Stereographic Projections for Designs on the Sphere

This paper is concerned with the use of the stereographic projection to map the points of a design on the sphere in three dimensions onto a two-dimensional stereogram. Details of the projection and its attendant stereogram are given and the application of the constructs to designs on the sphere is highlighted. Spherical $t$-designs are then introduced and stereograms of selected examples are presented. In addition, examples of regression models with design space the unit ball are considered, stereograms which represent the spherical component of such designs are given and an example in which stereograms are used to elucidate the geometric isomorphism of a suite of designs generated by an exchange algorithm is presented. Emphasis throughout the text is placed on the use of the stereographic projection in teaching and research within the framework of statistics.