A Symmetric GRG Kernel for Area

A Symmetric GRG Kernel for Area

Jun 17, 2026

A Symmetric GRG Kernel for Area

The binary radial formulation combines complementary internal and external sector triangles. The area kernel uses one internal contribution and two external contributions, unlike the circumference kernel’s two-to-one chord–tangent balance.

Geometric Elements

Resolve one Turn into N equal symmetric sectors. Let a be the internal triangle’s height, h half its chord, and e half the external tangent side. Define q = h/a.

$$
N=2^b,\quad b\ge2,\quad R>0,\qquad a^2+h^2=R^2,\quad e=R\frac ha.
$$
$$
I_b=ah=R^2\frac{q_b}{1+q_b^2},\qquad E_b=Re=R^2q_b.
$$

Two Area Kernels

$$
A_m=\frac N2(I_b+E_b),\qquad A_w=\frac N3(I_b+2E_b).
$$

Accuracy for the Defined Triangles

The current formulation gives an analytical verification using a local integral as an exact sector reference, not as a kernel input.

$$
F(q)=\int_0^q\frac{dt}{1+t^2},\qquad A_T(R)=NR^2F(q_b).
$$
$$
\begin{aligned}A_m-A_T&=NR^2\left(-\frac{q_b^3}{6}+\frac{3q_b^5}{10}+O(q_b^7)\right),\\A_w-A_T&=NR^2\left(\frac{2q_b^5}{15}+O(q_b^7)\right).\end{aligned}
$$

Regular geometric bisection yields a scale bound. Together with the expansion, it proves second-order total error for the arithmetic kernel and fourth-order total error for the weighted kernel in exact arithmetic.

$$
0\lt q_b\le\frac4N,\qquad A_m-A_T=O(N^{-2}),\qquad A_w-A_T=O(N^{-4}).
$$

The same analysis proves the following bracket for these sectors:

$$
A_{\mathrm{in}}\le A_m\le A_T\le A_w\le A_{\mathrm{out}}.
$$

Sampling and Scope

$$
\Delta u=\frac1{4N}.
$$

The earlier version introduced this quarter-step sampling parameter inside each module. The proof here concerns the complete sector triangles defined above. A different sample-based area construction requires its own explicit definition and proof.

GRG length and area kernel inversion A clean schematic comparison between the GRG circumference kernel, based on two internal chord contributions and one external tangent contribution, and the proposed GRG area kernel, based on one internal area contribution and two external area contributions. Length kernel 2 chord + 1 tangent Chord Chord Tangent Area kernel 1 internal + 2 external Internal External External weight inversion
Internal–external weights are reversed between the circumference and area kernels. This diagram is schematic; the triangle definitions determine the calculation.

Research Status

The cancellation is established for the regular triangular elements above. This does not establish original authorship of the weighting, numerical stability of an implementation, or a computational advantage. Those questions require separate literature and implementation comparisons.

Earlier revisions of this page described fourth-order behaviour as a candidate. The current analytical clarification identifies the geometric assumptions under which that behaviour follows.

A New GRG Formulation: Circle and Area from Binary Radial Modules

A New GRG Formulation: Circle and Area from Binary Radial Modules

Jun 16, 2026

A GRG Formulation: Circle and Area from a Binary Radial Framework

Generative Radial Geometry describes circular measures through a finite binary radial framework. Its working quantities are a radius, a full Turn, a binary resolution, and local chord–tangent constructions.

About the Current GRG Formulation

This page presents the documented binary radial formulation, with geometric definitions and an explicit verification of its measurement kernels. Earlier approaches, including Original Projection and previous Turns formulations, remain part of the research history.

The geometric construction, kernel accuracy, and computational efficiency are separate subjects. The analysis below establishes accuracy for regular symmetric sectors. It does not establish a performance advantage or prescribe an implementation.

The weighted area proposal is documented in A Symmetric GRG Kernel for Area. The geometric ratio notation and proofs below are an analytical clarification of the documented triangles and kernels.

