Dynamical geometric visualization of the generalized Stokes vectors for 3-D Stokes polarimetries

Three-dimensional (3-D) Stokes polarimetry is becoming increasingly important in advanced optics, with the nine-component generalized Stokes vector (GSV) serving as its native output. However, in contrast to conventional 2-D polarimetry, which uses the Poincare sphere and polarization ellipse for visualization, 3-D polarimetry still lacks a geometric framework for dynamically visualizing full GSV-encoded polarization structures. This article introduces two complementary GSV-based geometries. The first is the 3-D polarization spectral geometry, which is constructed from two algebraically independent Casimir-type 3-D polarization invariants and provides a parameterization of the 3-D polarization spectrum. The second is a triadic Poincar'e sphere (TPS) geometry, where the GSV is decomposed into three R^3 sub-vectors whose directions and Euclidean scales characterize the 3-D polarization textures and define the total degree of polarization (TDOP), respectively. The two geometries provide an effective dynamical visualization strategy for 3-D polarization structures and establish a geometric basis for GSV-based polarimetry.