Methods

Methods

Sep 12, 2025

Methods and Their Scope

Current Binary Radial Kernels

The current formulation defines regular equal sectors, local chord–tangent lengths, triangular areas, and weighted kernels. Its proofs concern those definitions.

Earlier Methods

Algorithm Specifications

A runnable method must identify initial values, refinement or direct evaluation, arithmetic format, outputs, and stopping conditions. A geometry identity alone is not a runtime benchmark.

Extensions

Adaptive sampling, general curves, multiple seeds, and three-dimensional constructions require their own definitions and validation. No error order or performance advantage is transferred automatically from the regular-circle proof.

Foundations

Foundations

Sep 12, 2025

Document Status and Current Reference

This page preserves an earlier formulation or research method. Its notation, formulas, and embedded demonstrations belong to that version and should not be treated as the specification of the current binary radial kernels.

Read the current formulation and its analytical verification. Historical performance statements and implementation behaviour require separate validation. Earlier references to circular constants remain in this historical document; they are not inputs to the current geometric kernel definitions.

Mathematical Foundations: Turns Trigonometry & Radial Discretization

Category: Generative Radial Geometry (GRG)

GRG measures angles in turns with period‑1 primitives c(u), s(u) (no π assumed in discrete steps). Constructions proceed on quarters/sectors, summing center‑seeded primitives (chords and radial strips). Below are the core identities used across methods.

Turns Trigonometry (period‑1)

Let a full rotation be one turn. The functions c(u), s(u) are period‑1 and satisfy the usual symmetries and addition rules:

$$ c(u+1)=c(u),\quad s(u+1)=s(u),\quad c(-u)=c(u),\quad s(-u)=-s(u) $$
$$ c(u+v)=c(u)c(v)-s(u)s(v),\qquad s(u+v)=s(u)c(v)+c(u)s(v) $$

Quarter symmetry and parity yield the relations needed for GRG’s inside‑out sums, without introducing π in the discrete formulas.

Chord, Sagitta, and Radial Strip Area

For a unit circle with step Δu between consecutive samples u_L and u_R=u_L+Δu:

$$ \operatorname{chord}(\Delta u)=2\,s\!\left(\tfrac{\Delta u}{2}\right) $$
$$ \operatorname{sagitta}(\Delta u)=1-c\!\left(\tfrac{\Delta u}{2}\right) $$

The area of the center‑seeded strip (triangle O–p(u_L)–p(u_R)) equals half the cross‑product magnitude, i.e. a period‑1 sine of the step:

$$ A_{\text{strip}}(\Delta u)=\tfrac{1}{2}\,\Big(c(u_L)\,s(u_R)-s(u_L)\,c(u_R)\Big) =\tfrac{1}{2}\,s(\Delta u) $$

Discrete Perimeter and Area on a Quarter

With uniform steps Δu=\tfrac{1}{4N} on a quarter (N samples):

$$ P_{\tfrac14}(N;\text{rule})=\sum_{k=1}^{N}\operatorname{chord}(\Delta u),\qquad P(N)=4\,P_{\tfrac14}(N;\text{rule}) $$
$$ A_{\tfrac14}(N)=\sum_{k=1}^{N}\tfrac{1}{2}\,s(\Delta u)=\tfrac{N}{2}\,s\!\left(\tfrac{1}{4N}\right),\qquad A(N)=4\,A_{\tfrac14}(N) $$

Rule choice (midpoint / right‑endpoint / averaged) governs bias for arc length; strip areas use the exact triangle formula per step, remaining π‑free at finite N.

π as an Emergent Limit (turns)

From the full polygon with Δu=1/N:

$$ \pi=\lim_{N\to\infty} N\,s\!\left(\tfrac{1}{2N}\right) $$

Background & Positioning

Category: Generative Radial Geometry (GRG)

Generative Radial Geometry (GRG) builds shapes from the inside via discrete radial scans. Measurements arise by summing chords and radial trapezoids without assuming π in advance. Angles are measured in turns (period‑1), using primitives c(u), s(u) with u \in [0,1). We present GRG as a digital‑native (2025) center‑seeded approach—π‑free and complementary to classical, formula‑first geometry.

Motivation

Classical geometry is top‑down, continuous, formula‑centric. GRG is inside‑out: pick a resolution N, rotate a central ray with step Δu, and accumulate discrete contributions that converge as N grows.

