Abstract: Some distribution-free tests for affine symmetry of a continuous bivariate distribution are proposed and studied here. Both rank order tests and tests of the Kolmogorov-Smirnov and Cramer-von Mises type (based on the empirical distribution) are considered and their asymptotic relative efficiency results are studied.

Abstract: Description and purpose The subroutine in this paper is used in the transformation of points in a plane between two referencing graticules. This is but a simplified case of the more general problem of transformations where, for example, points on a curved surface are transformed to a plane in map projections (Maling, 1973), or three-dimensional objects are projected onto a plane in photogrammetry (Masry and Faig, 1977) or architecture (Forrest, 1976). This problem is becoming of increasing importance in environmental sciences where information is captured from maps using digitisers. By use of the conventions of Cartesian coordinates, any point in the plane can be referenced by dropping perpendiculars onto each of the orthogonal axes and reading off the abscissa and ordinate intercept values. Any point P has a locational reference (x, y) relative to the origin (0, 0). If the axes are shifted, scaled, and rotated then the point will likely end up with different coordinate values which we might denote as (x*, y*). Two control points impart sufficient information to map (x, y) into (x*, y*). If the axes suffer a shear in the transformation, then a third control point is required to effect the mapping. An efficient method for a solution in this situation is provided by Saxena (1976). The transformation is complicated if an original shape is stored on an unstable base and the conformal transformation needs to rectify the distortion (Ahuja and Coons, 1968; Holmes et al, 1975). For a number of reasons which will become evident later it is convenient to express the location of a point P in vector form as