Page 48 - Modern Spatiotemporal Geostatistics

Spatiotemporal Geometry 29

space that exists at all times, but rather a "space-at-an-instant" within the
space/time continuum.
EXAMPLE 2.2: In Figure 2.2a the evolution of a geologic boundary B at differ-
ent times is represented by a set of lines. This set forms a space/time surface
in £. In Figure 2.2b, a two-dimensional contaminant distribution is broken
into three parts, sweeping out a three-dimensional volume in space/time.
The message conveyed by the above examples is that in most scientific
studies we are seeking to establish a theory of relationships between events.
This leads to our next postulate.
POSTULATE 2.2: Since events are associated with points of the space/
time continuum £, relationships between events are essentially relation-
ships between the points of £.
Beyond the continuum with its individual points, the spatiotemporal struc-
tures differ, as do the geometries which describe them. Generally four essential
characteristics of spatiotemporal geometry may be identified:
(i.) Geometric objects (individual points, lines, planes, vectors, and tensors)
that give geometry its objective features.
(ii.) Measurable properties of objects and spaces (angles, ratios, curvature,
and distance or metric) that give geometry its quantitative features.
(Hi.) Modes of comparison (equal, less than, greater than) that give geometry
its comparative features.
(iv.) Spatiotemporal relationships (inside, outside, between, before, after) that
give geometry its relative features.
C O M M E N T 2.1: I t is noteworthy that in their physical application these four

characteristics of spatiotemporal geometry may be considered independently.
For example, the spatiotemporal metric and the coordinate system used to
describe that metric could be independent of each other. As we shall see

below, this case has significant consequence in spatiotemporal analysis.
EXAMPLE 2.3: Traditionally, geometries are divided into Euclidean and non-
Euclidean. Table 2.2 summarizes some of the main features of these geome-
tries in the case of two-dimensional space. Euclidean geometry was invented
around 300 B.C. Euclid, in his epoch-making work, Elements (see Heath,
1956 [1908]), reduces the whole of geometrical science to an axiomatic form in
which all propositions and theorems are deduced from a small number of ax-
ioms and postulates. Non-Euclidean geometries are derived by modification of
one or more of the five basic Euclidean postulates. Well-known non-Euclidean
geometries have been constructed by modifying Euclid's infamous fifth postu-
late (given a straight line and a point not on that line, there is one and only one
line through that point parallel to the given line). In particular, the geometry
of Bolyai and Lobachevski asserts the existence of more than one parallel, and
the geometry of Riemann denies the existence of parallels altogether (Faber,