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.

• The paper introduces the 'Self-Directrix Theorem', which states that the radial map $f_R(z) = z/(1+|z|/R)$ transforms lines into arcs of conics with specific focal properties.
• The authors present an algebraic analysis, showing that the family of these maps forms a semigroup under composition and a partial group under signed curvature.
• The map yields a flat, finite-area metric on the plane, reducing the cone construction's geometric relevance to a disk modeled on Euclidean space, while the reciprocal lens identity aligns with specific projective configurations.

The paper studies the radial map $f_R(z)=z/(1+|z|/R)$, which the author derives from an elementary three-dimensional construction: joining a point of the complex plane to the center of a cone's base, intersecting that segment with the cone's lateral surface, and projecting orthogonally back to the plane. The same formula appeared earlier as a bounded, ray-preserving activation function for complex-valued neural networks (Georgiou and Koutsougeras, 1992), where it acts as a complex analogue of the real sigmoid; the present work deliberately sets that context aside and develops the map's geometry on its own terms. The central contribution is the Self-Directrix Theorem: $f_R$ maps every line not through the origin onto an arc of a conic whose focus is the origin and whose directrix is the original line itself. A generalization to all focal polar loci, together with an algebraic, projective, dynamical, metric, and axiomatic analysis of the map family, completes the framework.

The construction and height independence

Fix a right circular cone of base radius $R$ and height $h$, apex at the origin. For a point $z$ in the plane containing the apex, draw the segment from $z$ to the base center $(0,h)$; let $P$ be its intersection with the lateral surface, and let $f(z)$ be the foot of the perpendicular from $P$. Working in the axial cross-section through $f_R$0, two pairs of similar triangles yield the ratios $f_R$1 and $f_R$2; eliminating $f_R$3 and $f_R$4 gives

$f_R$5

and since $f_R$6 lies on the ray $f_R$7, the vector formula $f_R$8 follows. The result is independent of the cone's height — only the base radius $f_R$9 matters.

Regularity is treated carefully: because the formula involves $R$0, the map is smooth off the origin but not holomorphic anywhere. In real coordinates the Jacobian extends continuously across the origin with $R$1, so $R$2 is globally $R$3; however, one-sided second derivatives along any ray disagree ($R$4 versus $R$5), so $R$6 fails to be $R$7 at the origin. All "diffeomorphism" claims in the paper are therefore to be read in the $R$8 sense. The inverse $R$9 on the disk $h$0 has identical regularity.

The reciprocal lens identity

Inverting the magnitude law gives the identity that organizes nearly everything else:

$h$1

This is the same reciprocal-addition rule governing parallel resistors, series springs, and reduced mass, and it is formally analogous to a thin-lens equation under a specific sign convention — an analogy the author explicitly restricts to reciprocal-distance algebra rather than physical optics. Defining the "cone curvature" $h$2 (not to be confused with Gaussian curvature, which vanishes on the lateral surface away from the apex), the projection adds curvature $h$3 to the inverse modulus of every point.

The Self-Directrix Theorem

Since $h$4 depends on $h$5, it is neither holomorphic nor conformal (radial stretch $h$6 differs from tangential stretch $h$7 for all $h$8), so there is no a priori reason for it to preserve any classical curve family. The main theorem establishes that it nonetheless does:

Self-Directrix Theorem. For a line $h$9 at distance $z$0, $z$1 is an arc of the conic $z$2: focus $z$3, directrix $z$4 itself, eccentricity $z$5, semi-latus rectum $z$6. The image lies strictly between $z$7 and $z$8 on each ray, so the input line reappears unchanged as the directrix of its own image — hence "self-directrix."

