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

Abstract

The cone projection $f_R(z)=z/(1+|z|/R)$ 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 $1/|f_R(z)|=1/|z|+1/R$. Its main Euclidean feature is the \emph{Self-Directrix Theorem}: every line $\ell$ not through the origin maps to the focus-side arc of the conic with focus $O$, directrix $\ell$ \emph{itself}, eccentricity $R/d$, and semi-latus rectum $R$, so the single distance $d=\operatorname{dist}(O,\ell)$ 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 $1/e\mapsto1/e+\delta/R$; 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.