whether a point lies on a wide line segment I know how to define a point on the segment, but I have this piece is a wide interval. The line has a width.
I have $x_1$ $y_1$, $x_2$ $y_2$, width and $x_3$ $y_3$
$x_3$ and $x_4$ what you need to check.
perhaps someone can help, and function in $\Bbb{C}$ #

Assuming @Henry's picture summarizes the question asked, for a given thickness $T$ a necessary and sufficient condition is given by the following inequalities:
$$
0\le\overrightarrow{AC}\cdot\overrightarrow{AB}\le AB^2,\qquad\qquad
AB^2AC^2\le T^2AB^2+\left(\overrightarrow{AC}\cdot\overrightarrow{AB}\right)^2.
$$
The first condition ensures that the projection of $C$ on the line $(AB)$ lies between $A$ and $B$ and the second condition ensures that the distance between $C$ and the line $(AB)$ is at most $T$.
To prove this, note that $\overrightarrow{AC}$ must be $\overrightarrow{AC}=u\overrightarrow{AB}+t\overrightarrow{N}$ with $0\le u\le 1$, $t^2\le T^2$ and $\overrightarrow{N}$ a unitary vector orthogonal to $\overrightarrow{AB}$ and try to express the conditions on $u$ and $t$ in terms of $\overrightarrow{AC}$, $\overrightarrow{AB}$, their norms $AC$ and $AB$, and their scalar product $\overrightarrow{AC}\cdot\overrightarrow{AB}$ only.