Binary Turns and Resolution

A full rotation is one Turn. Resolve it into equal sectors. The nondegenerate chord–tangent formulas on this page require at least four sectors.

$$
N=2^b,\qquad b\ge2,\qquad R>0.
$$
$$
t_b=\frac1N,\qquad \text{half-sector width}=\frac1{2N}.
$$

One Symmetric Radial Sector

Let a be the distance from the centre to the internal chord, h half that chord, and e half the external tangent side. The tangent is perpendicular to the central radius. The two boundary rays intersect both the circle and this tangent.

$$
a^2+h^2=R^2,\qquad e=R\frac ha.
$$
Symmetric binary radial sector with chord and tangent side A circle with a symmetric radial sector. The internal chord lies between two points on the circle. The external tangent side lies on the tangent line perpendicular to the central radius. O R tᵦ tangent line external tangent side upper sector boundary lower sector boundary internal chord
Figure 1 — Symmetric sector: internal chord, central radius, and external tangent side.

Define dimensionless ratios from these constructed lengths:

$$
c_b=\frac aR,\qquad s_b=\frac hR,\qquad q_b=\frac ha.
$$
$$
c_b^2+s_b^2=1,\qquad q_b=\frac{s_b}{c_b},\qquad c_b=\frac1{\sqrt{1+q_b^2}}.
$$

These symbols denote geometric ratios. They do not specify a trigonometric software routine.

Circumference Kernels

$$
L_i=2h,\qquad L_t=2e.
$$

The arithmetic balance and the classical Snell–Huygens weighted balance give:

$$
\begin{aligned}C_m&=\frac N2(L_i+L_t)=NR(s_b+q_b),\\C_{SH}&=\frac N3(2L_i+L_t)=\frac{2NR}{3}(2s_b+q_b).\end{aligned}
$$

The weighting is a known local arc approximation. This formulation uses it as a kernel within the binary radial framework; the weighting itself is not claimed as an original GRG result.

Area Elements and Kernels

The internal triangle has base 2h and height a. The external triangle has base 2e and height R.

$$
\begin{aligned}I_b&=ah=R^2c_bs_b=R^2\frac{q_b}{1+q_b^2},\\E_b&=Re=R^2q_b.\end{aligned}
$$
$$
A_{\mathrm{in}}=NI_b,\qquad A_{\mathrm{out}}=NE_b.
$$

The arithmetic area estimate and the proposed internal–external weighting are:

$$
\begin{aligned}A_m&=\frac N2(I_b+E_b),\\A_w&=\frac N3(I_b+2E_b).\end{aligned}
$$
$$
A_w=\frac{NR^2}{3}\left(\frac{q_b}{1+q_b^2}+2q_b\right).
$$

Geometric Refinement Bound

The following identity is used to verify regular equal-sector bisection. It is a geometric consequence, not an attribution of a particular numerical algorithm to GRG.

At four sectors the half-sector triangle has equal legs, so q₂ = 1. Bisecting its direction gives the following ratio, obtained by normalising the boundary direction and adding it to the central unit direction:

$$
q_{b+1}=\frac{q_b}{1+\sqrt{1+q_b^2}}.
$$

The denominator is at least two. Induction therefore gives:

$$
0\lt q_b\le 2^{2-b}=\frac4N.
$$

This proves the scale bound needed for the global error estimates. It uses the regular bisection geometry; it does not require a precomputed full-Turn length or area.

Kernel Accuracy: Analytical Verification

Define the local integral below solely as an analytical reference for the exact sector measures. It is not an input to the measurement kernels.

$$
F(q)=\int_0^q\frac{dt}{1+t^2}.
$$

Parameterising the circular boundary by its slope gives exact sector arc length 2RF(q) and exact sector area R²F(q). Thus the fixed full-Turn reference measures satisfy:

$$
C_T(R)=2NRF(q_b),\qquad A_T(R)=NR^2F(q_b).
$$
$$
F(q)=q-\frac{q^3}{3}+\frac{q^5}{5}+O(q^7).
$$

Substitution into the circumference kernels yields:

$$
\begin{aligned}C_m-C_T&=NR\left(\frac{q_b^3}{6}-\frac{q_b^5}{40}+O(q_b^7)\right),\\C_{SH}-C_T&=NR\left(\frac{q_b^5}{10}+O(q_b^7)\right).\end{aligned}
$$

For area:

$$
\begin{aligned}A_m-A_T&=NR^2\left(-\frac{q_b^3}{6}+\frac{3q_b^5}{10}+O(q_b^7)\right),\\A_w-A_T&=NR^2\left(\frac{2q_b^5}{15}+O(q_b^7)\right).\end{aligned}
$$

The area weighting cancels the cubic local residual. Combining these expansions with the geometric scale bound proves, for fixed radius and exact arithmetic:

$$
\begin{aligned}C_m-C_T&=O(N^{-2}),& C_{SH}-C_T&=O(N^{-4}),\\A_m-A_T&=O(N^{-2}),& A_w-A_T&=O(N^{-4}).\end{aligned}
$$

These are asymptotic error statements, not claims of exact equality at finite resolution or guarantees about floating-point performance.

