I
n the previous chapter we illustrated examples
of geological surfaces, such as the top of the
Triassic Chinle Formation throughout the San
Rafael Swell in southern Utah, using structure
contours (Fig. 2.27). The more detailed shape of
the top of the blue-gray limestone bed near the
bottom of the Carmel Formation in the Chimney
Rock area is shown by the structure contours in
Fig. 2.29. The frontispiece of this chapter shows
two aerial photographs of the exposed surfaces of
Jurassic sandstone formations on the flank of the
Waterpocket Fold that defines the southeastern
margin of the San Rafael Swell. Notice how the
sandstone formations bend over the monoclinal
flexure and also bend as the strike of the beds
changes along the length of the fold. Monoclinal
flexures are a continuing focus of structural investigation in this region (Reches, 1978; Reches and
Johnson, 1978; Cooke et al., 2000; Johnson and
Johnson, 2000). Clearly these surfaces are not
planar, but what is their shape? A primary task for
structural geologists is to describe and characterize such surfaces and this may be accomplished in
a mathematically rigorous and complete manner
using concepts and tools from differential geometry,
the branch of mathematics that brings the power
of vector calculus to geometry (Gauss, 1827). Here
we review some of the elementary concepts of differential geometry that are helpful to quantify
the departure of geological surfaces from a plane
(Mallet, 2002).
Structural data typically are gathered at scattered exposures as point measurements and the
locations of these points should be identified
using geographic or local coordinates and position
vectors. The measured data include the local attitudes of planar and linear elements that approximate, for example, a foliation (Figs. 2.24a–c) or
lineation (Figs. 2.24d–f) at the point of measurement (Cloos, 1946; Turner and Weiss, 1963).
Plotting the attitudes of a set of structural elements on a stereographic projection enables one
to compare the orientations of different members
of the set. While serving a useful purpose in their
own right, stereographic projections provide
an incomplete characterization of foliations and
lineations, because these projections lack any
information about the spatial variations of orientations. It may be clear that a particular lineation
has a different plunge direction and plunge at different exposures, but how does the attitude vary
from one location to the next? The lineation may
approximate a three-dimensional curve, so we
need to understand how to describe the shape of
such curves. The spatial variation of plunge direction and plunge can be inferred qualitatively from
the distribution of attitude symbols on a structural map, but differential geometry provides the
tools for the quantification and analysis of these
spatial variations.
Relatively little use of differential geometry is
found in the twentieth-century literature of
structural geology, despite the obvious need to
describe the complex shapes of curved lineations
and surfaces, and the attractiveness of accomplishing this in a quantitative manner (Mallet,
2002). This literature provides few examples that
we can use to illustrate the concepts introduced
here. Furthermore, field techniques for deducing
the shapes of curves or surfaces from scattered
location and orientation data are just now being
devised and tested. Therefore this chapter focuses
on the principles and methods of differential
geometry that appear to have the greatest potential for application to structural geology. This
introduction is meant to encourage the use of
these principles and methods and thereby
provide, in the words of Reveil Netz (2000), “good
starting points for truly productive discussion” of
the geometry of geological structures.
Position vectors are used to describe points,
curves, and surfaces in differential geometry.
Many other vector quantities, such as the tangent
and curvature vector, are derived from the position vector and used extensively in this chapter.
Therefore it is necessary to understand the basic
concepts of vectors, and to be familiar with
specific techniques for manipulating vectors as
introduced in the previous chapter before reading
this chapter. For in-depth treatments of differential geometry that provide a rigorous mathematical basis, the reader is referred to textbooks on the
subject (Struik, 1961; Stoker, 1969). In particular,
the book by Lipschutz (1969) is a source for much
of the material in this chapter and provides many
useful exercises and worked examples.
76
CHARACTERIZING STRUCTURES USING DIFFERENTIAL GEOMETRY
n the previous chapter we illustrated examples
of geological surfaces, such as the top of the
Triassic Chinle Formation throughout the San
Rafael Swell in southern Utah, using structure
contours (Fig. 2.27). The more detailed shape of
the top of the blue-gray limestone bed near the
bottom of the Carmel Formation in the Chimney
Rock area is shown by the structure contours in
Fig. 2.29. The frontispiece of this chapter shows
two aerial photographs of the exposed surfaces of
Jurassic sandstone formations on the flank of the
Waterpocket Fold that defines the southeastern
margin of the San Rafael Swell. Notice how the
sandstone formations bend over the monoclinal
flexure and also bend as the strike of the beds
changes along the length of the fold. Monoclinal
flexures are a continuing focus of structural investigation in this region (Reches, 1978; Reches and
Johnson, 1978; Cooke et al., 2000; Johnson and
Johnson, 2000). Clearly these surfaces are not
planar, but what is their shape? A primary task for
structural geologists is to describe and characterize such surfaces and this may be accomplished in
a mathematically rigorous and complete manner
using concepts and tools from differential geometry,
the branch of mathematics that brings the power
of vector calculus to geometry (Gauss, 1827). Here
we review some of the elementary concepts of differential geometry that are helpful to quantify
the departure of geological surfaces from a plane
(Mallet, 2002).
Structural data typically are gathered at scattered exposures as point measurements and the
locations of these points should be identified
using geographic or local coordinates and position
vectors. The measured data include the local attitudes of planar and linear elements that approximate, for example, a foliation (Figs. 2.24a–c) or
lineation (Figs. 2.24d–f) at the point of measurement (Cloos, 1946; Turner and Weiss, 1963).
Plotting the attitudes of a set of structural elements on a stereographic projection enables one
to compare the orientations of different members
of the set. While serving a useful purpose in their
own right, stereographic projections provide
an incomplete characterization of foliations and
lineations, because these projections lack any
information about the spatial variations of orientations. It may be clear that a particular lineation
has a different plunge direction and plunge at different exposures, but how does the attitude vary
from one location to the next? The lineation may
approximate a three-dimensional curve, so we
need to understand how to describe the shape of
such curves. The spatial variation of plunge direction and plunge can be inferred qualitatively from
the distribution of attitude symbols on a structural map, but differential geometry provides the
tools for the quantification and analysis of these
spatial variations.
Relatively little use of differential geometry is
found in the twentieth-century literature of
structural geology, despite the obvious need to
describe the complex shapes of curved lineations
and surfaces, and the attractiveness of accomplishing this in a quantitative manner (Mallet,
2002). This literature provides few examples that
we can use to illustrate the concepts introduced
here. Furthermore, field techniques for deducing
the shapes of curves or surfaces from scattered
location and orientation data are just now being
devised and tested. Therefore this chapter focuses
on the principles and methods of differential
geometry that appear to have the greatest potential for application to structural geology. This
introduction is meant to encourage the use of
these principles and methods and thereby
provide, in the words of Reveil Netz (2000), “good
starting points for truly productive discussion” of
the geometry of geological structures.
Position vectors are used to describe points,
curves, and surfaces in differential geometry.
Many other vector quantities, such as the tangent
and curvature vector, are derived from the position vector and used extensively in this chapter.
Therefore it is necessary to understand the basic
concepts of vectors, and to be familiar with
specific techniques for manipulating vectors as
introduced in the previous chapter before reading
this chapter. For in-depth treatments of differential geometry that provide a rigorous mathematical basis, the reader is referred to textbooks on the
subject (Struik, 1961; Stoker, 1969). In particular,
the book by Lipschutz (1969) is a source for much
of the material in this chapter and provides many
useful exercises and worked examples.
76
CHARACTERIZING STRUCTURES USING DIFFERENTIAL GEOMETRY