Geology Reference
In-Depth Information
Fig. 12.1.
Orientation of vector
OC
in
right-handed
xyz
space;
θ
is
the azimuth of the vector and
δ
is the plunge. The direction
angles are
α
,
β
,
γ
12.2.1
Direction Cosines of a Line from Azimuth and Plunge
The direction cosines of a line in terms of its azimuth and plunge are obtained from
the relationships in Eq. 12.2:
cos
δ
sin
θ
= (OA / OC) (OE / OA) = OE / OC = cos
α
,
(12.3a)
cos
δ
cos
θ
= (OA / OC) (OG / OA) = OG / OC = cos
β
,
(12.3b)
-sin
δ
=cos
γ
.
(12.3c)
12.2.2
Azimuth and Plunge from Direction Cosines
To reverse the procedure and find the azimuth and plunge of a line from its direction
cosines, divide Eq. 12.3a by 12.3b and solve for
θ
, then solve Eq. 12.3c for
δ
:
θ
' = arctan (cos
α
/cos
β
) ,
(12.4)
= arcsin (-cos
) .
(12.5)
δ
γ
' returned by Eq. 12.4 will be in the range of ±90° and must be corrected
to give the true azimuth over the range of 0 to 360°. The true azimuth,
The value
θ
θ
, of the line can
be determined from the signs of cos
(Table 12.1). The direction cosines
give a directed vector. The vector so determined might point upward. If it is necessary
to reverse its sense of direction, reverse the sign of all three direction cosines. Note that
division by zero in Eq. 12.4 must be prevented.
An Excel equation that returns the azimuth in the correct quadrant has the form
α
and cos
β
=IF(cell B<=0,180+Cell E,IF(cell A>=0,cell E,360+cell E))
(12.6)
where cell A contains cos
α
, cell B contains cos
β
, and cell E contains Eq. 12.4.
