$D$-optimal designs for a two-dimensional polynomial model

The influence of an affine transformation of the design space on the number of support points in the $D$-optimal design has been studied for a two-dimensional polynomial regression model. For a full rank model of degree n, a result was obtained that determines the structure of the $D$-optimal plan. It is proven that for a region of design space that is symmetric about zero, the optimal plan is symmetric as well. This result allows for a significant reduction in the dimensionality of the optimization problem and forms the basis of an algorithm developed by the author for finding $D$-optimal plans for models of incomplete rank in nonsymmetric design regions. The D-efficiency of designs concentrated at equidistant points was investigated.