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

Abstract

The \emph{cone projection} $f_R(z)=z/(1+|z|/R)$ arises from an elementary spatial construction: join a point of the complex plane to the center of a cone's base, mark where that segment meets the lateral surface, and drop a perpendicular back to the plane. The resulting point is independent of the cone's height, so the construction defines a radial homeomorphism $f_R:\mathbb{C}\to D_R$ onto the open disk of radius $R$, governed by the reciprocal lens identity $1/|f_R(z)|=1/|z|+1/R$. The main Euclidean result is the \emph{Self-Directrix Theorem}: $f_R$ carries every line $\ell$ not through the origin onto an arc of a conic with focus $O$, directrix $\ell$ \emph{itself}, eccentricity $R/d$, and semi-latus rectum $R$. The single distance $d=\operatorname{dist}(O,\ell)$ determines ellipse, parabola, or hyperbola. Its generalization, the \emph{Confocal--Codirectrix Theorem}, carries each focal polar locus (the whole ellipse or parabola, and in the hyperbolic case the focus-side branch) to a focal arc (possibly the whole carrier ellipse) that keeps the focus $O$ and the directrix while strictly lowering the eccentricity, by $1/e\mapsto1/e+δ/R$. The same reciprocal lens identity organizes the rest: the family $\{f_R\}_{R>0}$ is closed under composition (curvatures add), extends to a one-parameter partial group with flow $\dot z=-|z|z/R$, and preserves cross-ratios along rays. Higher-dimensional, metric, and axiomatic results close the paper.