Lie's line–sphere correspondence (Lie quadric)
Statement
There is a natural contact-geometric correspondence between oriented lines in Euclidean 3-space and points on a quadric hypersurface in projective 5-space (the Lie quadric), extending Plücker line coordinates and encoding incidence of lines as contact structure. It underlies sphere geometry, minimal surfaces, and modern Lie-theoretic treatments of conformal and Lagrangian geometry.
Formula
ℝP³ as space of lines ⊃ Plücker quadric; Lie quadric Q ⊂ ℝP⁵ with contact structure ↔ line–sphere duality
Proof sketch
Synthetic and analytic constructions in line geometry; see classical sources (Lie, Klein) and modern treatments of the Lie quadric in contact topology and conformal geometry.
Source
Lie, M. S. Über Komplexe, insbesondere Linien- und Kugelkomplexe, mit Anwendung auf die Theorie partieller Differentialgleichungen, Math. Ann. 5 (1872), 145–256. See also Fischer, Mathematical Models, and modern surveys on the Lie quadric.
Domains
mathematics
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- Theorem: Lie's line–sphere correspondence (Lie quadric)
- Proof: Lie's line–sphere correspondence (Lie quadric)