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_Journal of Structural Geology 32 (2010) 685e692_ _Contents lists available at ScienceDirect_ _Journal of Structural Geology_ _journal homepage: www.elsevier.com locate jsg_ _A rapid B?zier curve method for shape analysis and point representation of asymmetric folds_ _Deepak C. Srivastava a,*, Vipul Rastogi b, Rajit Ghosha a _Department of Earth Sciences, IIT Roorkee, Roorkee 247 667, India_ _b Department of Physics, IIT Roorkee, Roorkee 247 667, India_ _article info_ _Article history: Received 5 August 2009_ _Received in revised form 30 March 2010_ _Accepted 9 April 2010_ _Available online 24 April 2010_ _Keywords: Asymmetric fold B?zier curve In?ection point Lift Point representation_ _abstract_ _Point representation of fold shapes is useful, in particular, for classi?cation of a large number of folds into different geometric populations. The methods for shape analysis and point representation of asymmetric folds are few and tedious, although several methods exist for the analysis of individual fold limbs or symmetric folds. This article gives a rapid method that uses the B?zier curve tool, available in any common computer graphics software, for the analysis of a complete asymmetric fold and its point representation in the two-dimensional frame._ _The new method is based on the reduction of variables in the parametric equations of a cubic B?zier curve. It makes the length of one B?zier handle zero, pins the end point of the other B?zier handle at the origin of the XeY frame and drags its control point along the Y-axis to ?t the B?zier curve on the given asymmetric fold. A Cartesian plot between normalised length of the B?zier handle and the lift, i.e., difference between the heights of the two in?ection points, gives the unique point that represents the given asymmetric fold shape. We test the validity of the new method on several computer simulated asymmetric folds and demonstrate its usefulness with the help of a natural example._ _? 2010 Elsevier Ltd. All rights reserved._ _1. Introduction_ _Structural geologists commonly require distinctions between the shapes of minor folds that may occur in different parts of a large scale fold or belong to one or more groups domains. Although any scheme that displays a large number of fold shapes on a single plot is useful for classi?cation of folds into different shape populations, the point representation in the two-dimensional Cartesian frame is ideal. This article proposes a B?zier curve method for the analysis and point representation of asymmetric fold shapes. The new method, an extension of the B?zier curve method for analysis of individual fold limbs (Srivastava and Lisle, 2004), is rapid and easier than existing methods for the fold shape analysis and the point representation (Tripathi and Gairola, 1999; Bastida et al., 2005). All the fold shapes referred in this article are the pro?le sections of single folded surfaces, e.g., contact surface between the adjacent beds lithological layers._ _* Corresponding author. Tel.: ?91 1332 285558; fax: ?91 1332 273560. E-mail address: dpkesfes@iitr.ernet.in (D.C. Srivastava)._ _0191-8141 $ e see front matter ? 2010 Elsevier Ltd. All rights reserved. doi:10.1016 j.jsg.2010.04.002_ _2. Geometrical attributes of asymmetric folds_ _2.1. Fold shapes_ _Loudon (1964) and Whitten (1966, pp. 586e590) define fold shapes by the statistical attributes, the fourth statistical moment, or Kurtosis. The interlimb angle or tightness, and the attributes such as dextral (z-shaped) or sinistral (s-shaped) shapes have long been used for description of asymmetric fold shapes (Fleuty, 1964; Ramsay, 1967, pp. 351e354). As two or more folds of equal interlimb angle can have different shapes, additional attributes are necessary for their complete description (Ramsay, 1967, pp. 350e351). In the most comprehensive account till date, Twiss (1988) shows that a complete description of an asymmetric fold shape requires, at least, six parameters: the aspect ratio, the folding angle (180 x14 e interlimb angle), the bluntness of the closure, the two inclination angles and the hinge tangent angle._ _Existing methods for the shape analysis of individual limbs or symmetric folds include the harmonic analysis (Chapple, 1968; Stabler, 1968; Hudleston, 1973; Ramsay and Huber, 1987, p. 314; Stowe, 1988), the conic section analysis (Aller et al., 2004), the power function analysis (Bastida et al., 1999), the super ellipses (Lisle, 1988) or the B?zier curve analysis (Srivastava and Lisle, 2004; Lisle et al., 2006). The results of limb-by-limb analysis can represent an asymmetric fold by a tie-line that joins the two points on a Cartesian plot, one corresponding to each fold limb (Fig. 15.14 in Ramsay and Huber, 1987, p. 317; Hudleston, 1973). Distinction between different populations of the asymmetric fold shapes is, however, dif?cult on the plots that contain a large number of intersecting tie-lines._ _2.2. Degree of asymmetry_ _The degree of asymmetry, an important geometrical attribute of asymmetric folds, has been used for interpretations regarding the large scale fold geometry and the buckling bending moment ratio during the process of buckle folding (Price, 1967; Price and Cosgrove, 1990, p. 328). Loudon (1964) and Whitten (1966, p. 588) define degree of asymmetry by the third statistical moment or skewness. Several other definitions of the degree of asymmetry exist. For example, Price (1967) defines the degree of asymmetry as the limb length ratio. This