Application of Brianchon's theorem to construction of geologic profiles

Brianchon's theorem states that the three diagonals joining opposite vertices of a hexagon circumscribed about a conic are concurrent. A corollary of this theorem applies to a pentagon so that the points of tangency of an inscribed conic may be located. Any five non-concurrent straight lines in a plane, no three of which are parallel, will ordinarily form some kind of a pentagon; and if considered as tangents to a conic, they will define its shape and position. If these five lines are also normals to the traces of parallel stratigraphic surfaces having a constant strike, the derived conic may be regarded as an evolute, from which a set of involutes can be drawn that will constitute a structural profile. A method is thus afforded for constructing a profile normal to the strike of the rocks, and for measuring stratigraphic thickness, by the utilization of five observations of dip along a suitable linear traverse. Graphical methods are also given for constructing a parabolic evolute from four observations and a circular evolute from three observations. Additional points on the conic evolutes are obtained by the application of Pascal's theorem. A mathematical analysis is presented of the relationship between a conic and five of its tangents; and the conic evolute, rather than its involutes, is recommended as a satisfactory record of the structure of a parallel fold. To obtain the equation of this evolute, the equations of the five tangents are first derived, using trilinear coordinates. Thereafter the tangential and trilinear equations of the general conic are deduced. Criteria are given for classifying the conic as a hyperbola, ellipse, parabola, or circle. These graphical and analytical methods are adaptations of the general method of evolute and involutes. They are offered, not as substitutes for the general method, but as quicker, though somewhat less accurate, means of obtaining similar results, where structural conditions justify their use. © 1948, The Geological Society of America, Inc.