The Cone Projection $f(z)={\displaystyle \frac{z}{1+\mathrm{\mid }z\mathrm{\mid }\mathrm{/}R}}$: Geometric structure and the Self-Directrix Theorem

The cone projection $f_R(z)=z\mathrm{/}(1+\mathrm{\mid }z\mathrm{\mid }\mathrm{/}R)$ is a radial homeomorphism from $\mathbb{C}$ onto the open disk $D_R$ of radius $R$, obtained by an elementary cone-and-perpendicular construction (independent of the cone’s height) and governed by the reciprocal lens identity $1\mathrm{/}\mathrm{\mid }{f}_R(z)\mathrm{\mid }=1\mathrm{/}\mathrm{\mid }z\mathrm{\mid }+1\mathrm{/}R$. Its main Euclidean feature is the \emph{Self-Directrix Theorem}: every line $\mathrm{\ell }$ not through the origin maps to the focus-side arc of the conic with focus $O$, directrix $\mathrm{\ell }$ \emph{itself}, eccentricity $R\mathrm{/}d$, and semi-latus rectum $R$, so the single distance $d=\mathrm{dist}(O,\mathrm{\ell })$ fixes the ellipse/parabola/hyperbola trichotomy. The \emph{Confocal–Codirectrix Theorem} extends this from lines to every focal polar locus of a fixed focus–directrix pencil, keeping the focus and directrix while lowering the eccentricity by $1\mathrm{/}e\mapsto 1\mathrm{/}e+\delta \mathrm{/}R$; the image of a circle, by contrast, is generally a circular quartic rather than a conic. The same lens identity organizes the remaining structure: a curvature-additive composition law and its flow, a raywise cross-ratio structure, and higher-dimensional, metric, and axiomatic results.