Continuous

Continuous

Sep 12, 2025

GRG — Continuous Model

Continuous Model: Radial Fields, Well-posedness & Stability

Category: Generative Radial Geometry (GRG)

We describe a continuous model that interprets GRG’s inside-out construction as integral curves of a center-seeded field on a domain Ω ⊂ ℝ². Let the seed be s ∈ Ω. For any point x ≠ s define the unit radial direction er(x) = (x−s)/‖x−s‖ and a tangential direction et(x) orthogonal to er. The guidance field blends radial and tangential components.

Radial Guidance Field

The model uses a field F: Ω\\{s} → ℝ² with radial bias

$$ \mathbf{F}(x) \;=\; a(x)\, \mathbf{e}_r(x) \;+\; b(x)\, \mathbf{e}_t(x), \qquad a(x) > 0 $$

Integral curves γ(τ) solve

$$ \dot{\gamma}(\tau) \;=\; \mathbf{F}\big(\gamma(\tau)\big), \qquad \gamma(0)=x_0,\quad x_0\in\Omega\setminus\{s\} $$

Each turn-angle u ∈ [0,1) selects an initial direction, producing a family of trajectories that reach the boundary. The discrete GRG construction samples these trajectories at uniform steps in turns.

Well-posedness

  • Existence & Uniqueness. If F is locally Lipschitz on Ω\\{s} then trajectories starting away from the seed have local existence and uniqueness while they remain in that domain. Global continuation and boundary arrival require additional hypotheses.
  • Termination. Trajectories stop when they hit ∂Ω. Dirichlet-type constraints prescribe a target boundary, while Neumann-type constraints control flux through ∂Ω.

Stability to Perturbations

If F has Lipschitz constant L, perturbing the seed by ε yields trajectories with deviation bounded by

$$ \|\gamma_\varepsilon(\tau) - \gamma(\tau)\| \;\le\; C\,\varepsilon\, e^{L\tau} $$

In practice we keep L moderate via smoothing of a(x), b(x) and by capping curvature of trajectories.

Regularization & Safety

  • Step lower bound. Minimum angular step Δu_{\min} to avoid local collapse.
  • Curvature damping. Add −λ\,κ(\gamma)\,\mathbf{e}_t to reduce spiraling near thin features.
  • Line-search. Backtrack the last advance if distance to boundary decreases below a threshold.

Multi-Seed and Holes

For non-simply connected domains or shapes with holes we blend fields from auxiliary seeds s_i with weights w_i(x):

$$ \mathbf{F}(x) \;=\; \sum_{i=1}^{m} w_i(x)\,\Big(a_i(x)\,\mathbf{e}_r^{(i)}(x) + b_i(x)\,\mathbf{e}_t^{(i)}(x)\Big), \qquad \sum_i w_i(x)=1,\ \ w_i(x)\ge 0 $$

Hole guards choose weights that vanish across cut-loci to prevent ambiguous paths.

Discrete Connection

Sampling each trajectory at uniform steps in turns yields the GRG primitives used for chord sums and radial strips. The continuous model justifies convergence as resolution increases. With turns-first sampling and dyadic refinement, the discrete GRG primitives converge monotonically as b increases.

Radial field with boundary, hole guard and trajectories Center-seeded field with trajectories to the boundary, auxiliary seed for a hole, and example paths. s (seed) aux seed ∂Ω hole guard
Center-seeded field with trajectories to the boundary, auxiliary seed for a hole, and example paths.

Practical Defaults

  • Field coefficients. a(x)=1 near the seed, tapering to maintain stable speed; b(x) small and smooth to avoid spirals.
  • Adaptive control. Increase samples where the curvature proxy exceeds a threshold; cap Δu.
  • Refinement (dyadic). Use the dyadic level b ∈ ℕ≥0 with N = 2^b samples and step Δu = 2^{-b}; increasing b yields monotone refinement.
  • Reconstruction. Quarter symmetry reconstructs the full figure; multi-seed blending covers non-simply connected domains.
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.

