An alternative method. Assume $PQ$ and $RS$ are the line segments. Let the direction cosines
of the vectors $\mathbf{u=}
\overrightarrow{PQ}$ and $\mathbf{v=}\overrightarrow{RS}$ be, respectively, $
\alpha _{u},\beta _{u},\gamma _{u}$ and $\alpha _{v},\beta _{v},\gamma _{v}$. The angle $\phi $ between the line segments is such that$^{1}$
$$
\begin{equation*}
\cos \phi =\alpha _{u}\alpha _{v}+\beta _{u}\beta _{v}+\gamma _{u}\gamma
_{v}.
\end{equation*}
$$
Hence the line segments are collinear if $\cos \phi =\pm 1$.
--
$^{1}$Formula 10.7 of Manual de Fórmulas e
Tabelas Matemáticas, Coleção Schaum, Portuguese translation of Schaum's Outline Series
Mathematical Handbook of Formulas and tables, 2/e by Murray Spiegel and John
Liu.
Two segments $p_1p_2$ and $p_3p_4$ are intersect iff
1) Their rectangles intersects, which can be written as
$\max(p_{1x},p_{2x})\geq \min(p_{3x},p_{4x})$ and
$\max(p_{3x},p_{4x})\geq \min(p_{1x},p_{2x})$ and
$\max(p_{1y},p_{2y})\geq \min(p_{3y},p_{4y})$ and
$\max(p_{3y},p_{4y})\geq \min(p_{1y},p_{2y})$.
2) $\langle[p_3-p_1,p_2-p_1],[p_4-p_1,p_2-p_1]\rangle\leq 0$
3) $\langle[p_1-p_3,p_4-p_3],[p_2-p_3,p_4-p_3]\rangle\leq 0$
see Introduction to Algorithms T. Cormen section 35 Computational geometry.
Best Answer
The name of your "inner path" is offset curve or parallel curve. See the earlier MSE question, Self-intersection removal in offset curves. To add to Jaap Scherphuis' comment, the offset curve may self-intersect. In general they are not easy to calculate.
Image from Paul Murrell, "Offset Curves for Variable-Width X-splines," 2017. Link to paper.