A MATHEMATICAL PROCEDURE FOR PROCESSING A VARIETY OF 3D STRUCTURAL GEOLOGY PROBLEMS INCLUDING ROTATIONS

Typical 3D problems in structural geology involve the measurement of angles between lines and/or planes in 3D space. Another class of problem involves the rotation of lines or planes about a rotational axis by a specified amount and sense of rotation. Traditionally undergraduates have been instructed to solve these types of problems with orthographic or stereographic methods. These traditional methods have proven to be excellent for forcing students to visualize 3D problems, however, there are disadvantages to these methods as well: (1) manually processed data are susceptible to plotting error, (2) accuracy may be lost, especially near the poles of a stereographic projection where great circles converge, and (3) students often need a procedure for verifying the correct answer without necessarily having the instructor inspect their solution.

A purely mathematical set of algorithms address the above problems by allowing students to rapidly calculate a correct answer on a computer or calculator. The mathematical method converts orientation data to 3D vectors where the positive X, Y, and Z axes are oriented due east and horizontal, due north and horizontal, and vertically down respectively. Because the desired answer is an orientation only, all orientation vectors are unit length. The directional angles alpha, beta, and gamma made by a vector with the X, Y, and Z axes respectively yield the vector components parallel to the axes by taking the cosine of the angle. The components can then be manipulated by algorithms that use the cross-product and dot-product to determine angular relationships, or process rotations. These algorithms are easily incorporated into spreadsheet applications for instantaneous solution. This allows students to verify answers, or for researchers to quickly process 3D problems in the field (app. dip, 3-point, rotation, etc.) on a calculator without the need for a stereonet or other plotting tools.