of the stereonet and marked with the north
direction at the top of the net. The overlay is then
rotated counterclockwise through the angle of
strike and the dip is scaled off from the reference
circle at east toward the center using the great
circles of the net for a scale. The great circle so
identified is traced on the overlay and the overlay
is rotated back to the original orientation to
produce the graphical representation of the
planar element shown in Fig. 2.18b.
To avoid the tedious and inaccurate process of
hand construction we seek analytical expressions
for plotting the circular arc representing a planar
element on a stereographic projection by computer. Consider a vertical plane passing through
the center of the sphere, C, and containing the
dip direction (Fig. 2.18c). The horizontal line is the
trace of the projection (equatorial) plane and the
line segment PQ is the trace of the planar
element. The point PЈ is the projection of P onto
the equatorial plane from a perspective at the
zenith, Z, and the point QЈ is the projection of the
point Q. The points PЈ and QЈ lie on the projected
circle and the line segment PЈQЈ is a diameter of
that circle. An arc of this circle is the stereographic projection of the half great circle, so we
seek the center and radius of this projected circle
in order to plot it.
The angle PCPЈ is the dip, d , of the planar
element and the angle CZP is defined as . By the
same argument leading to (2.68), the two angles
are related as ϭ 45ЊϪ d /2. Using this relation
and the triangle CZPЈC, the angle ZPЈC is related to
the dip as:
60
STRUCTURAL MAPPING TECHNIQUES AND TOOLS
(a)
W
E
N
S
Sphere
(b)
W
E
N
S
G
GЈ
C
C
GЈ
(c)
Z
C
PЈ
P
n
R
Q
QЈ
CЈ
(d)
W
S
f
d
C
CЈ
PЈ
QЈ
RЈ
k
h
␥
R
R'
b
Equatorial
plane
Zenith, Z
Projection
of half great
circle
Planar
element
Half great circle
Vertical
plane
Trace of
projection
Dip
direction
f d
Trace of
planar
element
Equatorial
plane
a s
S t r i k e
d i r e c t i o n
Projection
of half
great circle
a d
D i p
d i r e c t i o n
f
d
N, y
(x, y)
a d
E, x
D ip
d ir e c t io n
Projection
of half
great circle
Equatorial
plane
L i n e o f s t r i k e
Fig 2.18 Stereographic projection of planar element.
(a) Reference sphere. (b) Projection of half great circle onto
equatorial plane. (c) Vertical plane passing through center
of reference sphere and containing the dip direction.
(d) Equatorial plane with projection of whole circle.
direction at the top of the net. The overlay is then
rotated counterclockwise through the angle of
strike and the dip is scaled off from the reference
circle at east toward the center using the great
circles of the net for a scale. The great circle so
identified is traced on the overlay and the overlay
is rotated back to the original orientation to
produce the graphical representation of the
planar element shown in Fig. 2.18b.
To avoid the tedious and inaccurate process of
hand construction we seek analytical expressions
for plotting the circular arc representing a planar
element on a stereographic projection by computer. Consider a vertical plane passing through
the center of the sphere, C, and containing the
dip direction (Fig. 2.18c). The horizontal line is the
trace of the projection (equatorial) plane and the
line segment PQ is the trace of the planar
element. The point PЈ is the projection of P onto
the equatorial plane from a perspective at the
zenith, Z, and the point QЈ is the projection of the
point Q. The points PЈ and QЈ lie on the projected
circle and the line segment PЈQЈ is a diameter of
that circle. An arc of this circle is the stereographic projection of the half great circle, so we
seek the center and radius of this projected circle
in order to plot it.
The angle PCPЈ is the dip, d , of the planar
element and the angle CZP is defined as . By the
same argument leading to (2.68), the two angles
are related as ϭ 45ЊϪ d /2. Using this relation
and the triangle CZPЈC, the angle ZPЈC is related to
the dip as:
60
STRUCTURAL MAPPING TECHNIQUES AND TOOLS
(a)
W
E
N
S
Sphere
(b)
W
E
N
S
G
GЈ
C
C
GЈ
(c)
Z
C
PЈ
P
n
R
Q
QЈ
CЈ
(d)
W
S
f
d
C
CЈ
PЈ
QЈ
RЈ
k
h
␥
R
R'
b
Equatorial
plane
Zenith, Z
Projection
of half great
circle
Planar
element
Half great circle
Vertical
plane
Trace of
projection
Dip
direction
f d
Trace of
planar
element
Equatorial
plane
a s
S t r i k e
d i r e c t i o n
Projection
of half
great circle
a d
D i p
d i r e c t i o n
f
d
N, y
(x, y)
a d
E, x
D ip
d ir e c t io n
Projection
of half
great circle
Equatorial
plane
L i n e o f s t r i k e
Fig 2.18 Stereographic projection of planar element.
(a) Reference sphere. (b) Projection of half great circle onto
equatorial plane. (c) Vertical plane passing through center
of reference sphere and containing the dip direction.
(d) Equatorial plane with projection of whole circle.