A rapid Bézier curve method for shape analysis and point representation of asymmetric folds

Point representation of fold shapes is useful, in particular, for classification of a large number of folds into different geometric populations. The methods for shape analysis and point representation of asymmetric folds are a few and tedious, although several methods exist for the analysis of the individual fold limbs, or symmetric folds. This article gives a rapid method that uses the Bézier 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 Bézier curve. It makes the length of one Bézier handle zero, pins the end point of the other Bézier handle at the origin of the X-Y frame and drags its control point along the Y-axis to fit the Bézier curve on the given asymmetric fold. A Cartesian plot between normalised length of the Bézier handle and the lift, i.e., difference between the heights of the two inflection 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

Strain estimation from single forms of distorted fossils-a computer graphics and MATLAB approach

Most of the existing methods of strain analysis can estimate strain in a single form of distorted brachiopod, or trilobite provided independent evidence, such as the association of the fossil with cleavage and/or stretching lineation is available for inferring the direction of maximum principal strain. This article proposes a simple computer graphics based method and its MATLAB code that determine the minimum amount of strain in a single distorted fossil form even if data for inferring the maximum principal strain direction are lacking. Our method is a rapid computer-graphics alternative to some of the existing analytical methods. In a distorted fossil form of original bilateral symmetry, the relative senses of angular shears along the hinge line and the median line are mutually opposite to each other. It follows, therefore, that the maximum principal strain direction lies within the acute angle between the hinge and the median lines in the plane of the fossil. Using this principle, our method performs several simulations such that each simulation retrodeforms the distorted fossil by assuming a particular orientation, lying within the acute angle between the hinge line and the median line, as the potential direction of the maximum principal strain. Each simulation of retrodeformation yields a potential strain ratio. The distribution of all the potential strain ratios, obtained by assuming different orientations as the potential directions of the maximum strain, is typically a parabola-like curve with a distinct vertex that corresponds to the minimum amount of strain in the distorted fossil. An entirely computer graphical approach is somewhat time-intensive because it involves a large number of retrodeformational simulations. We, therefore, give a MATLAB code, namely, the Minstrain, that rapidly retrodeforms the fossil and determines the minimum strain with precision