Comparison of traditional and whole-space stereographic projections in block stability analysis

The stereographic projection used in engineering geological analysis is a hemispherical projection: a plane or a straight line in 3-D space is projected to be an arc or a point inside the equatorial circle. This kind of projection is widely used to judge the unstable blocks cut by discontinuities by analyzing the relationship of locations between the intersection of arcs projected by multiple sets of discontinuities and the arc projected by the free surface. Nevertheless, this kind of judgement lacks a rigous theoretical basis. The whole-space stereographic projection proposed in the block theory projects a whole plane and both its upper-half and lower-half space onto the equatorial plane. The removability and failure mode of blocks can be determined by analyzing the location relationship between the joint pyramid projected by multiple sets of discontinuities and the circle projected by the free surface. This approach is complete and accurate in theory. In this study, the shortcomings of the traditional stereographic projection used for slope stability analysis in "Geological engineering handbook" and "Design code for engineering slopes of water resources and hydropower projects" are pointed out by analyzing some specially designed examples, and the reasons for these shortcomings are released. Through the whole-space stereographic projection, the continuous variation of morphology of joint pyramid is exposed, the characteristics of the shape of projection zone projected by flat or sharp blocks are analyzed, and the significance of these issues is explored. Finally, the advantages of the whole-space stereographic projection are summarized, and the simplicity in using and the feasibility for popularization are pointed out.