Line Location with Vertical Distance

Abstract

In this section we consider line location problems with vertical distance d _ver The vertical distance between a point x = (x ₁, x ₂) and a non-vertical line l = l _s,−1,b with slope s ∈ ℝ and intercept b ∈ ℝ is given by the length of the vertical line segment between x and l and can be calculated by

$$ \begin{gathered} {d_{ver}}\left( {x,l} \right) = \mathop {\min }\limits_{y \in l} {d_{ver}}\left( {x,y} \right) \hfill \\ = {d_{ver}}\left( {x,\left( {{x_1},{y_2}} \right)} \right)withy = \left( {{x_1},{y_2}} \right) \in l \hfill \\ = \left| {{x_1}s - {x_2} + b} \right|. \hfill \\ \end{gathered} $$

If l is a vertical line and x ∉ l, then d _ver (x, l) = ∞, while d _ver (x, l) = 0 for x ∈ l.