Geology Reference
In-Depth Information
12.2.4
Azimuth and Plunge of a Line from the End Points
To find the azimuth and plunge of a line from the coordinates of two points, substitute
Eq. 12.9 into 12.4 and 12.5:
θ
' = arctan ((
x
2
-
x
1
)/(
y
2
-
y
1
)) ,
(12.11)
δ
=arcsin((
z
2
-
z
1
)/
L
) .
(12.12)
The value of
L
is given by Eq. 12.10. If
y
2
=
y
1
, there is a division by zero in Eq. 12.13
which must be prevented. The value of
θ
is obtained from Table 12.1.
12.2.5
Pole to a Plane
The pole to a plane defined by its dip vector has the same azimuth as the dip vector,
and a plunge of
+ 90. Substitute this into Eqs. 12.3 to obtain the direction co-
sines of the pole,
p
, in terms of the azimuth,
δ
p
=
δ
, and plunge,
, of the bedding dip:
θ
δ
cos
α
p
=cos(
δ
+ 90) sin
θ
=-sin
δ
sin
θ
,
(12.13a)
cos
β
p
=cos(
δ
+ 90) cos
θ
= -sin
δ
cos
θ
,
(12.13b)
cos
γ
p
=-sin(
δ
+90)=-cos
δ
.
(12.13c)
12.3
Attitude of a Plane from Three Points
The general equation of a plane (Foley and Van Dam 1983) is
Ax
+
By
+
Cz
+
D
= 0 .
(12.14)
The equation of the plane from the
xyz
coordinates of three points is
.
(12.15)
This expression is expanded by cofactors to find the coefficients
A
,
B
,
C
, and
D
:
A
=
y
1
z
2
+
z
1
y
3
+
y
2
z
3
-
z
2
y
3
-
z
3
y
1
-
z
1
y
2
,
(12.16a)
B
=
z
2
x
3
+
z
3
x
1
+
z
1
x
2
-
x
1
z
2
-
z
1
x
3
-
x
2
z
3
,
(12.16b)
