§§ 6–7: Geometry of Reciprocal Radii and Lie’s Sphere Geometry — Summary¶
Analytical Synopsis of Conformal Möbius Geometry, the Riemann Sphere, and Pentaspherical Contact
§ 6. Geometry of Reciprocal Radii (Möbius Inversion) & Complex Binary Forms¶
Structural Parallel Between Projective and Inversion Geometry: Whereas projective geometry adjoins a plane (or line) at infinity and treats circles/spheres as special conics/quadrics relative to the imaginary circle at infinity, the geometry of reciprocal radii (Möbius inversion) adjoins a single point at infinity $\infty$ and treats straight lines and planes as circles and spheres passing through $\infty$. Fixing $\infty$ immediately recovers the Euclidean Principal Group.
Projective Quadric Model: Stereographic projection identifies $2$-dimensional inversion geometry with projective geometry on a smooth quadric surface in $\mathbb{P}^3$, and $3$-dimensional inversion geometry with projective geometry on a $3$-quadric in $\mathbb{P}^4$ ($5$ homogeneous variables).
Complex Variable $z = x + iy$ & the Riemann Sphere: In the real plane, fractional linear transformations $z' = \frac{\alpha z + \beta}{\gamma z + \delta}$ (together with anti-conformal $\bar{z}$ inversions, added by Klein in 1893 fn. 26) generate the full planar Möbius group, identifying complex binary invariant theory with real projective geometry on the Riemann sphere.
§ 7. Two Higher Circle/Sphere Extensions and Lie’s Sphere Geometry¶
Transferring planar circles to spatial planes cutting the unit sphere $x_1^2 + x_2^2 + x_3^2 + x_4^2 = 0$, Klein extends inversion geometry in two distinct ways:
Non-Contact Circle Geometry ($\operatorname{PGL}_4$ on Circles): Allowing all spatial collineations (dropping the sphere) induces planar circle transformations mapping pencils and orthogonal bundles of circles to pencils and bundles, isomorphic to ordinary $3$-space projective geometry (circles form a $3$-dimensional Körper, whereas points are no longer preserved).
Lie’s Contact Sphere/Circle Geometry ($\operatorname{PO}_5$ / $\operatorname{PO}_6$): Introducing the oriented radius coordinate $i u_5 = \sqrt{u_1^2 + u_2^2 + u_3^2 + u_4^2}$ embeds oriented circles (including points as radius-$0$ circles and oriented lines) into the quadric $\sum_{k=1}^5 u_k^2 = 0$ (and spatial oriented spheres into $\sum_{k=1}^6 u_k^2 = 0$ in $\mathbb{P}^5$). Linear transformations preserving this quadric form Lie's sphere geometry, a genuine group of contact-transformations preserving oriented tangency of curves and surfaces while mixing points, lines/planes, and circles/spheres into a single transitive Körper.