Let $L$ be a line in $\mathbb{R}^n$, that is, $\dim L = 1$ ($L$ arises as the face of a polytope/intersection of halfspaces). We're given a fixed objective vector $\mathbf{c}\in\mathbb{R}^n$.
Consider now a hyperplane with normal vector $\mathbf{a}\in\mathbb{R}^n$ and offset $b$, that is, $\mathbf{a}^T\mathbf{x}\le b$. Let $v(\mathbf{a}, b)$ denote the intersection of this hyperplane with $L$. I'm trying to understand what $\mathbf{c}^T v(\mathbf{a}, b)$ looks like as a function of $\mathbf{a}$ and $b$. It seems attractive to say that it's linear in $\mathbf{a} = (a_1,\ldots, a_n)$ and $b$, but I'm not sure that this is true. Even in two dimensions, the intersection of the lines $x-y=0$ and $a_1 x + a_2 y = b$ is at $(b/(a_1+a_2), b/(a_1+a_2))$, so $\mathbf{c}^T v(\mathbf{a}, b)$ is certainly not linear.
Is there a simple characterization of $\mathbf{c}^T v(\mathbf{a}, b)$? Is it a polynomial in $\mathbf{a}, b$ of bounded degree?
Let assume that $L = \left\{\mathbf d + \lambda \mathbf u: \lambda \in \mathbb R\right\}$, where $\mathbf d, \mathbf u\in \mathbb R^n$. Since $v\left(\mathbf a, b\right)$ is on both $L$ and the hyperplan, you can then write $\mathbf a^Tv(\mathbf a, b) = b$ and $v(\mathbf a, b) = \mathbf d + \lambda \mathbf u$. So $$ v\left(\mathbf a, b\right) = \mathbf d + \frac{b - \mathbf a^T\mathbf d}{\mathbf a^T\mathbf u}\mathbf u. $$
Which is not exactly linear but fractionnal on the values of $(\mathbf a, b)$.