Geology Reference
In-Depth Information
Table 12.1.
Relationship between signs
of the direction cosines and
the quadrant of the azimuth
of a line
12.2.3
Direction Cosines of a Line on a Map
The direction cosines of a line defined by its horizontal,
h
, and vertical,
v
, dimensions
on a map can be found by letting
δ
=arctan(
v
/
h
) in Eqs. 12.3:
cos
α
= cos (arctan (
v
/
h
)) sin
θ
,
(12.7a)
cos
β
= cos (arctan (
v
/
h
)) cos
θ
,
(12.7b)
cos
γ
= -sin (arctan (
v
/
h
)) .
(12.7c)
Both
v
and
h
are taken as positive numbers in Eqs. 12.7 and the resulting dip is
positive downward.
The direction cosines of a line defined by the
xyz
coordinates of its two end points
are obtained by letting point 1 be at O and point 2 be at C in Fig. 12.1. Then
x
2
-
x
1
= OE ,
(12.8a)
y
2
-
y
1
= OG ,
(12.8b)
z
2
-
z
1
= AC .
(12.8c)
Substitute Eqs. 12.8 into 12.3 to obtain
cos
α
=(
x
2
-
x
1
) / OC ,
(12.9a)
cos
β
=(
y
2
-
y
1
) / OC ,
(12.9b)
cos
γ
=-sin
δ
=-AC/OC=-(
z
2
-
z
1
)/OC=(
z
1
-
z
2
) / OC ,
(12.9c)
where OC =
L
, g iven by
L
=[(
x
2
-
x
1
)
2
+(
y
2
-
y
1
)
2
+(
z
2
-
z
1
)
2
]
1/2
.
(12.10)
Using the convention that point 1 is higher and point 2 is lower, a downward-di-
rected bearing is positive in sign.
