necessary to reproduce the surface, and they do
not provide adequate measures for quantitatively
comparing the shapes of the different surfaces or
the spatial variations in shape of a single surface.
Descriptions of folded surfaces using differential
geometry do not have these shortcomings.
Metamorphic rocks composed of the systematic arrangement of layers of different rock types
(Fig. 2.24a), a set of sub-parallel fractures (Fig.
2.24b), or a set of similarly oriented platy mineral
grains (Fig. 2.24c) are said to have a penetrative
planar fabric (Turner and Weiss, 1963, p. 97). The
name indicates that the constituents locally
define a surface that resembles a plane, but the
normal to this planar fabric may systematically
change orientation from exposure to exposure. In
these cases it may be possible to represent this
spatial variation in orientation with a set of
curved surfaces such that a given surface is everywhere tangent to the locally planar fabric.
Procedures to define individual continuous surfaces in three-dimensional space from scattered
measurements of a penetrative planar fabric have
not been devised, so that is a noteworthy challenge for structural geologists. With such a characterization in hand one could model fabrics
using continuum mechanics and test hypotheses
concerning their orientation using differential
geometry.
3.2.2 Parametric representations of
curved surfaces
We began our discussion of curved lines by appealing to the intuitive notion of a set of points
arranged in an orderly and continuous fashion to
form a curve in three-dimensional space. Because
individual points in three-dimensional space are
identified by their position vectors, p, this led to
the definition of a curved line as a continuous
vector function of a single scalar variable t, called
the arbitrary parameter of the curve, such that c
ϭ c(t). As t increases in value the heads of successive position vectors trace out the curved line. The
analytical description of curved surfaces also may
be approached from the intuitive notion of a set
of points arranged in some continuous fashion in
three-dimensional space. However, sufficiently
close to any particular point, the neighboring
points are distributed such that they resemble a
plane, not a line. This leads to the definition of a
curved surface as a continuous vector function of
two scalar variables (u, v), called the parameters of
the surface, such that s ϭ s(u, v). The two parameters may be thought of as the coordinates of
points on a plane, called the parameter plane, and
those points map onto the surface according to
the vector function s(u, v). As the two parameters
vary, the heads of the successive position vectors
sweep out the curved surface in three-dimensional space.
To understand the analytical definition of a
curved surface we begin by describing the coordinate systems used for the two-dimensional parameter plane and the three-dimensional curved
surface (Fig. 3.15). The two Cartesian axes (Ou, Ov)
and the associated base vectors (e u , e v ) define the
parameter plane on which the two coordinates
are the parameters u and v. The three Cartesian
axes (Ox, Oy, Oz) and the associated base vectors (e x ,
e y , e z ) comprise the system for the curved surface.
The position vectors, s, for the curved surface can
be written as a function of the two parameters
(Lipschutz, 1969, p. 128):
(3.54)
The three scalar functions [s x (u,v), s y (u,v), s z (u,v)] are
the components of the vector function, s(u, v),
with respect to the base vectors (e x , e y , e z ). These
functions, along with the base vectors, determine
the position vectors for all points on the curved
surface. The vector equation (3.54) is called the
parametric representation of the surface. Compare
the facts that a single variable parameterizes the
curved line (3.1) and a pair of variables parameterizes the curved surface (3.54).
Any point in the parameter plane (Fig. 3.15a)
may be defined by a two-dimensional position
vector w ϭ ue u ϩ ve v with respect to the base
vectors e u and e v . Thus, the position vectors for the
curved surface, s, in three-dimensional space
are determined by a vector function of the twodimensional vector variable, w, that is s ϭ s(w).
However, because the components of the vector w
are the two parameters (u, v), we can speak of the
surface as a function of these two scalar parameters, s ϭ s(u, v), and that is what we will do in the
following discussion. An individual point (u o ,v o ) in
the parameter plane (Fig. 3.15a) maps onto the
s(u, v) ϭ s x (u, v)e x ϩ s y (u, v)e y ϩ s z (u, v)e z
3.2 THE CONCEPT AND DESCRIPTION OF CURVED SURFACES
93
not provide adequate measures for quantitatively
comparing the shapes of the different surfaces or
the spatial variations in shape of a single surface.
