<p>Thin annular membranes under torsional loading exhibit periodic wrinkling phenomena, leading to surface irregularities. In this study, we introduce a homeomorphic transformation approach to equivalently address the torsional wrinkling of circular membranes by mapping them to rectangular domains. We establish a numerical analysis framework that integrates an implicit difference method with a perturbation technique to solve the nonlinear governing equations. This framework effectively connects the reference rectangular topology with the target annular topology. Based on this geometric equivalence, we analyze the correlation between rectangular shear and annular torsion. Furthermore, under a variable separation assumption, we derive semi-analytical and semi-empirical explicit expressions specifically for the critical load and critical wavelength of torsional wrinkling. The proposed homeomorphic transformation significantly extends the applicability of traditional difference methods. Comparisons with numerical results and existing literature demonstrate the accuracy of the derived expressions.</p>