The proof is short and clean: writing $z$9 and using the inverse relation, the condition $z$0 becomes $z$1, whence $z$2, i.e., $z$3. The single parameter $z$4 produces the full trichotomy: ellipse if $z$5, parabola if $z$6, hyperbola if $z$7. Throughout the sweep the focus stays at $z$8 and the semi-latus rectum stays fixed at $z$9, so all image conics pass through the same latus-rectum endpoints $(0,h)$0 (with $(0,h)$1 a unit vector parallel to $(0,h)$2); these endpoints are limiting, never attained, since $(0,h)$3 is open. In the hyperbolic case the image reaches only the central portion of the focus-side branch.

The Confocal–Codirectrix Theorem

A line is the degenerate member of a larger family. Fixing focus $(0,h)$4, axis direction $(0,h)$5, and directrix distance $(0,h)$6 determines a pencil of focal polar loci $(0,h)$7 parametrized by eccentricity. Rewriting reciprocally,

$(0,h)$8

exposes the mechanism: the lens identity shifts only the constant term $(0,h)$9, leaving the cosine coefficient untouched. Consequently:

Confocal–Codirectrix Theorem. $P$0 carries each focal arc $P$1 onto $P$2 over the same angular subarc, with $P$3 and $P$4. The focus, axis, and directrix are invariant ($P$5), while the eccentricity strictly decreases via $P$6.

Three cases arise. For an ellipse ($P$7), the image is the entire carrier ellipse, lying strictly inside $P$8. For a parabola ($P$9), the carrier of the image is an ellipse internally tangent to $f(z)$0, with exactly one point removed — the tangency point at distance exactly $f(z)$1. For a hyperbola branch ($f(z)$2), the image is an open arc with limiting endpoints on $f(z)$3 in the asymptotic directions. Here "confocal" means sharing the single distinguished focus $f(z)$4 and its directrix, not both foci classically; since $f(z)$5 changes, the second focus moves.

Two structural remarks deserve emphasis. First, the Self-Directrix Theorem is recovered as the limiting member $f(z)$6 of each fixed-directrix pencil, showing the phenomenon is not special to lines. Second, iteration sharpens the picture: since $f(z)$7, the $f(z)$8-fold image has parameters $f(z)$9 with $P$0 fixed, so carriers become asymptotically circular while collapsing to $P$1 — the limit is the point, not a nondegenerate circle. A corollary gives a clean criterion: the carrier of the image is elliptical whenever $P$2, regardless of the input type.

Algebraic structure: semigroup, partial group, flow

Composition follows immediately from the lens identity: $P$3 with $P$4. Thus $P$5 is an abelian semigroup isomorphic to $P$6 via curvature; radii combine by the parallel law $P$7. It is not a group — no identity exists among the $P$8.

Extending to signed curvature yields radial Möbius maps $P$9 on maximal domains $f_R$00. These form a one-parameter partial group: $f_R$01 holds exactly on the intersection domain, and each $f_R$02 is a homeomorphism $f_R$03. The qualifier matters: positive curvatures compress the plane into a disk, negative ones expand a disk to the plane, so no single space carries the whole group action.

Iteration gives $f_R$04 with $f_R$05 — decay rate independent of starting radius. The continuous interpolation $f_R$06 is the global forward flow of the vector field

$f_R$07

which is globally $f_R$08 but not $f_R$09 at the origin. Backward solutions blow up in finite time $f_R$10. In the reciprocal-radius coordinate $f_R$11 the flow becomes uniform translation $f_R$12 — the dynamical form of the lens identity. Notably, $f_R$13 is not a Banach contraction: its derivative tends to the identity at the origin and its global Lipschitz constant is 1; the contraction toward $f_R$14 is orbit-wise, asymptotically $f_R$15, not exponential.

Projective structure on rays

The radial action $f_R$16 is the one-dimensional Möbius transformation with matrix $f_R$17. Hence cross-ratios of quadruples on a common ray are preserved — a projective signature of the construction, though emphatically not a statement about arbitrary collinear quadruples in the plane, since on a full line through $f_R$18 the dependence on $f_R$19 breaks the signed-coordinate Möbius property. A special case connects forward and backward images: for $f_R$20, the four points $f_R$21, $f_R$22, $f_R$23, $f_R$24 form a harmonic range, $f_R$25. This is the projective face of the symmetric reciprocal-radius shifts $f_R$26.

