An Algebraic Unification of Astrolabe-HyperG Algebra with Lorentz Geometry via Stereographic Projection and Hessian Duality

This paper introduces the Astrolabe-HyperG algebra ahg ( 2 , 1 ) , a four-dimensional reductive Lie algebra unifying stereographic projection with Lorentzian geometry. The hyper G-matrix decomposition A − T = D 1 B D 2 , with diagonal calibrations D 1 , D 2 and B normalizable to S L ( 2 , R ) up to reflection, is interpreted via the astrolabe’s rete, alidade, and tympan. The algebra is isomorphic to sl ( 2 , R ) ⊕ R ; its semisimple part is so ( 2 , 1 ) , but the full algebra is not. This paper introduces the Astrolabe-HyperG algebra ahg ( 2 , 1 ) , a four-dimensional reductive Lie algebra unifying stereographic projection with Lorentzian geometry. The hyper G-matrix decomposition A − T = D 1 B D 2 , with diagonal calibrations D 1 , D 2 and B normalizable to S L ( 2 , R ) up to reflection, is interpreted via the astrolabe’s rete, alidade, and tympan. The algebra is isomorphic to sl ( 2 , R ) ⊕ R ; its semisimple part is so ( 2 , 1 ) , but the full algebra is not. A lemma links the bordered Hessian of constrained optimization to this decomposition. The Fan product of dual matrices extends the structure, while conic spectral decomposition yields spacelike, timelike, and lightlike cases under the duality between conics and dual vectors. Applications to geometric optics and planar robotics illustrate the theory. Three conjectures and the Astrolabe-HyperG Principle are stated. Applications to geometric optics and planar robotics illustrate the theory. Three conjectures and the Astrolabe-HyperG Principle are stated.