We analyze the bulk elastic transformation of volume holograms as a general approach for three-dimensional pupil engineering. The physical relationship between transformation and the resulting point spread function is discussed by deriving the corresponding analytical expressions. For affine transformations, an analytical solution is directly possible. However, for nonaffine transformations, the analytical solution is not straightforward and we must turn to quasi-analytical solutions using the approximation of the stationary phase. Transformational volume holography offers richer design flexibility and real-time adjustment capabilities for imaging systems.