Fold Geometry - 3-D Structural Geology - page 117

cos =(sin

δ 1 cos θ 1 cos δ 2 -cos δ 1 sin δ 2 cos θ 2 )/ N ,

(5.7a)

α

cos β = (cos δ 1 sin δ 2 sin θ 2 -sin δ 1 sin θ 1 cos δ 2 )/ N , (5.7b)

cos γ =(sin δ 1 sin θ 1 sin δ 2 cos θ 2 -sin δ 1 cos θ 1 sin δ 2 sin θ 2 )/ N , (5.7c)

θ 2 ) 2 N = [(sin δ 1 cos θ 1 cos δ 2 -cos δ 1 sin δ 2 cos

δ 2 ) 2

+(cos δ 1 sin δ 2 sin θ 2 -sin δ 1 sin θ 1 cos

θ 2 ) 2 ] 1/2 , +(sin δ 1 sin θ 1 sin δ 2 cos θ 2 -sin δ 1 cos θ 1 sin δ 2 sin (5.8)

and the azimuth and plunge of the first plane is

θ 1 , δ 1 and of the second plane is θ 2 , δ 2 .

The value ' given by Eq. 5.5 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,

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. 5.5 must be prevented.

α and cos β

5.4
Axial Surfaces

The axial surface geometry is important in defining the complete three-dimensional
geometry of a fold (Sect. 1.5) and in the construction of predictive cross sections
(Sect. 6.4.1).

5.4.1
Characteristics

The axial surface of a fold is defined as the surface that contains the hinge lines of all
horizons in the fold (Fig. 5.13; Dennis 1967). The axial trace is the trace of the axial
surface on another surface such as on the earth's surface or on the plane of a cross
section. The orientation of an axial surface within a fold hinge is related to the layer

Fig. 5.13.
Characteristics that may be
associated with an axial sur-
face. The axial surface con-
tains the hinge lines of succes-
sive layers (the defining prop-
erty). The surface may be the
boundary between fold limbs
(different shading ) and may
be the plane across which the
sense of layer-parallel shear
( double arrows ) reverses di-
rection