Geology Reference
In-Depth Information
Direction Cosines and Vector Geometry
12.1
Introduction
The orientations of structural elements are most readily obtained analytically by 3-D vector
geometry. A line is represented by a vector of unit length and a plane by its pole vector or
its dip vector. The direction cosines of a line describe the orientation of the unit vector parallel
to the line. Structural information such as bearing and plunge is converted into direction
cosine form, the necessary operations performed, and then the values converted back to
standard geological format. This chapter gives the basic relationships and outlines some of
the ways vector geometry can be applied to structural problems involving lines and planes.
12.2
Direction Cosines of Lines
Direction cosines define the orientation of a vector in three dimensions (Fig. 12.1).
The direction angles between the line OC and the positive coordinate axes
x
,
y
,
z
are
α
,
,
, respectively. The direction cosines of the line OC are
β
γ
cos
α
= cos EOC ,
(12.1a)
cos
β
= cos GOC ,
(12.1b)
cos
γ
= cos (90 + AOC) .
(12.1c)
In the geological sign convention, an azimuth of 0 and 360° represents north, 90° east,
180° south, and 270° west. Dip (of a plane) or plunge (of a line) is an angle from the
horizontal between 0 and 90°, positive downward. From the geometry of Fig. 12.1:
cos
δ
= OA / OC ,
(12.2a)
cos
θ
= OG / OA ,
(12.2b)
sin
θ
= OE / OA ,
(12.2c)
cos
α
= OE / OC ,
(12.2d)
cos
β
= OG / OC ,
(12.2e)
cos
γ
=cos(90+
δ
)=-sin(
δ
) .
(12.2f )
