Using the topological equivalence between the Riemann sphere \(\mathbb {S}\) and the extended complex plane \(\overline{\mathbb {C}} = \mathbb {C} \cup \{\infty \}\) , where \(\mathbb {C}\) is the field of complex numbers, we establish 2D-bijective representations of 3D point clouds. Points of 3D point clouds are mapped into the Riemann sphere \(\mathbb {S}\) , and a stereographic projection is implemented to map the points into the complex plane \(\mathbb {C}\) . The way the 3D objects are mapped into \(\mathbb {S}\) may be varied for various applications. To prove the accuracy and efficiency of the proposed 2D representation of 3D objects, we apply this correspondence to 3D point cloud encryption. We utilize chaotic permutations, chaotic circuits, and Latin cubes in addition to the stereographic projection representation to construct our scheme. The permutation steps using chaotic maps and Latin cubes are carried out on the object data points in both \(\mathbb {S}\) and \(\overline{\mathbb {C}}\) , while the chaotic circuits are applied to 2D projections of the 3D objects. To the best of our knowledge, no earlier work employed stereographic projections for 3D object encryption. Experimental simulations of this method show high encryption strength and strong confusion and diffusion properties based on quantitative and statistical measures.
