a surface that is the target for mapping may be
written:
(3.59)
Here we consider the (x, y)-plane as the horizontal
plane of a local Cartesian coordinate system established in the region being mapped, and values of
g(x, y) are the measured elevations of exposures of
the surface relative to the local origin. Similarly,
the parameter plane can be superimposed on the
UTM grid for the region in which case (3.58) is
transformed such that s x ϭ easting, s y ϭ northing,
and s z ϭ elevation.
To gain further insights about the Monge
patch consider the parameter plane (Fig. 3.17a)
with a rectangular grid of lines, u ϭ constant and
vϭconstant, parallel to the coordinate axes and
use the following parametric representation of a
curved surface which is in the form (3.58) of a
Monge patch (Lipschutz, 1969, p. 185):
(3.60)
Using the first two components to eliminate the
parameters u and v from the expression for the
third component, and noting that the components of the vector are equivalent to the coordinates, the equation z ϭ x
2 ϩ y
2 is found which is the
same elliptic paraboloid illustrated in Fig. 3.16.
However, in contrast to that representation, the uand v-parameter curves for (3.60) are the intersections of the surface with planes parallel to the
(x, z)-plane and the (y, z)-plane, respectively.
Note how three sets of curved lines are used to
represent the elliptic paraboloid in the wireframe diagram (Fig. 3.17b). The intersections of
the curved surface with planes parallel to the (x, z)plane form a set of parabolas, as do the intersections of the curved surface with planes parallel to
the (y, z)-plane. The intersections of the curved
surface with planes parallel to the (x, y)-plane form
a set of circles. A particular u-parameter curve,
defined by the vector function c(u, 0.7), and a particular v-parameter curve, defined by c(0.8, v), are
highlighted in Fig. 3.17b. The point (0.8, 0.7) on the
parameter plane maps to the point p(0.8, 0.7) on
the curved surface at the intersection of the two
highlighted curves. In this way every coordinate
line and every point in the two-dimensional parameter plane have a corresponding curve and a
s(u, v) ϭ ue x ϩ ve y ϩ (u 2 ϩ v 2 )e z
s(x, y) ϭ xe x ϩ ye y ϩ g(x, y)e z
corresponding point on the curved surface in
three-dimensional space.
3.2.3 The tangent plane, tangent vector,
and unit normal vector
We continue to use the more general parametric
representation of a surface (3.54) to develop the
theoretical concepts necessary to characterize
surfaces, but recognize the Monge patch (3.58) as
a useful representation for mapping. Because the
parametric representation of a surface, s(u, v),
describes a vector function of two variable parameters, there is a partial derivative associated
with each parameter. To calculate the partial
derivative Ѩs/Ѩu, for example, one takes the derivative with respect to u of each component of the
vector function while holding v constant, and
then uses these as the components of a new vector
function. Thus, the partial derivatives of s(u, v)
with respect to the two parameters are (Lipschutz,
1969, p. 126):
(3.61)
Recall from (3.8) that for the natural representation of a curve, c(s), the derivative with respect to
the arc length, s, is the unit tangent vector.
Because the partial derivative Ѩs/Ѩu is taken with
v ϭ constant, this is equivalent to taking the derivative along any one of the u-parameter curves, for
example, c(u, 0.7) as shown in Fig. 3.17b. Thus, the
partial derivative, Ѩs/Ѩu, is a vector that is tangent
to a u-parameter curve and points in the direction
of increasing u. Similarly, Ѩs/Ѩv, is a vector that is
tangent to a v-parameter curve and points in the
direction of increasing v. These tangent vectors
are not necessarily unit vectors because the u- and
v-parameter curves are not necessarily the natural
representations of these curves.
As an example consider the partial derivatives
of the parametric representation for the elliptic
paraboloid (3.60) illustrated in Fig. 3.17b:
(3.62)
Note that the tangent vectors, Ѩs/Ѩu, for the uparameter curves lie in planes that are parallel to
the (x, z)-plane. Consider the particular u-parameter
Ѩs
Ѩu
ϭ e x ϩ 2ue z ,
Ѩs
Ѩv
ϭ e y ϩ 2ve z
Ѩs(u,v)
Ѩv
ϭ
Ѩs x
Ѩv
e x ϩ
Ѩs y
Ѩv
e y ϩ
Ѩs z
Ѩv
e z
Ѩs(u, v)
Ѩu
ϭ
Ѩs x
Ѩu
e x ϩ
Ѩs y
Ѩu
e y ϩ
Ѩs z
Ѩu
e z
96
CHARACTERIZING STRUCTURES USING DIFFERENTIAL GEOMETRY
written:
(3.59)
Here we consider the (x, y)-plane as the horizontal
plane of a local Cartesian coordinate system established in the region being mapped, and values of
g(x, y) are the measured elevations of exposures of
the surface relative to the local origin. Similarly,
the parameter plane can be superimposed on the
UTM grid for the region in which case (3.58) is
transformed such that s x ϭ easting, s y ϭ northing,
and s z ϭ elevation.
