Clustering Moving Object Trajectories. How many different trajectories are there between two endpoints?
You probably go from home to office using a known route that you learned from experience is the best in some sense. Maybe it minimizes your morning commute time or the travel cost, or perhaps it is the most convenient. When I go to work every morning, I drive my car to a public parking lot and then take the subway. The commute back home starts with the tube and then the car, but I always use a different route this time. Driving home using the morning route would mean substantial traffic jams, so I take a detour that shortens my commute. My route choices reflect some conditions that I am subject to, and knowing these would probably be beneficial to the traffic authorities for better planning. But how could they know this just by looking at traffic patterns?
The study of trajectories is fundamental for the comprehension of moving object behavior. We not only care about the start and end locations but especially care about the path, where the object moved through between endpoints. Once we know the trajectories, we can gather statistics about the moving object behavior and use it for future inference. For instance, we determine the typical fuel consumption of a vehicle when traveling between two known locations given the general path. We can later decide if a new trip between the same locations fell within the known distribution or was overly aggressive.
A spatial trajectory is a trace generated by a moving object in geographical spaces, usually represented by of a series of chronologically ordered points, e.g., p1 → p2 → ··· → pn, where each point consists of a geospatial coordinate set and a timestamp such as p = (x,y,t). 
Moving objects create trajectories, temporal sequences of locations that define curves in space. We usually collect trajectory information using a sampling process, collecting positions at discrete time intervals. This process happens when you allow your smartphone to collect location information from your whereabouts. Instead of a continuous line, the device is collecting data that form a polyline. By assigning a timestamp to each vertex, the polyline assumes a definite direction, from smaller to larger time values.
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