Understanding the transition between the projective disk and Poincaré disk models is vital in various contexts, we propose analogous conversion from the Poincaré disk model to the projective disk model using a distinct approach. Within the framework of \(\text {SL}(3,{\mathbb {R}})\) action from an Erlangen perspective, this paper explores the correspondence mapping between the Poincaré disk and the projective unit disk. We investigate the fixed subgroup of the projective unit circle in \(\mathbb{R}\mathbb{P}^2\) and under specific conditions, an isomorphism between the isotropy subgroup of the projective unit circle and \(\text {PSL}(2,{\mathbb {R}})\) is established by using the Iwasawa decomposition of the Lie group \(\text {SL}(2,{\mathbb {R}})\) . This results in mappings from the projective plane to the upper half planes of elliptic, parabolic, and hyperbolic types, each corresponding to the actions of \(\text {SL}(2,{\mathbb {R}})\) geometry. Additionally, we derive a mapping from the projective disk to the conformal disk within the complex plane, which in turn provides a geometric interpretation of the resulting mapping. We anticipate that this conversion possesses the potential to reveal valuable insights, encouraging exploration of the multidimensional case as well.
