Constant bed-length folding: Three-dimensional geometrical implications

The folding of a sheet without stretching of lines within it is referred to, in the language of differential geometry, as an isometric bending. As can be readily verified by flexing sheets of paper, the no-stretch condition imposes important restrictions on the curvature changes which points on the sheet can undergo during folding. These constraints are embodied in Gauss's Theorema Egregium which states that “the total curvature (equal to the product of the two principal curvatures) at any point remains invariant under isometric bendings”. It follows from this theorem that there is a limited range of folded geometries that an initially planar non-stretching sheet can adopt. These developable surfaces, which include cylindrical and conical folds, have the property that points of equal dip and strike of the surface are arranged in straight lines. This property allows a simple check to be made of the validity of the constant bed-length assumption in the case of natural fold structures. For any fold represented by structure contours, points of equal strike on the structure are linked by isotrend lines which will be straight if the structure is developable. Curved isotrend lines indicate that the structure has a geometry incompatible with the constant bed-length model. The patterns of isotrend lines constructed for a fold help to indicate parts of the structure where layer stretching or faulting is likely. On the other hand, patterns consisting of straight isotrend lines can be used for the prediction of the structure in adjacent areas. Isotrend analyses to date suggest that bed-length balancing of cross-sections is generally an invalid procedure