Continuous

Sep 12, 2025

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 e_r(x) = (x−s)/‖x−s‖ and a tangential direction e_t(x) orthogonal to e_r. 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.

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

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:

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

Left: chord and sagitta for step Δu. Right: one center‑seeded strip with area ½·s(Δu).

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

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

Quarter with chord (orange), sagitta (dashed), and one radial trapezoid (blue).

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

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.

Turns & Precision

Sep 6, 2025

Turns & Precision

This page formalises the angle unit used across ERG/GRG. Angles are measured in turns (fractions of a full rotation). We also define a binary fixed-point representation for hardware/software precision.

Angle unit (turns)

A full rotation is $1$ turn. Quarter-turn $= \tfrac{1}{4}$, half-turn $= \tfrac{1}{2}$, etc. Trigonometric functions are period-1 in turns: $c(u),\, s(u)$ with $c(0)=1,\ s(0)=0,\ c(\tfrac{1}{4})=0,\ s(\tfrac{1}{4})=1$.

Equal-angle partition (N sectors)

Choose the number of sectors $N$ and set the step $\Delta u = \tfrac{1}{N}$. Sampling angles: $u_k = \tfrac{k}{N}$, $k=0,\dots,N-1$.

Binary fixed-point turns (B bits)

Store angles as $u = \tfrac{k}{2^{B}}$ turns with $k \in \{0,\dots,2^{B}\!-\!1\}$. The angular resolution is $\Delta u_B = 2^{-B}$. To get exactly $N$ equal sectors, pick $N$ dividing $2^{B}$ (e.g. $N=2^{B}$). Then use indices $k = m\,j$ with $m = \tfrac{2^{B}}{N}$, $j=0,\dots,N-1$.

Example with $B = 3$ bits ⇒ $2^{3}=8$ ticks. Increasing $B$ doubles the ticks and the precision.

Why more bits ⇒ more accuracy

Circumference (chords): $ C_N \;=\; 2\,N\,r\, s\!\bigl(\tfrac{1}{2N}\bigr) $ is the inscribed $N$-gon perimeter; as $N$ doubles, the step halves and the error typically drops ≈ $\times 4$.
Area (sectors): $ A_N \;=\; \tfrac{N}{2}\,r^{2}\, s\!\bigl(\tfrac{1}{N}\bigr) $ behaves similarly (step-squared).

Quick reference

Inline: $ \Delta u = \tfrac{1}{N},\quad u_k = \tfrac{k}{N},\quad u = \tfrac{k}{2^{B}},\quad \Delta u_{B} = 2^{-B} $.
Display:

$$ C_{N} \;=\; 2\,N\,r\, s\!\Bigl(\frac{1}{2N}\Bigr), \qquad A_{N} \;=\; \frac{N}{2}\,r^{2}\, s\!\Bigl(\frac{1}{N}\Bigr). $$

Implementation notes

Keep an angle register $k$ modulo $2^{B}$ (wrap-around on $B$ bits).
For LUTs of size $2^{M}$ with $M \le B$, index by $k \gg (B-M)$.
Step-doubling check: compare $N$ vs $2N$ to gauge the remaining gap.

ERG – Radial (Turns)

Sep 6, 2025

ERG — Radial (Turns)

Method block (turns-based). Angles are measured in turns (fractions of a full rotation, u∈[0,1)). We write c(u), s(u) for period-1 cosine/sine in turns (so c(0)=1, s(0)=0; c(1/4)=0, s(1/4)=1). No sexagesimal units and no external constants appear in the primary formulas.

Radial node

For a circle of radius r and an angle u (turns):

$$ P(u) \;=\; \bigl(r\,c(u),\; r\,s(u)\bigr), \qquad u\in[0,1). $$

Uniform partition over the full turn: $u_k = k/N$, step $\Delta u = 1/N$.

Fig. A — Radial point $P(u)=(r\,c(u),\,r\,s(u))$ with angle $u$ expressed in turns.

Circumference (equal-angle chords)

Each chord between $P(u_k)$ and $P(u_{k+1})$ has length $2r\,s(\Delta u/2)$. Sum of $N$ identical chords → product:

$$ \boxed{\,C_N \;=\; 2\,N\,r\,s\!\Bigl(\tfrac{1}{2N}\Bigr)\,} $$

Interpretation: inscribed regular $N$-gon; $C_N$ increases with $N$ and converges to the circle’s circumference.

Fig. B — Equal-angle chords tile the full turn (regular $N$-gon).

