Engineering Rock Mechanics

Appendix B:

Hemispherical

projection

Hemispherical projection methods

These methods enable three-dimensional orientation data to be displayed

in two dimensions and manipulated graphically.

Fundumentd geometry

Directions are vectors with unit length. We assume that these vectors
emanate from the origin of a Cartesian co-ordinate system. It is conveni-
ent to use east/north/down for rock mechanics.

0

Directions are measured in terms of

the angles

a = trend

/3 = plunge

a is measured with a compass,

p is measured with a clinometer.

Note that OB = sin acosp

oc = cos acosp

I AD = sin p.

Because every vector has unit length, the tips lie on the surface of a
sphere. We are usually only interested in downward-directed vectors,
lying on the lower hemisphere.

Projection onto two dimensions

One way to form a two-dimensional plot associated with vectors on the

lower hemisphere is to project the tips of the vectors onto the horizontal