Page 58 - Modern Spatiotemporal Geostatistics

Spatiotemporal Geometry 39

Figure 2.8. Gaussian coordinate system.
defined by the pair (ui,uz). Neither the u\ curves nor the U2 curves are
equally spaced; the important idea of Gauss is that the spacing between them
is not a consideration at all. The HI curves never cross other HI curves, and
the U2 curves never cross other U2 curves. The u\ curves intersect the u^
curves, but not necessarily at right angles. This intersecting grid allows us to
locate points, but not to measure distances between them directly. Indeed,
unlike Euclidean coordinates, the idea of distance is not essential to Gaussian
coordinates. The values of HI and MI are just numbers that assign an order
to the network curves. So, when we say that the Gaussian coordinates of a
point are (3,2), we provide no information about distance from the origin or
any other point.

Figure 2.9. A mesh of the Gaussian coordinate system shown in Figure 2.8.
Let us now consider one specific mesh of the network in Figure 2.8
bounded by the Gaussian u\ curves 3 and 4 and u 2 curves 2 and 3 (Fig.
2.9). We may consider the local coordinate system OQR, and within the
mesh under consideration the point P can be assigned the Gaussian coordi-
nates (3 + dui, 2 + du z), where du\ = OPi/OQ and du 2 = OP 2/OR are
ratios (which, however, do not give us the distances OP\ and OP?)- If from
actual measurements the lengths of OQ and OR are found to be (n and
&2r respectively, the corresponding distances can be given by (OPi, OP?) =
(£n dui, £22 duz). This discussion leads to the following definition.

DEFINITION 2.3: In the Gaussian coordinate system, the coordinates
of a point P on a curved surface are defined by