Radius, Rotation, Arc

Radius, Rotation, Arc

Sep 7, 2025

Generative Radial Geometry: Radius, Rotation, and Arc Length

Generative Radial Geometry (GRG) provides a discrete, π-free, constructive method to generate circles, spheres, and related shapes. Instead of assuming π and continuous formulas, GRG builds geometry from radial divisions, summing chords and trapezoids. This creates a radically different perspective compared to the classical approach. Angles are expressed as fractions of the turn ($u\in[0,1)$).

Why GRG Matters

Classical geometry relies on π and continuous trigonometric functions. GRG, in contrast, is digital-native: it works with finite steps, exact points on the circumference, and simple arithmetic operations. This makes it both conceptually new and computationally efficient.

Core Comparison

The table below highlights the key differences between the two approaches.

Aspect Classical Approach Generative Radial Geometry (GRG)
Definition of angle Radians \((s=r\theta)\) with \(\pi\). Angle = discrete fraction \(k/N\) of the turn \((u\in[0,1))\), generated internally.
Mathematical tools Trigonometry, sine/cosine, continuous integrals. Finite sums of chords and trapezoids; additions and square roots.
Points on the circle Require trig functions (numerically approximated). Exact by construction on generated radial sets \((x^2+y^2=R^2)\).
Lengths and areas Closed forms: \(s=r\theta\), \(A=\pi R^2\). Generated sums: \(L=\sum \ell_i\), \(A=\sum A_i\); \(\pi\) emerges as a limit.
Local error None in theory, but depends on \(\pi\)/trig numeric precision. Error only from finite sums; boundary points remain exact.
Extension to 3D New formulas needed (e.g., \(V=\tfrac{4}{3}\pi R^3\)). One constructive scheme extends to surfaces and volumes.
Computational efficiency Expensive trig; slower on low-level hardware. Roots + sums; friendly to SIMD/GPU and embedded devices.
Concept of \(\pi\) \(\pi\) assumed a priori. \(\pi\) appears from the discrete generative process.
Conceptual view Geometry as a static entity to be measured. Geometry as an internal, generative process.

Three GRG Modes

On this page we use three operational modes. Two are Cartesian and compute area on a quarter (by symmetry). One is radial and computes circumference on the full turn.

Mode Target Partition Element Estimate Formula (sketch) Notes
Δx Trapezoidal (Cartesian) Area Quarter, \(x_i=i\,\Delta x\), \(\Delta x=R/N\) Trapezoids between heights \(h_i=\sqrt{\,R^2-x_i^2\,}\) Lower (under-tiling) \(A_Q \approx (\Delta x/2)\sum(h_i+h_{i+1})\); total area \(=4\,A_Q\) Monotone ↑ with \(N\)
Projection Corridor (Cartesian) Area Quarter, same \(\Delta x\) Riemann-like bars at right / midpoint Lower/Upper (squeezing) \(A_Q^{\downarrow}=\Delta x\sum h_i\), \(A_Q^{\uparrow}=\Delta x\sum h_{i+\tfrac12}\); total \(=4\times\) Certified bounds
Chord-Sum at \(\Delta u\) (Radial) Circumference Full turn, \(\Delta u=1/N\) Chords between consecutive radial points Lower (inscribed polygon) \(L \approx N\,R\,\sqrt{\,2-2\,c(1/N)\,}\) No quarter; defined on the whole circle
Note on “quarters”. The quarter is used only for the Cartesian area modes (by symmetry). For circumference we operate on the full turn (radial partition), so “quarters” are not meaningful there.

Further Evaluations & Options

  • Certified area bounds. The corridor (right vs midpoint) yields lower/upper bounds that squeeze the exact area as \(N\) grows.
  • Stronger circumference bracketing. Pair the inscribed chord-sum (lower) with a circumscribed tangent-sum (upper) to bound the true length.
  • Adaptive refinement. Increase \(N\) only where variation is largest (near endpoints for Cartesian area; uniform \(\Delta u\) usually suffices for chords).
  • Hybrid pipelines. Use circumference from chords to cross-check the discrete derivative of area w.r.t. \(R\).
  • 3D extension. The same logic generalizes to spherical belts (length) and frusta/patch sums (area/volume) with discrete radial partitions.
  • Hardware efficiency. All modes need only sums and square roots → ideal for embedded, SIMD, and GPU kernels.

