We previously demonstrated that radial basis functions may be preferred as a descriptor of free-form shape for a single mirror magnifier when compared to other conventional descriptions such as polynomials [Opt. Express 16, 1583 (2008)]. A key contribution is the application of radial basis functions to describe and optimize the shape of a free-form mirror in a dual-element magnifier with the specific goal of optimizing the pupil size given a 20 degrees field of view. We demonstrate a 12 mm exit pupil, 20 degrees diagonal full field of view, 15.5 mm eye clearance, 1.5 arc min resolution catadioptric dual-element magnifier design operating across the photopic visual regime. A second contribution is the explanation of why it is possible to approximate any optical mirror shape using radial basis functions.