Representation of geological discontinuities by means of structural diagrams, often called stereoplots, is standard practice. When the dip directions of such discontinuities form a roughly circular cluster about the horizontal, bimodal pole distributions are observed (and vice versa). This is problematic in the interpretation of any rock-engineering or structural-geology problem where one does not want to be misled by single fracture sets being represented by two projected sets and vice versa. In this paper, the mathematical relations are developed that substantiate this unimodal-bimodal relationship. The paper goes beyond this in analytically developing a few other relationships between dip directions and poles. Examples from several practical cases illustrate the theoretical relationships. The mathematical relations provide a rational proof for the bimodal-unimodal relation and allow one to generalize it, The practically important consequence of this paper is to make the reader aware of the possibility of making mistakes in using stereoplots. This can be avoided if both dip and pole distributions are plotted, particularly for cases where dip or pole clusters represent joint sets that are verv steep or very flat.