Conclusions

Classical geometry excels in closed-form elegance. GRG complements it with a digital-native, constructive view: exact boundary points, sums that converge without assuming π, and a single scheme that scales from 2D to 3D. Here we use three GRG modes—two Cartesian for area on a quarter, one radial for circumference on the full turn—unlocking certified bounds, adaptive strategies, and hardware-friendly implementations.

GRG Trapezoidal

GRG Trapezoidal

Sep 6, 2025

Earlier Formulation

This page preserves a historical GRG method. Its notation and formulas belong to that version. The current binary radial formulation and verification are available here. Historical computational claims require separate validation.

Δx Trapezoidal

This GRG block computes the circle’s area from the inside using uniform $\Delta x=r/N$ on the quarter. On the graph $y(x)=\sqrt{r^{2}-x^{2}}$, adjacent samples form trapezoids strictly under the arc; multiply by $4$ for the full circle. No angles are required and no external constants appear in the primary formulas.

Setup

$$ \Delta x \;=\; \frac{r}{N}, \qquad x_k \;=\; k\,\Delta x \quad (k=0,\dots,N), \qquad y_k \;=\; \sqrt{\,r^{2}-x_k^{2}\,}. $$
Quarter circle with Δx trapezoids (N=5) Axes, quarter arc of radius 100, and five Δx strips forming interior trapezoids strictly under the arc. x y
Fig. 1 — Δx trapezoids strictly inside the quarter arc (N=5).

Quarter-area sum (→ ×4 for the circle)

$$ A_{\text{quarter}}(N) \;=\; \sum_{k=0}^{N-1} \frac{\Delta x}{2}\,\bigl( y_k + y_{k+1} \bigr), \qquad A_{\text{circle}}(N) \;=\; 4\,A_{\text{quarter}}(N). $$
Single Δx trapezoid between x_k and x_{k+1} One strip showing the trapezoid under the quarter arc; bracket indicates Δx. x_k x_{k+1} y_k y_{k+1} Δx
Fig. 2 — One Δx trapezoid; the dashed arc shows the exact curve above it.

Why use this block

  • Monotone under-approximation (trapezoids are strictly inside the quarter arc).
  • Smooth convergence $\sim O(N^{-2})$ and numerically stable (simple sums and square roots).
  • No angle handling; easy to implement with fixed-point or vectorized code.

How to use

  • Choose $N$ (e.g., $200$–$1000$) to balance speed and accuracy.
  • Step-doubling check: compare results at $N$ and $2N$ to gauge the remaining gap.
  • Optional diagnostic: plot the trapezoids on the quarter to see the under-tiling.

Notes

  • The last strip degenerates to a triangle since $y_N=0$ at $x_N=r$.
  • Perimeter pairs naturally with equal-angle chords; see the Core Turns block for the chord formula.

See also: Core Turns · Original Projection · Applications / Circle

Δx Trapezoidal

Area from the inside: sample the quarter with uniform $\Delta x = r/N$, use trapezoids under the arc on $y(x)=\sqrt{r^{2}-x^{2}}$, then multiply by $4$ for the full circle. No angles required.

Formula box — Quarter setup

$$ \Delta x \;=\; \frac{r}{N}, \qquad x_k \;=\; k\,\Delta x \ \ (k=0,\dots,N), \qquad y_k \;=\; \sqrt{\,r^{2}-x_k^{2}\,}. $$

Formula box — Quarter area (→ ×4)

$$ A_{\text{quarter}}(N) \;=\; \sum_{k=0}^{N-1} \frac{\Delta x}{2}\,\bigl[\,y_k + y_{k+1}\,\bigr] $$
$$ A_{\text{circle}}(N) \;=\; 4\,A_{\text{quarter}}(N) $$

Properties

  • Monotone under-approximation: each trapezoid lies strictly inside the quarter arc.
  • Convergence $\sim O(N^{-2})$. Numerically stable: sums and square roots only.
  • The last strip degenerates to a triangle because $y_N=0$ at $x_N=r$.

How to use

  • Choose $N$ (e.g., $200$–$1000$) based on speed/accuracy needs.
  • Step-doubling check: compare results at $N$ and $2N$ to estimate the remaining gap.
  • Optional: plot the trapezoids on the quarter to visualize the under-tiling.