Add a section for finding the optimal route, #4.

apx-flare-timing.tex

@@ -107,6 +107,146 @@ \subsection{Extracting Inputs}
 </Fs>
 \end{lstlisting}
 
+\newpage
+\subsection{Finding an Optimal Route}
+
+To find the best route \texttt{flare-timing} constructs a graph and finds the
+shortest path connecting the nodes. It puts nodes on the turnpoint cylinder arc
+boundaries and uses the haversine distance as the cost of connecting nodes in
+the network. It would be expensive to construct and evaluate a large network
+with the accuracy required so in an iterative process, as the arc of the circle
+is closed, the density of nodes is increased. All happening on the FAI sphere,
+this is the edge to edge distance.
+
+\begin{lstlisting}[caption={Edge to edge distance of the optimal route, \texttt{edgeToEdge} node of \texttt{*.task-length.yaml}.}]
+taskRoutes:
+  edgeToEdge:
+    distance: 159.373683
+    legs:
+    - 10.078208
+    - 42.525217
+    - 0
+    - 64.949832
+    - 41.820427
+    legsSum:
+    - 10.078208
+    - 52.603424
+    - 52.603424
+    - 117.553256
+    - 159.373683
+    waypoints:
+    - lat: -33.36047067
+      lng: 147.93206999
+    - lat: -33.43411056
+      lng: 147.86878018
+    - lat: -33.7159199
+      lng: 147.55846831
+    - lat: -33.7159199
+      lng: 147.55846831
+    - lat: -33.13199024
+      lng: 147.57575486
+    - lat: -33.35857718
+      lng: 147.93468357
+\end{lstlisting}
+
+The naive way to measure task length would just be to connect the centers of each control zone. This is the point to point distance.
+
+\begin{lstlisting}[caption={Which fixes are considered flown, \texttt{flying}
+nodes of \texttt{*.task-length.yaml}.}]
+taskRoutes:
+  pointToPoint:
+    distance: 169.10714
+    legs:
+    - 57.427511
+    - 69.547668
+    - 42.131961
+    legsSum:
+    - 57.427511
+    - 126.975179
+    - 169.10714
+    waypoints:
+    - lat: -33.36137
+      lng: 147.93207
+    - lat: -33.75343
+      lng: 147.52864998
+    - lat: -33.12908
+      lng: 147.57322998
+    - lat: -33.36099999
+      lng: 147.93149998
+\end{lstlisting}
+
+Knowing that FS uses a plane to work out the shortest route in two dimensions,
+on the the Universal Transverse Mercator projection.  We can also do that with
+our graph algorithm.
+
+\begin{lstlisting}[caption={Points on the plane, distances on the sphere, \texttt{projection/spherical} nodes of
+\texttt{*.task-length.yaml}.}]
+taskRoutes:
+  projection:
+    spherical:
+      distance: 159.373683
+      legs:
+      - 10.078208
+      - 42.525217
+      - 0
+      - 64.949832
+      - 41.820427
+      legsSum:
+      - 10.078208
+      - 52.603424
+      - 52.603424
+      - 117.553256
+      - 159.373683
+      waypoints:
+      - lat: -33.36047067
+        lng: 147.93206999
+      - lat: -33.43411056
+        lng: 147.86878018
+      - lat: -33.7159199
+        lng: 147.55846831
+      - lat: -33.7159199
+        lng: 147.55846831
+      - lat: -33.13199024
+        lng: 147.57575486
+      - lat: -33.35857718
+        lng: 147.93468357
+\end{lstlisting}
+\begin{lstlisting}[caption={Points and distances on the plane, \texttt{projection/planar} nodes of
+\texttt{*.task-length.yaml}.}]
+taskRoutes:
+  projection:
+    planar:
+      distance: 159.144781
+      legs:
+      - 10.065441
+      - 42.4942
+      - 0
+      - 64.761082
+      - 41.820427
+      legsSum:
+      - 10.065441
+      - 52.559642
+      - 52.559642
+      - 117.320723
+      - 159.14115
+      mappedPoints:
+      - easting: 586715.834
+        northing: 6308362.198
+      - easting: 580759.282
+        northing: 6300248.47
+      - easting: 551744.701
+        northing: 6269201.551
+      - easting: 551744.701
+        northing: 6269201.551
+      - easting: 553704.761
+        northing: 6333932.964
+      - easting: 586960.882
+        northing: 6308569.955
+      mappedZones:
+      - latZone: H
+        lngZone: 55
+\end{lstlisting}
+
 \newpage
 \subsection{Finding Zone Crossings}