definition, however, does not account for the fold shape, because two asymmetric folds with the same limb length ratio may have different shapes. Similarly, the degree of asymmetry, expressed as the angle between bisector of the folding angle (180 x14 e interlimb angle) and the median trace (Twiss and Moores, 1992, p. 207) also does not give any information about the fold shape._ _In yet another definition, the degree of asymmetry is the sum of the differences in the aspect ratio Dsize and, the shape Dshape of the two limbs (Tripathi and Gairola, 1999). The scope of this definition is, however, limited because several combinations of the two differences, Dsize and Dshape, may yield the same degree of asymmetry. Bastida et al. (2005) improve this scheme by defining the degree of asymmetry as the ratio of ‘shape asymmetry’ and ‘size asymmetry’ and show that an asymmetric fold shape can be represented as a point on the two-dimensional plot of the two types of asymmetry. They define the ‘shape asymmetry’ as the ratio of the normalised areas of the opposite limbs, where the normalised area itself is the ratio of limb area and rectangular area that bounds the limb, and the ‘size asymmetry’ as the amplitude ratio of the two limbs. We propose a simple and rapid alternative method that traces the asymmetric fold shape by a B?zier curve and represents it as a point in the twodimensional frame._ _3. The B?zier curve method_ _B?zier curves are named after B?zier (1966), who used them for designing curvatures in the automobile industry. De Paor (1996) first demonstrated the potential of B?zier curves in structural geological applications, such as section construction and balancing, modeling fault displacement problems and representation of strain variations in thrust sheets and ductile shear zones. It is now well established that the B?zier curve tool, available in most computer graphics software, is a powerful tool for rapid and accurate analysis of fold shapes (Wojtal and Hughes, 2001; Srivastava and Lisle, 2004; Coelho et al., 2005; Lisle et al., 2006; Liu et al., 2009a,b)._ _3.1. Rationale_ _A B?zier curve consists of one or more polynomial segments such that each segment is defined by the following parametric equations (Davies et al., 1986; De Paor, 1996; Srivastava and Lisle, 2004)._ _x?t? ? ?1 ? t?3x0 ? 3?1 ? t?2tx1 ? 3?1 ? t?t2x2 ? x3_ _(1)_ _y?t? ? ?1 ? t?3y0 ? 3?1 ? t?2ty1 ? 3?1 ? t?t2y2 ? y3_ _(2)_ _These equations are derived from the coordinates of the four points: two end points P0 (x0, y0) and P3 (x3, y3), and two control points, P1 (x1, y1), P2 (x2, y2) of the two B?zier handles, P0P1 and P2P3, respectively. Parameter t varies from 0 to 1 from the first end point P0 to the second end point P3, and it satisfies 0 < t < 1 at all other points on the curve (Fig. 1a)._ _The shape of B?zier curve depends on eight variables, i.e., the coordinates of the four points, P0, P1, P2, and P3 (Fig. 1a). Such a curve cannot be directly represented as a point on the two-dimensional XeY plane, because the number of variables is too large. The number of variables can, however, be reduced by defining a cubic B?zier curve in a XeY Cartesian frame such that: (i) first B?zier handle, P0P1 lies on the Y-axis with its end point P0 and the control point P1 at (0, 0) and (0, c), respectively, (ii) the control point P2 of second B?zier handle, P2P3 coincides with its end point P3 (a, b). The length of second B?zier handle P2P3 is, therefore, zero (Fig. 1b). With these constraints, the shape of B?zier curve depends on only two parameters: (i) L, the ratio of length c of the first B?zier handle and width a of the fold, L ? c/a, and (ii) R, the_ Ключевые слова: srivastava lisle, handle, lift, tectonophysics, variation, point b?zier, in?ection, symmetric fold, classication, description, image, limb-by-limb analysis, de?nition, inection point, cartesian plot, prole, design, parameter, paor, chapple, frame, representation asymmetric, jsg, fold shape, hudleston, simulated, strain, drag, dragged, progressive increase, geological, yield, surface, join, angle, origin, validity, stowe, tool, cluster-i fold, control point, parabola, length, bzier curve, folds, stabler, shape analysis, b?zier curve, geometrical analysis, unique, degree, moores, york, structural, graphic, rotation, huber, ramsay huber, simulation, coelho, horizontal, point representation, equal, large, dene, ratio, graphic software, rs, control, interlimb, geology, tightness, substitution, represent, black dot, method, natural fold, davies, shape asymmetry, structural geology, plot, test, limb, folding angle, vertical separation, folded, press, b?zier handle, ramsay, y-axis, denition, fleuty, aller, in?ection point, software, fold, roorkee, price, application, doe, asymmetry, lisle, two-dimensional, result, equation, sine curve, vertical, long, parametric equation, article, bastida, conic, rapid, cartesian, sharp crested, layer, twiss, reduction, srivastava journal, journal structural, attribute, complete, asymmetric, number, variable, analysis, cosgrove, point, represents, analysis fold, asymmetric fold, degree asymmetry, lie, combination, bzier handle, geometrical, distortion, classi?cation, whitten, curve method, natural, representation, analysis point, length b?zier, scheme, gairola, large number, department, coordinate, shape, sine, size asymmetry, normalised area, srivastava, journal structural geology, b?zier, journal, ii, unique point, loudon, interlimb angle, graphic simulation, curve, difference, tangent, coordinate point, straight