To gain further insights about the Monge
patch consider the parameter plane (Fig. 3.17a)
with a rectangular grid of lines, u ϭ constant and
vϭconstant, parallel to the coordinate axes and
use the following parametric representation of a
curved surface which is in the form (3.58) of a
Monge patch (Lipschutz, 1969, p. 185):
(3.60)
Using the first two components to eliminate the
parameters u and v from the expression for the
third component, and noting that the components of the vector are equivalent to the coordinates, the equation z ϭ x
2 ϩ y
2 is found which is the
same elliptic paraboloid illustrated in Fig. 3.16.
However, in contrast to that representation, the uand v-parameter curves for (3.60) are the intersections of the surface with planes parallel to the
(x, z)-plane and the (y, z)-plane, respectively.
Note how three sets of curved lines are used to
represent the elliptic paraboloid in the wireframe diagram (Fig. 3.17b). The intersections of
the curved surface with planes parallel to the (x, z)plane form a set of parabolas, as do the intersections of the curved surface with planes parallel to
the (y, z)-plane. The intersections of the curved
surface with planes parallel to the (x, y)-plane form
a set of circles. A particular u-parameter curve,
defined by the vector function c(u, 0.7), and a particular v-parameter curve, defined by c(0.8, v), are
highlighted in Fig. 3.17b. The point (0.8, 0.7) on the
parameter plane maps to the point p(0.8, 0.7) on
the curved surface at the intersection of the two
highlighted curves. In this way every coordinate
line and every point in the two-dimensional parameter plane have a corresponding curve and a
s(u, v) ϭ ue x ϩ ve y ϩ (u 2 ϩ v 2 )e z
s(x, y) ϭ xe x ϩ ye y ϩ g(x, y)e z
corresponding point on the curved surface in
three-dimensional space.
3.2.3 The tangent plane, tangent vector,
and unit normal vector
We continue to use the more general parametric
representation of a surface (3.54) to develop the
theoretical concepts necessary to characterize
surfaces, but recognize the Monge patch (3.58) as
a useful representation for mapping. Because the
parametric representation of a surface, s(u, v),
describes a vector function of two variable parameters, there is a partial derivative associated
with each parameter. To calculate the partial
derivative Ѩs/Ѩu, for example, one takes the derivative with respect to u of each component of the
vector function while holding v constant, and
then uses these as the components of a new vector
function. Thus, the partial derivatives of s(u, v)
with respect to the two parameters are (Lipschutz,
1969, p. 126):
(3.61)
Recall from (3.8) that for the natural representation of a curve, c(s), the derivative with respect to
the arc length, s, is the unit tangent vector.
Because the partial derivative Ѩs/Ѩu is taken with
v ϭ constant, this is equivalent to taking the derivative along any one of the u-parameter curves, for
example, c(u, 0.7) as shown in Fig. 3.17b. Thus, the
partial derivative, Ѩs/Ѩu, is a vector that is tangent
to a u-parameter curve and points in the direction
of increasing u. Similarly, Ѩs/Ѩv, is a vector that is
tangent to a v-parameter curve and points in the
direction of increasing v. These tangent vectors
are not necessarily unit vectors because the u- and
v-parameter curves are not necessarily the natural
representations of these curves.
As an example consider the partial derivatives
of the parametric representation for the elliptic
paraboloid (3.60) illustrated in Fig. 3.17b:
(3.62)
Note that the tangent vectors, Ѩs/Ѩu, for the uparameter curves lie in planes that are parallel to
the (x, z)-plane. Consider the particular u-parameter
Ѩs
Ѩu
ϭ e x ϩ 2ue z ,
Ѩs
Ѩv
ϭ e y ϩ 2ve z
Ѩs(u,v)
Ѩv
ϭ
Ѩs x
Ѩv
e x ϩ
Ѩs y
Ѩv
e y ϩ
Ѩs z
Ѩv
e z
Ѩs(u, v)
Ѩu
ϭ
Ѩs x
Ѩu
e x ϩ
Ѩs y
Ѩu
e y ϩ
Ѩs z
Ѩu
e z
96
CHARACTERIZING STRUCTURES USING DIFFERENTIAL GEOMETRY