Key Idea (Quarter Aggregation)

Build a quarter from center‑seeded primitives, then use symmetry.

$$\operatorname{chord}(\Delta u)=2\, s\!\left(\tfrac{\Delta u}{2}\right)$$

Quarter arc length follows from summing chords (upper/lower/midpoint/average rules). Area uses radial trapezoids on the quarter, scaled ×4. The one‑step sagitta is

$$\operatorname{sagitta}(\Delta u)=1-c\!\left(\tfrac{\Delta u}{2}\right)$$

π as an Emergent Limit (in turns)

Let the full polygon have N chords with Δu = 1/N. The chord between consecutive samples is 2·s(Δu/2), hence the polygon perimeter is C_N = 2N·s(1/(2N)). Define π (in turns) as half the circumference of the unit circle. Then:

$$\pi \,=\, \lim_{N\to\infty} N\, s\!\left(\tfrac{1}{2N}\right)$$

Dyadic variant: letting N = 2^B, we also have π = lim_{B→∞} 2^{B}·s(2^{-(B+1)}).

What GRG Adds

  • Deterministic control: explicitly choose N, the step Δu, and the rule (midpoint / right‑endpoint / average / radial‑trapezoidal Δu).
  • Turns‑based formalism: period‑1 trigonometry with c(u), s(u); quarter aggregation unifies perimeter and area.
  • Inside‑out convergence: measurements as discrete sums of center‑seeded primitives.

Page Takeaways

  • GRG is a center‑seeded, π‑free discretization framework (2025).
  • Angles are measured in turns with period‑1 primitives c(u), s(u).
  • Quarter aggregation gives simple, reproducible formulas for perimeter and area.
Overview

Overview

Sep 12, 2025

GRG Overview

Generative Radial Geometry studies finite radial constructions and their measurement kernels. The current two-dimensional reference uses equal binary sectors, internal chord triangles, and external tangent triangles.

Current Reference

Definitions, kernel formulas, bounds, and exact-arithmetic error proofs.

Research History

Original Projection, Cartesian trapezoidal methods, and earlier Turns formulations retain their own definitions and scope. They are not interchangeable specifications.

Evidence

Geometric accuracy, floating-point stability, computational performance, and originality are separate questions. See Results and the benchmark protocol for their status.

Inside-Out GRG

Inside-Out GRG

Aug 24, 2025

GRG — Looking from the Inside: a discrete, π-free language for shapes

Generative Radial Geometry (GRG) — Looking from the Inside

GRG conventions. Angles are measured in turns \(u\in[0,1)\). Radial computations (perimeter and radial area) run on the full turn; the Cartesian area uses the quarter (and then ×4). Trigonometric forms are optional cross-checks, not inputs to the pipeline.

Work-in-progress. GRG is a discrete, π-free way to measure and compare shapes by radial generation. The turning point was this: I looked at the circle from its inside—a point that sends rays outward and measures what they meet. That inside→out switch is small to say and large to practice. The door is open; we will walk through it.

Why GRG matters

  • Discrete by nature: everything runs in finite steps (N)—heights, chords, bands.
  • π-free computation: the pipeline never needs π to run; π appears only when we compare to classical values.
  • Immediate comparability: two shapes at the same resolution are directly comparable with the same grammar.
  • Multi-scale: increase N; the very same procedure refines itself (coarse → fine).
  • GPU-friendly: short loops and sums, ideal for digital geometry, CAD/graphics, and simulation.

Looking from the inside

Not from a grid of squares, not from external coordinates—from a center that emits rays. A shape becomes a radial profile (a finite vector of values) rather than a formula to solve. From there, measurement and comparison happen discretely and immediately.

The joined method (circle)

Perimeter (radial, full turn): split the full turn \(u\in[0,1)\) into \(4N\) equal steps \(\Delta u=1/(4N)\). Join successive GRG border nodes by chords; the quarter view in the figure is for clarity.

Area (Cartesian, quarter): split one radius into \(N\) parts; at each split, drop the perpendicular to the circle (the “height”). Adjacent heights form a trapezoid; summing them tiles the quarter of a disk (the last one degenerates to a triangle). Multiply by 4 for the full disk.

Figure — joined method (circle)

Blue = angular sectors (perimeter, shown on the quarter). Red = single-radius trapezoids (area on the quarter). Dashed red rays start from the height intersections and generally do not coincide with the blue rays.

Figure 1. Quarter circle built from angular sectors (blue) and single-radius trapezoids (red); dashed rays start at height intersections.

