Stereographic projection and its uses

Stereographic projection is a method of representing three-dimensional orientations (planes and lines) on a two-dimensional circle. A reference sphere is cut by the plane or line through its centre, and the point or trace where it meets the lower hemisphere is projected onto the horizontal equatorial plane. A plane appears as a great circle, its normal as a pole, and a line as a point. The equal-angle (Wulff) and equal-area (Schmidt) nets are used.

Uses in engineering geology

1. Slope stability: kinematic analysis of plane, wedge and toppling failure; selecting safe cut-slope angle and direction (Markland test, friction circle).
2. Joint analysis: plotting poles and contouring them to find joint sets and their mean orientation (rose and contour diagrams).
3. Structural geology: finding true dip from two apparent dips, apparent dip in any direction, angle between planes or lines, and intersection of two planes (line of intersection and its plunge).
4. Folds and faults: locating fold axis (π and β diagrams), axial plane and plunge; analysing fault slip direction.
5. Tunnels and underground openings: deciding tunnel axis relative to joint sets, and identifying unstable roof wedges.
6. Dams, bridges and foundations: checking stability of abutments and foundation rock against sliding along discontinuities.
7. Rock excavation and quarrying: planning orientation of faces and blasting.
8. Rotation of structures: restoring beds to original position (tilt correction) to interpret the geological history.