Page 67 - Modern Spatiotemporal Geostatistics

48 Modern Spatiotemporal Geostatistics — Chapter 2

Table 2.3. Common orthogonal curvilinear coordinates.

Curvilinear Rectangular Polar Cylindrical Spherical

Sl Sl r r e P
S2 S2 0 ip
S 3 S3 - S3 e
9n 1 1 1 i
922 1 r 2 P 2 r 2 2

933 1 - 1 (r sirup)
e\ e\ e r e r e p
ei 62 ee ee e v
— ee
e 3 e 3 e 3

Therefore, the Euclidean metric tensor is

The distance for the {s t} system is given by

and the conditions of Comment 2.4 are satisfied.

A summary of the commonly used orthogonal curvilinear (Euclidean) co-
ordinate systems is provided in Table 2.3. As mentioned, Euclidean metrics are
special cases of Equation 2.16. For a metric of the form of Equation 2.16 to
be considered Euclidean, a transformation of coordinates must exist such that
Equation 2.16 can be put in Cartesian form (Eq. 2.12). In Euclidean space, the
general arc-length formula—which is valid in various coordinate systems—is
given by

T
where s(v) gives the arc-length of the curve Si = Si(v), I < i < n, and
a < v < b. Equation 2.17 yields

and by introducing the differential ds t = \dsi(v)/dv\ dv and ds 2 = \ds\ , we
find Equation 2.16. Thus, in Euclidean space, Equation 2.16 is equivalent to
Equation 2.17. Equation 2.17 is independent of the particular parameterization
of the curve.