decomposed into areas that are locally shaped like
one or the other of the six different characteristic
shapes (Bergbauer and Pollard, 2003). Performing
this analysis on the modeled bedding surface
shows that the surface is composed primarily of
domal and saddle-like areas, which reflect the
gentle undulations superimposed on the broader
fold shape. The modeled bedding surface does
not contain any cylindrically shaped areas of
significant extent. We suggest that the characterization of folded surfaces using differential geometry will provide new insights concerning the
process of folding (Fisher and Wiklerson, 2000;
Lisle, 2000; Bergbauer and Pollard, 2004).
3.4 Concluding remarks
The objective of this chapter is to introduce structural geologists to the elementary concepts of differential geometry that serve to characterize
curves (lineations) and surfaces in three-dimensional space. One could imagine that these concepts and the tools that follow from them might
capture the attention of structural geologists,
much as the concepts and tools related to descriptive geometry and stereographic projection did in
the second half of the twentieth century. This is
not the intention of the authors. We view differential geometry as the appropriate mathematical
machinery to characterize structures, but this
characterization is just one step in an investigation which ultimately must include consideration
of the constitutive properties of rock and models
of deformation based on the equations of motion
(Guiton et al., 2003).
Justifications for learning differential geometry are several. The structures encountered in
Earth’s crust are three dimensional with spatial
variations in size and shape that only can be
accounted for using a geometry that involves the
spatial derivatives of such things as orientation and
curvature. Plotting orientation data on a stereographic projection eliminates the opportunity
to visualize and analyze these spatial changes.
Furthermore, to proceed with modeling one needs
to write boundary conditions that refer explicitly
to geometry of surfaces. Finally, we now have
precise field data on the three-dimensional shape
of surfaces from new technology such as GPS and
we need to know how to describe these surfaces and
how to compare them to a model result.
3.4 CONCLUDING REMARKS
119
one or the other of the six different characteristic
shapes (Bergbauer and Pollard, 2003). Performing
this analysis on the modeled bedding surface
shows that the surface is composed primarily of
domal and saddle-like areas, which reflect the
gentle undulations superimposed on the broader
fold shape. The modeled bedding surface does
not contain any cylindrically shaped areas of
significant extent. We suggest that the characterization of folded surfaces using differential geometry will provide new insights concerning the
process of folding (Fisher and Wiklerson, 2000;
Lisle, 2000; Bergbauer and Pollard, 2004).
3.4 Concluding remarks
The objective of this chapter is to introduce structural geologists to the elementary concepts of differential geometry that serve to characterize
curves (lineations) and surfaces in three-dimensional space. One could imagine that these concepts and the tools that follow from them might
capture the attention of structural geologists,
much as the concepts and tools related to descriptive geometry and stereographic projection did in
the second half of the twentieth century. This is
not the intention of the authors. We view differential geometry as the appropriate mathematical
machinery to characterize structures, but this
characterization is just one step in an investigation which ultimately must include consideration
of the constitutive properties of rock and models
of deformation based on the equations of motion
(Guiton et al., 2003).
Justifications for learning differential geometry are several. The structures encountered in
Earth’s crust are three dimensional with spatial
variations in size and shape that only can be
accounted for using a geometry that involves the
spatial derivatives of such things as orientation and
curvature. Plotting orientation data on a stereographic projection eliminates the opportunity
to visualize and analyze these spatial changes.
Furthermore, to proceed with modeling one needs
to write boundary conditions that refer explicitly
to geometry of surfaces. Finally, we now have
precise field data on the three-dimensional shape
of surfaces from new technology such as GPS and
we need to know how to describe these surfaces and
how to compare them to a model result.
3.4 CONCLUDING REMARKS
119