Page 60 - Modern Spatiotemporal Geostatistics

Spatiotemporal Geometry 41

The generalization of the Gaussian coordinate system to higher dimen-
sional spaces requires the use of the concept (proposed by Riemann) of a
manifold, which is a natural extension of Euclidean space (Riemann originally
introduced the concept of a continuous manifold as a continuum of elements,
such that a single element is defined by n continuous variable magnitudes; this
definition includes the analytical conception of space in which each point is
defined by n coordinates). In simple terms, an n-dimensional manifold is a set
of points such that each point can serve as the origin of a local coordinate
system (also called a coordinate patch) that covers only a portion of the space
and offers an internal visualization of the surface. A set of coordinate patches
that covers the whole space of interest is called an atlas. Since it requires
two Gaussian coordinates to locate a point on an ordinary surface, the sur-
face is called a two-dimensional manifold. If Cartesian coordinates are used,
a relation among three of them is needed to describe such a manifold. Rie-
mann extended Gauss' intrinsic geometry of two-dimensional surfaces (n = 2)
to n-dimensional manifolds with n > 2, leading to the Riemannian coordinate
system, as follows.

DEFINITION 2.4: The Riemannian coordinate system in an n-
dimensional manifold consists of a network of Uj-curves (i = l,...,n)
so that a one-to-one correspondence is established between each point
(
P of the manifold and the n-tuples HI, .. .,u n).
The Riemannian coordinates consistently assign to each point on a man-
ifold a unique n-tuple; i.e., they individuate, but neither relate nor measure.
If explicit relations or measures are required by the physical situation of con-
cern, extra constructions are introduced, like the metrical structure of a small
n-dimensional region around each point (see the following section). In phys-
ical applications, one may choose to change or transform the parametric net
MI, ...,«„ to suit some specified objective. Then, one may have to work with
restricted regions (or patches) of a manifold to meet the requirements for such
a transformation of parameters. In a Riemannian system, the jobs of Cartesian
lines are divided between the coordinate patches fixing positions and the curves
connecting points on a manifold. A rigorous mathematical presentation of the
Riemannian theory of space, which is utilized in differential geometry, may be
found in Chavel (1995).
Other useful non-Euclidean coordinate systems include the so-called geo-
desies coordinate system. According to this system, a point P on a surface
is defined by choosing an origin O and then measuring the angle <j)p and the
distance \OP\ to point P by means of the corresponding geodesic (Fig. 2.10).
Depending on the shape of the surface, some constraints may apply. In the
case of the Earth, e.g., a maximum limit should be set for the distance such
that \OP\ < c/2, where c is the circumference of the Earth (considered as a
sphere). In addition, the Glebsch coordinates, the Boozer-Grad coordinates,
the Hamada coordinates, and the toroidal coordinates (e.g., D'haeseleer et
al, 1991) are systems of coordinates with certain particular physical properties