USING BEZIER CURVES TO ANALYZE THE SHAPES OF FOLDED SURFACES

The profile shapes of folds hold clues to the behavior of rock and the character of deformation. Two methods to describe folds objectively are commonly used. A ratio of the first and third terms in a Fourier series, b₁/b ₃, describes quarter wave segments of upright folds. Fourier analysis requires that the curve y=f(x) depicting a fold is one-to-one, i.e. each x value has a single y value. Fourier analysis cannot treat asymmetric folds or folds with negative interlimb angles. Twiss analyzed fold shape by circumscribing half-wavelength folds by trapezoids and determining three parameters that define its shape: the aspect ratio P=amplitude/half wavelength; the folding angle F=the maximum angle between limbs; and the bluntness b, which measures relative curvature at the hinge. This system treats asymmetric folds and folds with negative interlimb angles, but it is cumbersome. We propose a method to analyze fold shapes based on Bèzier curves fit to fold half-waves. Bèzier curves are parametric functions of the position coordinates of four 'pulling points.' Two pulling points fall on the fold's inflection points, and therefore fall on its median line. Two other pulling points define a trapezoid that circumscribes the fold. The relative positions of pulling points to define: (1) Fold altitude a=the distance from the median line to a point midway between the pulling points not on the median line. This relates to fold amplitude. (2) Shift angle q=the inclination of the line connecting the midpoint of the median line to a point midway between the pulling points not on the median line. This measures fold asymmetry. (3) Rotation angle a=the inclination of the line connecting the pulling points not on the median line. (4) Fold width d=half the distance between the pulling points not on the median line. This determines the shape of fold. For symmetrical folds (q=90° & a=0°), the ratio d/b, where b=half the distance between inflection points, characterizes fold shape. Distinct values of the d/b ratio correlate directly to distinct b₁/b ₃ values. Thus, we can characterize the shapes of folds as well as Fourier analysis does. For asymmetric folds, our method requires four parameters to define fold shape. This is no improvement over Twiss's analysis, but our method, which utilizes an Excel spreadsheet to calculate parameters, is easier to implement.