AbstractA conformal (orthomorphic) mapping projection of the spheroid can be constructed to give minimum scale error over a given arbitrary area, and in this respect has an advantage over more regular projections such as the transverse Mercator or the Lambert conformal conic. Geodetic coordinates on the spheroid are first transformed into isometric coordinates, and the latter are then transformed into the rectangular cartesian coordinates of the desired projection by means of a polynomial expression in complex variables. The total distortion of the projection is expressed as the integral of the squared scale error over the given area. After fixing the values of the rectangular coordinates and of the meridian convergence at the origin of the projection, the remaining coefficients of the complex polynomial are adjusted to minimise the total distortion. This set of coefficients can be used directly in formulae to carry out the direct and inverse transformations between geodetic and rectangular coordinates, and to calculate the scale factor, the meridian convergence, and the geodesic curvatures of projected curves (including Ineridians and parallels) at any point. In the reduction of the observations of local surveys in rectangular coordinates, the minimum scale error property means that corrections to bearings and distances are often negligible, or if required they can be interpolated from small-scale contour maps.As an example, coefficients have been calculated for a projection designed to give minimum distortion over the land area of New Zealand, using a complex polynomial to order six. The range of scale error for this projection is about 4×10−4, less than can be obtained with any conventional projection.