Descriptions of folded surfaces using differential
geometry do not have these shortcomings.
Metamorphic rocks composed of the systematic arrangement of layers of different rock types
(Fig. 2.24a), a set of sub-parallel fractures (Fig.
2.24b), or a set of similarly oriented platy mineral
grains (Fig. 2.24c) are said to have a penetrative
planar fabric (Turner and Weiss, 1963, p. 97). The
name indicates that the constituents locally
define a surface that resembles a plane, but the
normal to this planar fabric may systematically
change orientation from exposure to exposure. In
these cases it may be possible to represent this
spatial variation in orientation with a set of
curved surfaces such that a given surface is everywhere tangent to the locally planar fabric.
Procedures to define individual continuous surfaces in three-dimensional space from scattered
measurements of a penetrative planar fabric have
not been devised, so that is a noteworthy challenge for structural geologists. With such a characterization in hand one could model fabrics
using continuum mechanics and test hypotheses
concerning their orientation using differential
geometry.
3.2.2 Parametric representations of
curved surfaces
We began our discussion of curved lines by appealing to the intuitive notion of a set of points
arranged in an orderly and continuous fashion to
form a curve in three-dimensional space. Because
individual points in three-dimensional space are
identified by their position vectors, p, this led to
the definition of a curved line as a continuous
vector function of a single scalar variable t, called
the arbitrary parameter of the curve, such that c
ϭ c(t). As t increases in value the heads of successive position vectors trace out the curved line. The
analytical description of curved surfaces also may
be approached from the intuitive notion of a set
of points arranged in some continuous fashion in
three-dimensional space. However, sufficiently
close to any particular point, the neighboring
points are distributed such that they resemble a
plane, not a line. This leads to the definition of a
curved surface as a continuous vector function of
two scalar variables (u, v), called the parameters of
the surface, such that s ϭ s(u, v). The two parameters may be thought of as the coordinates of
points on a plane, called the parameter plane, and
those points map onto the surface according to
the vector function s(u, v). As the two parameters
vary, the heads of the successive position vectors
sweep out the curved surface in three-dimensional space.
To understand the analytical definition of a
curved surface we begin by describing the coordinate systems used for the two-dimensional parameter plane and the three-dimensional curved
surface (Fig. 3.15). The two Cartesian axes (Ou, Ov)
and the associated base vectors (e u , e v ) define the
parameter plane on which the two coordinates
are the parameters u and v. The three Cartesian
axes (Ox, Oy, Oz) and the associated base vectors (e x ,
e y , e z ) comprise the system for the curved surface.
The position vectors, s, for the curved surface can
be written as a function of the two parameters
(Lipschutz, 1969, p. 128):
(3.54)
The three scalar functions [s x (u,v), s y (u,v), s z (u,v)] are
the components of the vector function, s(u, v),
with respect to the base vectors (e x , e y , e z ). These
functions, along with the base vectors, determine
the position vectors for all points on the curved
surface. The vector equation (3.54) is called the
parametric representation of the surface. Compare
the facts that a single variable parameterizes the
curved line (3.1) and a pair of variables parameterizes the curved surface (3.54).
Any point in the parameter plane (Fig. 3.15a)
may be defined by a two-dimensional position
vector w ϭ ue u ϩ ve v with respect to the base
vectors e u and e v . Thus, the position vectors for the
curved surface, s, in three-dimensional space
are determined by a vector function of the twodimensional vector variable, w, that is s ϭ s(w).
However, because the components of the vector w
are the two parameters (u, v), we can speak of the
surface as a function of these two scalar parameters, s ϭ s(u, v), and that is what we will do in the
following discussion. An individual point (u o ,v o ) in
the parameter plane (Fig. 3.15a) maps onto the
s(u, v) ϭ s x (u, v)e x ϩ s y (u, v)e y ϩ s z (u, v)e z
3.2 THE CONCEPT AND DESCRIPTION OF CURVED SURFACES
93