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_Journal of Structural Geology 32 (2010) 166–171_ Contents lists available at ScienceDirect Journal of Structural Geology journal homepage: www.elsevier.com locate jsg The hinge lines of non-cylindrical folds R.J. Lisle a,*, N. Toimil b, J. Aller b, N. Bobillo-Ares c, F. Bastida a School of Earth and Ocean Sciences, Cardiff University, Cardiff CF10 3YE, UK b Departamento de Geolog??a, Universidad de Oviedo, Spain c Departamento de Matema?ticas, Universidad de Oviedo, Spain Article info Article history: Received 4 February 2009 Received in revised form 6 October 2009 Accepted 31 October 2009 Available online 6 November 2009 Keywords: Fold analysis Differential geometry Hinge lines Ridge lines Deformation Curvature Abstract Hinge lines are loci of high curvature points on folded surfaces. They are significant geometrical features of geological folds, and the arrangement of hinge lines constructed for the surface serves to characterize important aspects of the fold pattern. Since the current definition of hinge line is only appropriate for cylindrical folds, we propose a new definition for use with folds of general shape. Like the concept of ridge lines used in differential geometry, the new definition uses the lines of curvature (principal curvature trajectories) as a reference frame for comparing curvatures across the surface. A hinge line passes through points of extreme principal curvature magnitude observed along the corresponding principal curvature trajectory. Two types of hinge lines are defined and methods for constructing hinge lines are suggested. ? 2009 Elsevier Ltd. All rights reserved. 1. Introduction Recently developed surveying methods allow the mapping of folded geological surfaces in three dimensions. 3D seismic reflection methods are employed to map subsurface structures, whereas GPS and laser scanning methods are being increasingly used to survey well-exposed structural surfaces (Bergbauer and Pollard, 2004; Pearce et al., 2006). Whilst these data provide the potential for gaining new insights concerning the process of folding (Pollard and Fletcher, 2005), they also highlight the inadequacy of a number of existing methods of geometrical analysis. The latter were mainly devised to cater for folds outcropping at the surface where information on the folded surface is little more than two-dimensional. New 3D methods for describing and analysing folded surfaces, founded on the concepts of differential geometry, are being devised to address the above problems (Pollard and Fletcher, 2005; Lisle and Toimil, 2007; Mynatt et al., 2007). For example, approaches have been proposed for dissecting a general folded surface into individual folds, and for distinguishing antiformal and synformal folds (Lisle and Toimil, 2007). * Corresponding author. Fax: ?44 29 2087 4326. E-mail address: lisle@cardiff.ac.uk (R.J. Lisle). 0191-8141 $ – see front matter ? 2009 Elsevier Ltd. All rights reserved. doi:10.1016 j.jsg.2009.10.011 Hinge lines, the subject of this paper, are key geometrical features of folds. The patterns of hinge lines are important in relation to the structural location of hydrocarbon fields (e.g. Al-Mahmoud et al., 2009), fracture prediction in hydrocarbon reservoirs (e.g. Stephenson et al., 2007), prediction of the direction of subsurface elongation of ore-bodies (e.g. Duuring et al., 2007), the analysis of structures produced by multiple folding events (Ramsay, 1967), and to folding processes in shear zones (Ghosh et al., 1999; Alsop and Carreras, 2007) or around diapirs (Jackson et al., 1990). In the present paper, we examine the existing definition of the hinge line. The term, hinge line, refers to the locus of points of maximum curvature on the folded surface (Fig. 1). Sets of hinge lines drawn on a folded surface serve to illustrate the folding pattern, refolded geometries, and relationships between different fold sets. However, we discover that the existing definition of hinge line is inadequate for general use, and therefore a new definition based on the concepts of differential geometry is proposed. In devising the new definition, our aim is to provide a conceptual framework for the practical construction of hinge lines on folded surfaces whilst honouring the essential meaning of the existing term. 167 R.J. Lisle et al. Journal of Structural Geology 32 (2010) 166–171 Generally lie on the true hinge line of the fold (Schryver, 1966). Fleuty’s (1964) suggestion to determine hinge points on serial sections parallel to the fold’s profile plane provides no solution to this problem because non-cylindrical folds do not possess profile planes or natural cross-sections. In summary, current definitions do not permit the drawing of hinge lines on non-cylindrically folded surfaces. This issue presents a real problem for fold analysis, especially as recent technological advances in mapping suggest that non-cylindrical folds are the norm rather than the exception. Fig. 1. Cylindrically folded surfaces possess true profile planes, e.g. section A. Points of maximum curvature of the folded surface observed in the profile plane, H, lie on the fold’s hinge line, H–H. In general, points of greatest curvature observed on other oblique section planes, e.g. H0 on section B, do not lie on the hinge line. This reason, literature relating to face recognition (Gordon and Vincent, 1992), analysis of medical images (Thurion and Gourdon, 1996), computer-aided design (Lukacs and Andor, 1998) and food engineering (Sivertsen et al., 2009) are pertinent to the issues addressed in this paper. 