minor changes

tex/chapters/grid.tex

@@ -91,17 +91,18 @@
 		$u,v$, examined by the shortest-path algorithm.
 		
 		To downvote vertices near walls, we need to determine the distance of each vertex from its nearest wall.
-		We therefore derive an inverted version $G' = (V', E')$ of the graph $G$, just describing walls and other
+		We therefore derive an inverted version $G' = (V', E')$ of the graph $G$, just describing walls and 
-		obstacles. A nearest-neighbour search \cite{Cover1967} within $V'$ provides the vertex $v'$
+		obstacles. A nearest-neighbour search \cite{Cover1967} $\fNN{v}{V'}$ within $V'$ provides the vertex
-		nearest to $v$:
+		nearest to $v$.
-		\begin{equation}
+		%\begin{equation}
-			v' = \fNN{v}{V'} \enskip .
+		%	v' = \fNN{v}{V'} \enskip .
-		\end{equation}
+		%\end{equation}
 		To get a smooth gradient around walls, the avoidance-factor 
 		is derived using a normal distribution with a deviation of \SI{0.5}{\meter}:
 		%
 		\begin{equation}
-			\fWA{v} = \mathcal{N}( \fDistance{v}{\fNN{v}{V'}} \mid 0.0, 0.5^2) \\
+			\fWA{v} = \mathcal{N}( \| v - \fNN{v}{V'} \| \mid 0.0, 0.5^2)
+			\enskip .
 			\label{eq:wallAvoidance}
 		\end{equation}
 		%
@@ -195,10 +196,11 @@
 		Hereafter, every node in the grid knows the distance and shortest path to the pedestrian's target.
 		
 		To get realistic path suggestions, we use the importance-factors to adjust the edge-weight
-		$\delta(v_1, v_2)$ for the Dijkstra:
+		$\delta(e_{1,2})$ for the Dijkstra:
 		%
 		\begin{equation}
 			\begin{split}
+				\delta(e_{1,2}) =
 				\delta(v_1, v_2) = 
 					\frac
 						{ \| v_1 - v_2 \|  }