Unifying Projective Geometry: Polarity, Circular Points, and Poncelet Orthocenter Loci — E8 Intelligence Research

FINDING: Projective geometry's polarity/involution framework unifies conic duality, circular points at infinity, and isogonal conjugation — with a new result showing Poncelet triangle orthocenter loci are homothetic to the generating ellipse. | MATH: Projective plane over reals: points at infinity form line \\(L_\\infty\\); circular points \\(I=(1:i:0), J=(1:-i:0)\\) lie on \\(L_\\infty\\). A polarity w.r.t. conic \\(C\\) maps point \\(P\\) to line \\(p\\) via \\(p = C^{-1}P\\) (in homogeneous coordinates). Involution on a line: \\(x \\mapsto (ax+b)/(cx-a)\\), trace-zero condition \\(a^2+bc=0\\) for projective involution. Isogonal conjugation in triangle = composition of two polarities w.r.t. the two circular points (a classical result). Poncelet result (arXiv:2508.02368): For nested ellipses \\(\\mathcal{E}, \\mathcal{E}_c\\), orthocenter locus is \\(\\mathcal{E}\\) homothetic by factor \\(k\\) and rotated \\(90^\\circ\\); isogonal conjugate of fixed point traces a conic. | CONNECTION: The circular points \\(I,J\\) are Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com