deletions | additions       diff --git a/Methodology Complementary Data.tex b/Methodology Complementary Data.tex  index 6f4565e..d244101 100644  --- a/Methodology Complementary Data.tex  +++ b/Methodology Complementary Data.tex 
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where a point located at ($x$,$y$) in cartesian space is snapped to the nearest position on the nearest alignment $l$, and is represented by the curvilinear distance $S$ along this alignment from its begining and the offset \( \gamma \), orthogonal to this alignment, measuring the distance between the original point and it's snapped position. These coordinates are useful for measuring following behaviour, lane changes, and lane deflection.  Many approaches exist to trajectory clustering, some methods are supervised, many more are supervised unsupervised  (e.g. k-means \cite{MacQueen_1967}). Supervised trajectory clustering is labour intensive and potentially a source of bias, but allows for tight control of scenery description and analysis oversight. Learned Unsupervised  clustering is systematic but naive as this form of clustering can only make use of trajectory data to infer position. The methodology is primarily based in manual trajectory spatial relationship. Supervised  clustering (called alignments), although along  a series of splines, called alignments, is chosen for its simple implementation and tight control over interpretation. A  hybrid approach, which refines spatial positioning of manually defined alignments through traditional unsupervised  clustering approaches, is proposed for as  future work. \subsubsection{Network Topology}  Once trajectories are clustered, a network topology is constructed in order to be able to inteligently propagate future possible positions of moving objects through the network. In simple networks (i.e. two alignments), these movements are implicitly defined simply by observing lane change ratios, but in more complex networks, such as the network shown in \ref{fig:complex-network}, movements may involve multiple lane changes and therefore may require a more general approach. A recursive tree model is employed.  Alignment extremities are linked to other nearby alignments, creating diverging or converging branches, as are momentarily adjacent alignments. Alignments Alternatively, alignments  which run in parralel over a distance of more than 15 metres are instead grouped into corridors over which lane changes may occur. occur freely.  This creates a series of links and nodes with implicit direction which can be searched to determine all possible future positions of a moving object inside of this network. This serves to reduce calculation processing  times of spatial relationships relationship calculations  between objects (triage) and provides more inteligent interpretation ofthe  spatial relationships. \subsubsection{Geometric data/inventory}