Application of Bezier Splines to the Analysis of Folds

We present a semi-analytical method for fitting Bezier splines to half-wavelength fold profiles and examine its utility for characterizing fold shapes. Bezier splines consist of polynomial parametric equations whose coefficients are defined by two anchor points, located at the ends of the curve, and two intermediate points that “pull” the curve on its path between the anchor points. Our semi-analytical fitting technique minimizes total distance from points in a half-wave profile to a spline by iterative estimation of the path locations and the pulling points. Experimental results from a variety of folds indicate that the method generally converges to the best estimate in a few iterations.

We use this method to analyze half-wavelength folds and outline a method of characterizing fold shapes based on the characteristics of the best-fit third-order Bezier spline. Our algorithm places anchor points at inflection points in a fold profile and finds optimum positions of pulling points. Their relative positions fix four fold properties: (1) Fold altitude, the length of a line extending from a point midway between the anchor points to a point midway between the pulling points, which corresponds to fold amplitude; (2) Shift angle, the inclination of the line used to define fold altitude; (3) Rotation angle, the inclination from horizontal of the line connecting the pulling points; and lastly, (4) Fold width, which is half the distance between the two pulling points. These four parameters fully characterize fold shape and can readily be compared to previous fold characterization schemes. Because Bezier functions are parametric, they can represent profiles of folds of any orientation or asymmetry. Fold trains can be represented by either (a) higher order Bezier splines which have more pulling points or (b) a series of cubic splines which share anchor points (i.e. "patched" splines).