maps. Formulae to compute the coordinates on
the UTM grid given geographic coordinates on an
ellipsoidal datum, and to calculate the geographic
coordinates on the ellipsoidal datum given the
UTM coordinates have been derived (Richardus
and Adler, 1972). These take several pages to write
down and are not repeated here. Many GPS
systems have these formulae built in and make
the computations at the push of a button.
2.2 Local coordinates and position
vectors
Points and sets of points can be defined only relative to
(i.e., as functions of ) a coordinate system, never
absolutely. The coordinate system is the unavoidable
residue of the eradication of the ego in that geometrico-physical world which reason sifts from the given
using “objectivity” as its standard – a final scanty
token in this objective sphere that existence is only
given and can only be given as the intentional content
of the processes of consciousness of a pure, sensegiving ego (Weyl, 1987).
Students first learn to deal with points and sets
of points described relative to a chosen coordinate system in elementary courses in mathematics. This procedure was apparently conceived by
René Descartes in the early seventeenth century
and has become one of the most powerful tools
ever developed for scientists and engineers
(Davis and Hersh, 1986; Aczel, 2000). In honor of
Descartes’ contribution the most familiar coordinate system we use is referred to as the Cartesian
coordinate system. With this coordinate system and
the concept of a position vector one can locate
outcrops or the point where samples are collected in the field relative to a local origin. In
other words the origin is located in the region
being mapped or on the actual outcrop for very
large-scale mapping, instead of at an arbitrary
point determined by a global projection such as
the UTM grid. To put the map or data in a global
context one transforms the position vectors from
the local coordinate system to the UTM system.
Here we introduce the position vector, several
local coordinate systems, and equations for the
transformation from one coordinate system to
another because these are basic tools for structural mapping.
It is useful to understand the concept of
vectors in general, and to be familiar with specific
techniques for manipulating vectors, because
structural geologists employ them for modeling
as well as mapping. This should come as no surprise because the evolution of geologic structures
is primarily a physical process and the physical
laws that describe such a process are written as
vector equations. This section focuses on position
vectors, but also serves as a summary of some
general vector concepts that we build upon in
later sections where, for example, we use vectors
to characterize the shapes of folded geological surfaces and to visualize velocity fields within a
deforming rock mass. Other general vector concepts are introduced as needed in later sections
and chapters.
2.2.1 Locating data using local
coordinates and position vectors
Geographical coordinates, based on the UTM
grid, may not be the best choice for mapping in
the field, but their use is becoming universal for
the final presentation of structural maps. For
large-scale maps awkwardness arises because the
central line of longitude for each UTM zone is
500 000 m to the east of the origin (Fig. 2.4c), so a
typical easting would have six digits with meter
precision and eight digits if centimeter precision
were required. Similar numbers of digits are
required for the northing unless, for example, the
location is near the equator in the northern
hemisphere. To reduce this cumbersome number
of digits it is practical to select a local origin
within the mapped region. Local origins are also
commonly employed in modeling and data analysis. Furthermore, a local origin may be necessitated by the use of surveying equipment that
references locations to the instrument itself
rather than the UTM system. Topographical, geological, and structural maps can be prepared
using a coordinate system consisting of the
easting, northing, and elevation for which the
datum is the local elevation of the instrument
and the easting and northing are measured in the
34
STRUCTURAL MAPPING TECHNIQUES AND TOOLS
the UTM grid given geographic coordinates on an
ellipsoidal datum, and to calculate the geographic
coordinates on the ellipsoidal datum given the
UTM coordinates have been derived (Richardus
and Adler, 1972). These take several pages to write
down and are not repeated here. Many GPS
systems have these formulae built in and make
the computations at the push of a button.
2.2 Local coordinates and position
vectors
Points and sets of points can be defined only relative to
(i.e., as functions of ) a coordinate system, never
absolutely. The coordinate system is the unavoidable
residue of the eradication of the ego in that geometrico-physical world which reason sifts from the given
using “objectivity” as its standard – a final scanty
token in this objective sphere that existence is only
given and can only be given as the intentional content
of the processes of consciousness of a pure, sensegiving ego (Weyl, 1987).
Students first learn to deal with points and sets
of points described relative to a chosen coordinate system in elementary courses in mathematics. This procedure was apparently conceived by
René Descartes in the early seventeenth century
and has become one of the most powerful tools
ever developed for scientists and engineers
(Davis and Hersh, 1986; Aczel, 2000). In honor of
Descartes’ contribution the most familiar coordinate system we use is referred to as the Cartesian
coordinate system. With this coordinate system and
the concept of a position vector one can locate
outcrops or the point where samples are collected in the field relative to a local origin. In
other words the origin is located in the region
being mapped or on the actual outcrop for very
large-scale mapping, instead of at an arbitrary
point determined by a global projection such as
the UTM grid. To put the map or data in a global
context one transforms the position vectors from
the local coordinate system to the UTM system.
Here we introduce the position vector, several
local coordinate systems, and equations for the
transformation from one coordinate system to
another because these are basic tools for structural mapping.
It is useful to understand the concept of
vectors in general, and to be familiar with specific
techniques for manipulating vectors, because
structural geologists employ them for modeling
as well as mapping. This should come as no surprise because the evolution of geologic structures
is primarily a physical process and the physical
laws that describe such a process are written as
vector equations. This section focuses on position
vectors, but also serves as a summary of some
general vector concepts that we build upon in
later sections where, for example, we use vectors
to characterize the shapes of folded geological surfaces and to visualize velocity fields within a
deforming rock mass. Other general vector concepts are introduced as needed in later sections
and chapters.
2.2.1 Locating data using local
coordinates and position vectors
Geographical coordinates, based on the UTM
grid, may not be the best choice for mapping in
the field, but their use is becoming universal for
the final presentation of structural maps. For
large-scale maps awkwardness arises because the
central line of longitude for each UTM zone is
500 000 m to the east of the origin (Fig. 2.4c), so a
typical easting would have six digits with meter
precision and eight digits if centimeter precision
were required. Similar numbers of digits are
required for the northing unless, for example, the
location is near the equator in the northern
hemisphere. To reduce this cumbersome number
of digits it is practical to select a local origin
within the mapped region. Local origins are also
commonly employed in modeling and data analysis. Furthermore, a local origin may be necessitated by the use of surveying equipment that
references locations to the instrument itself
rather than the UTM system. Topographical, geological, and structural maps can be prepared
using a coordinate system consisting of the
easting, northing, and elevation for which the
datum is the local elevation of the instrument
and the easting and northing are measured in the
34
STRUCTURAL MAPPING TECHNIQUES AND TOOLS