Page 56 - Modern Spatiotemporal Geostatistics

Spatiotemporal Geometry 37

Figure 2.7. Transformation in R 2 (combination of translation with a rotation)
that leads from one Cartesian coordinate system to another.

Passive transformations, on the other hand, relate different sorts of co-
ordinate systems (e.g., a Cartesian system and a polar system). An arbitrary
orthogonal system {si} may be expressed by means of a passive transforma-
tion of the general form (Eq. 2.4) where, for future notational convenience, the
(li,..., ~s n) denote the rectangular coordinates. In the special case that T
is a linear transformation, the s» are called affine coordinates. A few example
transformations follow.
EXAMPLE 2.8: In the Euclidean polar coordinate system, n = 2 and s =
(si, s^) = (r, 9), with r > 0, in which case the following transformations are
established

The inverse for s 2 = 6 above is valid in the first and fourth quadrants of the
¥1 «2 plane, while other solutions can be obtained over the remaining two
quadrants (likewise for the 9 coordinate in the cylindrical and spherical coordi-
nate systems below). In cylindrical coordinates, n = 3 and s = (si, s 2, SB) =
(r, 0, 33), in which case the following transformations hold true

In spherical coordinates, n = 3 and s = (sj, s^, $3) = (p, <p, 0), which are
related to the rectangular coordinates by