Theories of strain analysis from shape fabrics: A perspective using hyperbolic geometry

A parameter space is proposed for unifying the theories of two-dimensional strain analysis, where strain markers are approximated by ellipses with a prescribed area. It is shown that the theories are unified by hyperbolic geometry, the oldest and simple non-Euclidean geometry. The hyperboloid model of the geometry is used for this purpose. Ellipses normalized by their areas are represented by points on the unit hyperboloid, the curved surface in a non-Euclidean space. Dissimilarity between ellipses is defined by the distance between the points that represent the ellipses. The merit of introducing the geometry comes from the fact that this distance equals the doubled natural strain needed to transform one ellipse to another. Thus, the introduction is natural and convenient for strain and error analyses. Equal-area and gnomonic projections of the hyperboloid are introduced for the Rf/ϕ and kinematic vorticity analyses, respectively. In our formulation, the strain ellipse optimal for a set of Rf/ϕ data is obtained as the centroid of the points corresponding to the data on the hyperboloid, and the dispersion of the points shows the uncertainty of the optimal strain. By means of a bootstrap method, the confidence region of the strain is drawn upon the surface, and equal-area projection from the surface to a Euclidean plane shows the dispersion of the points and the size of the confidence region. In addition, our formulation provides a new graphical technique for kinematic vorticity analysis using the gnomonic projection. The technique yields the optimal kinematic vorticity number with its uncertainty.