Normalized to determinant 1, the matrix $f_R$27 is a parabolic element of $f_R$28, conjugate via $f_R$29 to the horizontal translation $f_R$30; it preserves the horocycles based at its fixed boundary point $f_R$31. The author is careful to scope this correctly: the hyperbolic interpretation concerns the raywise radial coordinate only, not $f_R$32 as a planar conformal map.

Alternative constructions

Several equivalent realizations of the lens identity are collected. A purely planar trapezoid construction recovers $f_R$33 as half the harmonic mean of the bases — the classical crossed-ladders configuration — with base-width independence mirroring height independence. Through unit-circle inversion $f_R$34, the map factors as $f_R$35, where $f_R$36 is radial translation by $f_R$37: the cone projection is inversion-conjugated rigid translation, which explains why both the raywise reciprocal-radius law and the transversal $f_R$38-shift are translations.

Higher dimensions, metric, and characterization

All results extend verbatim to $f_R$39: the same cone construction yields $f_R$40, a $f_R$41 homeomorphism onto the ball $f_R$42, $f_R$43-equivariant, with the composition law, partial group, flow, and cross-ratio properties transferring directly. Hyperplanes map to patches of quadrics of revolution with the same focus/directrix/eccentricity behavior, proved meridian-by-meridian.

Metrically, declaring $f_R$44 an isometry pulls the Euclidean disk metric back to a flat length metric on the plane,

$f_R$45

under which $f_R$46 has diameter $f_R$47 (unattained), total area $f_R$48, and completion the closed disk. This complements stereographic compactification: there the boundary at infinity is a single point with round metric; here it is a circle with flat metric and boundary circles of intrinsic length $f_R$49.

Finally, a characterization theorem shows the axioms determine the map uniquely: any continuous ray-preserving map whose radial action is a real Möbius transformation fixing $f_R$50, with finite positive limit $f_R$51 at infinity and tangent to the identity at the origin, must equal $f_R$52. The normalizations are necessary — dropping ray preservation admits radius-dependent twists, and dropping the first-order condition leaves a one-parameter residual family.

Limitations and open questions

The paper is candid about scope. The map is only $f_R$53 at the origin, so the pullback tensor does not give a smooth Riemannian metric globally, and all diffeomorphism statements are $f_R$54-sense. The optical analogy is formal only, under a nonstandard sign convention. The hyperbolic case requires separate treatment of the far branch, whose image obeys a different sign law; in the special case $f_R$55 it maps onto the open chord of the directrix inside $f_R$56 rather than the full line, and no full-conic version of the hyperbolic case can hold since a homeomorphism cannot identify a two-component locus with a one-component one. The neural-network origins are acknowledged but not pursued; whether the self-directrix geometry has consequences for complex-valued learning dynamics is left unexamined. Open questions include the behavior of the far hyperbolic branch beyond the special case $f_R$57, and possible characterizations replacing the strong Möbius hypothesis in the axiomatic theorem with weaker regularity assumptions.

Conclusion

The paper demonstrates that a single elementary radial map, derivable from a height-independent cone construction and governed throughout by the reciprocal lens identity, carries substantial coherent structure: closure under composition with curvature addition, a partial-group flow generated by $f_R$58, raywise projective and harmonic structure, a flat finite-area metric realizing the plane as a disk, and — centrally — the Self-Directrix and Confocal–Codirectrix Theorems, in which the input curve's focus and directrix survive the projection while eccentricity alone decreases. The treatment is rigorous about regularity, domain restrictions, and the precise arcs attained, and the axiomatic characterization confirms that the cone projection is not one example among many but the unique map satisfying its defining geometric conditions.