Unit III: Space and Geometry — Scholarly Notes

Variational Optics of the Heated Disk, Erlanger Programm & Beltrami–Klein Bridge

1. Fermat’s Principle in the Heated Disk ($n \propto (R^2 - r^2)^{-1}$)

In Chapter IV (FR pp. 84–87 / EN pp. 65–68; Fig. 6, Concordance PC.09), Poincaré endows the open disk $r < R$ (normalized to $R = 1$) with two coupled physical laws:

  1. Thermal Ruler Law: A ruler of nominal length $ds_H$ placed at Euclidean radius $r = \sqrt{x^2 + y^2}$ contracts to Euclidean length $ds_E \propto (1 - r^2)\,ds_H$, inducing the Poincaré conformal metric:

    \[ds_H = \frac{2\,\sqrt{dx^2 + dy^2}}{1 - (x^2 + y^2)} \quad (K = -1).\]

  2. Optical Refraction Law: With refractive index $n(x, y) = \frac{2}{1 - (x^2 + y^2)}$, Fermat’s principle of least optical time $\delta \int n(x, y)\,ds_E = 0$ minimizes the exact same hyperbolic arc-length functional $\int ds_H$! Hence every optical ray in Poincaré’s heated enclosure is an exact circular arc orthogonal to $\partial\Omega$, constructed symbolically over $\mathbb{Q}$ in geometor.poincare.disk.construct_heated_disk_optical_rays().

2. Klein’s Erlanger Programm & the Straightedge Beltrami–Klein Bridge (Fig. 7)

Chapter IV’s thesis—that “the object of geometry is the study of a particular group”—is Poincaré’s epistemological synthesis of Felix Klein’s 1872 Erlanger Programm (Klein 1872, digitized in geometor/klein) and Sophus Lie’s classification of Riemann–Helmholtz–Lie mobility groups (cf. Helmholtz 1868).

How does Klein and Beltrami’s projective disk model (Beltrami 1868, interactive CindyJS applets in geometor/codes), where hyperbolic lines are straight Euclidean chords $UV$, relate to Poincaré’s conformal disk model, where hyperbolic lines are orthogonal circular arcs $\overset{\frown}{UV}$?

In geometor.poincare.disk.construct_beltrami_klein_bridge() (Fig. 7, Concordance PC.10), we prove a remarkable purely algebraic straightedge theorem over $\mathbb{Q}$:

  • Let $U, V \in \partial\Omega$ be the shared ideal endpoints of a hyperbolic line, so the orthogonal Poincaré circle satisfies $x^2 + y^2 - 2(c_x x + c_y y) + 1 = 0$ and the radical axis (straight Beltrami–Klein chord $UV$) satisfies $c_x X + c_y Y = 1$.

  • For any point $P = (x, y)$ on the Poincaré circular arc $\overset{\frown}{UV}$, $c_x x + c_y y = \frac{1 + x^2 + y^2}{2}$.

  • Therefore, projecting $P$ radially from the disk center $O(0, 0)$ via the Weierstrass map

    \[K(P) = \frac{2P}{1 + \|P\|^2} = \left(\frac{2x}{1 + x^2 + y^2},\; \frac{2y}{1 + x^2 + y^2}\right)\]

    satisfies $c_x K_x + c_y K_y = \frac{2(c_x x + c_y y)}{1 + x^2 + y^2} = 1$, proving that the Euclidean straight line $OP$ meets the straight chord $UV$ at the exact Beltrami–Klein image $K(P)$! Furthermore, the conformal quadrance $\tau(P_1, P_2)$ induces the exact Cayley–Klein chordal quadrance $q_{\mathrm{CK}}(K_1, K_2) = \frac{4\tau}{(1 + \tau)^2}$ on the projective model.