Area (equal-angle sectors)

The $N$ isosceles sectors (sides $r,r$ and angle $\Delta u$) each have area $\tfrac12 r^2 s(\Delta u)$. Sum → product:

$$ \boxed{\,A_N \;=\; \tfrac{N}{2}\,r^2\,s\!\Bigl(\tfrac{1}{N}\Bigr)\,} $$

$A_N$ increases with $N$ and converges to the circle’s area.

Fig. C — Area by sectors at equal angular step (sum of identical pieces).

How to use

Pick a precision N (e.g., 128–1024). If you store angles as fixed-point turns k/2^B, choose N dividing 2^B.
Step-doubling check: compare results at N vs 2N to gauge the remaining gap.

Pseudocode (2D)

Input: r, N
C = 2 * N * r * s( 1 / (2*N) )
A = 0.5 * N * r^2 * s( 1 / N )

Notes

Turns keep formulas simple and hardware-friendly (fixed-point angle indices, LUTs, SIMD).
Use this block when you want a pure angular build; for Cartesian builds see Δx Trapezoidal and Original Projection pages.

Angle unit & precision (turns, binary fixed-point)

We measure angles in turns: a full rotation is 1 turn. Hence quarter-turn = 1/4, half-turn = 1/2, etc. Trigonometric functions are period-1 in turns: c(u), s(u) with c(0)=1, s(0)=0, c(1/4)=0, s(1/4)=1.

Equal-angle partition

Choose an integer N (number of sectors) and set the angular step Δu = 1/N. The sampling angles are u_k = k/N, k=0,…,N−1.

Binary fixed-point turns (B bits)

For hardware/software precision, store angles as u = k / 2^B turns with k ∈ {0,…,2^B−1}. The angular resolution is Δu_B = 1 / 2^B. If you want exactly N equal sectors, pick N that divides 2^B (e.g., N = 2^B), and use indices k = m·j with m = 2^B / N, j = 0,…,N−1.

Why more bits ⇒ more accuracy

Circumference (chords): C_N = 2 N r · s(1/(2N)) is the perimeter of the inscribed regular N-gon and increases to the circle’s circumference. With N=2^B, the step halves when B→B+1 and the error typically drops ~×4 (quadratic in the step).
Area (sectors): A_N = ½ N r² · s(1/N) is the area of N equal sectors and increases to the circle’s area; same step-squared behaviour.

Example with B = 3 bits ⇒ 2³ = 8 ticks. Increasing B doubles the ticks and the precision.

Pseudocode (B-bit angles)

Input: r, B
N = 1 << B            // N = 2^B sectors
C = 2 * N * r * s( 1.0 / (2 * N) )
A = 0.5 * N * r^2 * s( 1.0 / N )

Implementation tip: if you keep an angle register k modulo 2^B, additions are wrap-around on B bits. For LUTs of size 2^M with M ≤ B, you can index by k >> (B−M).

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

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

Fig. 1 — Δx trapezoids strictly inside the quarter arc (N=5).

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

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

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.

Continuous

Continuous Model: Radial Fields, Well-posedness & Stability

Radial Guidance Field

Well-posedness

Stability to Perturbations

Regularization & Safety

Multi-Seed and Holes

Discrete Connection

Practical Defaults

Foundations

Document Status and Current Reference

Mathematical Foundations: Turns Trigonometry & Radial Discretization

Turns Trigonometry (period‑1)

Chord, Sagitta, and Radial Strip Area

Discrete Perimeter and Area on a Quarter

π as an Emergent Limit (turns)

Background & Positioning

Motivation

Key Idea (Quarter Aggregation)

π as an Emergent Limit (in turns)

What GRG Adds

Page Takeaways

Overview

GRG Overview

Current Reference

Research History

Evidence

Turns & Precision

Turns & Precision

Angle unit (turns)

Equal-angle partition (N sectors)

Binary fixed-point turns (B bits)

Why more bits ⇒ more accuracy

Quick reference

Implementation notes

ERG – Radial (Turns)

ERG — Radial (Turns)

Radial node

Circumference (equal-angle chords)

Area (equal-angle sectors)

How to use

Pseudocode (2D)

Notes

Angle unit & precision (turns, binary fixed-point)

Equal-angle partition

Binary fixed-point turns (B bits)

Why more bits ⇒ more accuracy

Pseudocode (B-bit angles)

GRG Trapezoidal

Earlier Formulation

Δx Trapezoidal

Setup

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

Why use this block

How to use

Notes

Δx Trapezoidal

Formula box — Quarter setup

Formula box — Quarter area (→ ×4)

Properties

How to use

Recent posts

Recent comments

Zenodo:Zenodo — GRG Community

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Zenodo — GRG Community