Discrete comparability (the practical power)

At the same resolution N, any two shapes share the same grammar (heights, chords). This makes them comparable immediately—π-free inside the computation. We use simple, discrete indices relative to a baseline circle at the same N:

  • Area Index \(a_N\)
  • Perimeter Index \(c_N\)
  • Isoperimetric-like ratio \(q_N=a_N/c_N^2\) (the circle maps to 1 at the same N)
Minimal math — turns-first (π-free, node-based)

Quarter nodes (turns):

\[ \Delta u \;=\; \frac{1}{4N} \] \[ u_k \;=\; k\,\Delta u \quad (k=0,\dots,N) \]

Quarter inscribed perimeter (node distances):

\[ C_{\mathrm{in}}^{(Q)}(N) \;=\; \sum_{k=0}^{N-1} \bigl\|\,P_{k+1}-P_k\,\bigr\| \]

Full inscribed perimeter:

\[ C_{\mathrm{in}}(N) \;=\; 4\,C_{\mathrm{in}}^{(Q)}(N) \]

Radial area (full turn, shoelace on 4N nodes):

\[ A_{\mathrm{in}}(N) \;=\; \frac{1}{2} \sum_{j=0}^{4N-1} \bigl(x_j\,y_{j+1}-x_{j+1}\,y_j\bigr) \]

Cartesian area (quarter, Δx) → ×4:

\[ \Delta x \;=\; \frac{R}{N} \] \[ x_i \;=\; i\,\Delta x \] \[ y_i \;=\; \sqrt{\,R^2 - x_i^{\,2}\,} \] \[ A_{\mathrm{trap}}^{(Q)}(N) \;=\; \sum_{i=1}^{N} \frac{y_{i-1}+y_i}{2}\;\Delta x \] \[ A_{\mathrm{trap}}(N) \;=\; 4\,A_{\mathrm{trap}}^{(Q)}(N) \]

Discrete indices (baseline = circle at the same N):

\[ a_N \;=\; \frac{A_{\text{shape}}(N)}{A_{\text{circle}}(N)} \] \[ c_N \;=\; \frac{C_{\text{shape}}(N)}{C_{\text{circle}}(N)} \] \[ q_N \;=\; \frac{a_N}{c_N^{\,2}} \]
Classical cross-check (optional)

Inscribed perimeter (quarter step in turns):

\[ \Delta u \;=\; \frac{1}{4N} \] \[ s_N \;=\; 2R\,\sin\!\bigl(\pi\,\Delta u\bigr) \] \[ C_{\mathrm{in}}(N) \;=\; 4\,N\,s_N \;=\; 8\,N\,R\,\sin\!\Bigl(\frac{\pi}{4N}\Bigr) \]

Circumscribed perimeter (outer bound):

\[ C_{\mathrm{out}}(N) \;=\; 8\,N\,R\,\tan\!\Bigl(\frac{\pi}{4N}\Bigr) \]

Radial area via triangles (full turn):

\[ A_{\mathrm{tri}}(N) \;=\; 2\,N\,R^{2}\, \sin\!\Bigl(\frac{\pi}{2N}\Bigr) \]

Cartesian area (quarter, Δx) — same as above, then ×4.

Figure — 3D comparator (hemisphere bands)

Clean comparator for surfaces: baseline hemisphere on the left, target surface on the right. Blue = GRG parallels on each latitude; red = bands (average of neighboring circumferences). Below, π-free indices (schematic bars).

Figure 2. Hemisphere bands comparator: left = baseline, right = target. Blue parallels indicate perimeter along latitude; red band shows area via average neighboring circumferences times slant. Bars below sketch π-free indices.

From the circle to the sphere (concept)

We stack latitude bands along the vertical axis and apply the same grammar on each parallel (GRG circumference on the band’s ends, averaged across the slant). Errors shrink fast with balanced vertical and angular splits. The comparator above is designed to show differences band-by-band without clutter.

Applications

  • Graphics/CAD: π-free arc lengths and areas that run fast and parallel.
  • Sampling/tiling on spheres: controllable bands, equal-area approximations.
  • Gear-like and elliptical profiles via directional radial generation.
  • Education: a concrete bridge from discrete geometry to classical values.

Status & invitation

GRG is young and grows with this site. The aim is simple: a practical, comparable, discrete language for shapes and surfaces. If the idea helps, let’s refine proofs, diagrams, and applications together.