2. Existing definitions of the term hinge line Formal definitions of the term, fold hinge line, vary, though the consensus view is that it is a line on a folded surface along which the curvature of the surface reaches a local maximum. As such, it separates adjacent regions of lesser curvature called fold limbs. A historical review of the concept by Wilson and Cosgrove (1982) reports that the definition of hinge line as the locus of points of large curvature has prevailed at least since the early part of the 20th century (e.g. Haug, 1924; Bonte, 1953; Stockwell, 1950; Wilson, 1961). Some authors have defined the term in a more specific sense by limiting its application to cylindrical, or approximately cylindrical, folds (Wegman, 1929; Clark and McIntyre, 1951). Such hinges are straight lines, or nearly so. Fleuty (1964), however, considered this latter usage to be too restrictive, suggesting that for non-cylindrical folds the hinge line can be drawn through points of maximum curvature observed on serial cross-sections through the folded surface. This definition corresponds to that by Turner and Weiss (1963), Ramsay (1967) and Marshak and Mitra (1988) and results in hinges that are not necessarily straight lines. When applied to folds that do not deviate greatly from the cylindrical type, the above definition of Turner and Weiss (1963) is workable and generally leads to satisfactory results. However, from a theoretical point of view the definition is flawed, and this leads to practical problems in locating the hinges of non-cylindrical folds. The essence of the problem lies in the choice of the orientation of the plane of section used to determine hinge points. It is well known that the curvature and the location of the point of greatest curvature observed on a 2D section are strongly influenced by the orientation of the section plane through a folded surface (Fig. 1). For instance, even in the case of cylindrical folds, the points of maximum curvature observed on an oblique section do not 3. General definition of hinge line Cylindrical folds possess natural profile planes; planes perpendicular to the fold axis (generator). On a serial set of such planes, points of maximum curvature can be identified and then linked to form the hinge line. However, this procedure is not possible in the case of non-cylindrical folds because they lack true profile planes for the observation of curvature variations. To overcome this problem, we propose a modified definition of hinge line based on the concept of ridge lines used in the field of differential geometry (e.g. Koenderink, 1990, p. 291). The ridge lines referred to here are not to be confused with ridges in the topographic sense. At any point, P, on a curved surface the curvature of the surface can be observed in section planes normal to the plane tangential to the surface at P. In general, these values of normal curvatures vary with the direction chosen for the normal section. In fact, these normal curvatures change systematically as the normal section plane is turned, and reach extreme values in two orthogonal directions of the section plane (Fig. 2). These are the principal curvatures k1 and k2 at P, where k1 > k2 and where convex-upward curvature is positive. The principal curvatures are associated with two perpendicular directions in the surface called the principal curvature directions. Lines of curvature, or principal curvature trajectories, are curves drawn on the surface whose tangents at any point are parallel to one of the principal curvature directions (Fig. 3). Therefore, two sets of principal trajectories, corresponding to k1 and k2 directions, respectively, form an orthogonal mesh on the surface. Ridge lines are the locus of points where the k1 and k2 reach extreme values along their respective trajectories (Koenderink, 1990; Belyaev and Anoshkina, 2005, p. 50). The curvature trajectories therefore provide a convenient reference frame for the assessment of Fig. 2. The definition of principal curvatures, k1 and k2, and the principal curvature directions at a point P on a folded geological surface. 168 R.J. Lisle et al. 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