Trajectory Data Warehouses - Data Warehouse Systems: Design and Implementation - page 473

The term Buffer(RouteT1, 0.6, 0:10) defines an elliptic cylinder, or cylindroid,
around the trajectory of the truck, with a half-axis of 0.6 over the spatial
dimension and a half-axis of 10min over the temporal dimension. In this
case, the initial points of the trajectories RouteT1 and RouteT2 will satisfy
the query, since they are at distance 1; that means two circles with radius
0.6 and with centers, respectively, in point (0,0) at 8:00 and in point (1,0)
at 8:05 have non-null intersection if the time tolerance is 10min. The same
occurs with the ending points of both trajectories.
As we have seen, aggregation operations can also be lifted. For example,
Union(RouteT1, RouteT2) will result in a single temporal geometry composed
of the two lines in Fig. 12.3 .
We define four operations for computing the rate of change for points.
Operation Speed yields the usual concept of speed of a temporal point at any
instant as a temporal real, defined as follows:

f distance ( f ( t + δ ) ,f ( t ))
δ

f ( t ) = lim
δ→ 0

.

Operation Direction returns the direction of the movement, that is, the angle
between the x -axis and the tangent to the trajectory of the moving point.
Operation Turn yields the change of direction at any instant, defined as
follows:

f direction ( f ( t + δ ) ,f ( t ))
δ

f ( t ) = lim
δ→ 0

.

Finally, Derivative returns the derivative of the movement as a temporal real.
We gave the definition of Derivative in the previous section. Note that we
can get the acceleration of a temporal point P by Derivative(Speed(P)) .For
example:
￿ Speed(RouteT1) yields a temporal real with values 16.9 at [8:00, 8:10] and
0 at [8:10, 8:25].
￿ Direction(RouteT1) yields a temporal real with value 45 at [8:00, 8:10].
￿ Turn(RouteT1) yields a temporal real with value 0 at [8:00, 8:10].
￿ Derivative(RouteT1) yields a temporal real with value 1 at [8:00, 8:10].

Notice that during the stop of the truck, the direction and turn are undefined.

12.2.2 Temporal Field Types

Temporal fields represent phenomena that vary both on time and space. As
shown in Fig. 12.5 a, a temporal field can be conceptualized as a function
that assigns a value to each point of a spatiotemporal space. Temporal fields
are obtained by composing the temporal and field constructors. For example,