Area Bounds

The geometric inner and outer triangles bound each exact sector. The two balanced area estimates provide a tighter bracket as well:

$$
A_{\mathrm{in}}\le A_m\le A_T\le A_w\le A_{\mathrm{out}}.
$$

For a direct verification, write the normalised kernels as M(q) and W(q). Both differences vanish at zero, and:

$$
\begin{aligned}\frac{d}{dq}\bigl(F(q)-M(q)\bigr)&=\frac{q^2(1-q^2)}{2(1+q^2)^2},\\\frac{d}{dq}\bigl(W(q)-F(q)\bigr)&=\frac{2q^4}{3(1+q^2)^2}.\end{aligned}
$$

Both derivatives are nonnegative for 0 ≤ q ≤ 1, which covers the sectors considered here. Consequently the bracket follows for every b ≥ 2, rather than only for tested resolutions.

Refinement and Quarter-Step Sampling

$$
N_{b+1}=2N_b,\qquad t_{b+1}=\frac{t_b}{2},\qquad \Delta u=\frac{t_b}{4}=\frac1{4N}.
$$

The related area article specifies quarter-step sampling within a module. The verification above concerns the complete sector triangles. An implementation using internal samples must specify how those samples evaluate the same elements; the sampling parameter alone does not define an additional area kernel.

Binary refinement of a radial circle Three circles divided into increasing binary sector counts, showing refinement from coarse radial modules to a smoother circular construction. b = 2, N = 4 b = 3, N = 8 b = 4, N = 16
Figure 2 — Binary sector refinement at N = 4, 8, and 16.

Computational Efficiency and Research Scope

Binary resolution is compatible with binary angular indices and power-of-two scaling. A performance advantage for the complete calculation must nevertheless be measured.

A reproducible comparison must use equal accuracy targets, include generation or retrieval of the local measures, account for precomputation and memory, and specify arithmetic precision, rounding, hardware, and execution time. Stability in fixed-point arithmetic requires a separate error analysis.

The results established here concern the documented regular chord–tangent geometry and its kernels. Historical originality and implementation efficiency are separate investigations.

Constructive Figures

Constructive Figures

Jan 7, 2026

Constructive Figures in GRG: Dyadic Quarter, Mirrored Circle, and Sector Area

This page collects three minimal, constructive drawings used throughout GRG: a dyadic quarter fan, a fully mirrored circle built from four quarters, and a sector-based area accumulator. All figures are π-free inside the loop; conversions happen only at setup or I/O.

1) Dyadic quarter fan (turns-first sweep)

Angle advances in turns. Refinement uses powers of two so that a quarter turn has N = 2^b rays with a step \Delta u = 2^{-b-2}. This aligns with binary hardware.

$$ N = 2^{\,b}, \qquad \Delta u = 2^{-\,b-2} \quad \text{(quarter sampling)} $$
b = 9 → N = 512 rays per full turn (128 per quarter)
Dyadic quarter fan Rays from the center advance by Δu turns (N = 2^b per quarter). Built by recurrence; no π inside the loop.
Quarter sweep with N = 2^b rays. Mirror to complete the circle.

2) Full circle via mirrored quarters (total symmetry)

Build one quarter, then mirror/rotate it three times to assemble the full circle with exact bilateral and rotational symmetry.

Full circle from four mirrored quarters Quarter geometry is cloned with 90° rotations to assemble a full circle with exact symmetry.
Four identical quarters enforce perfect symmetry and cache-friendly construction.

3) Sector-based area accumulation (π-free loop; classical limit)

Area is the sum of radial sectors. The loop aggregates in turn units; the classical value appears in the limit.

$$ \Delta A = \tfrac12 r^2\,\Delta\theta \qquad \text{with}\quad \Delta\theta = 2\pi\,\Delta u $$ $$ A \;=\; \sum \Delta A \;\xrightarrow[\Delta u \to 0]{}\; \pi r^2 $$
Sector area accumulation Radial sectors of angle Δθ tile the disk. Each sector contributes ½ r² Δθ. Summation converges to π r².
Sum of ½ r² Δθ over dyadic steps converges to the classical area.

Notes on determinism and implementation

  • π outside the loop: the runtime loop uses recurrences; π appears only in calibration or reporting.
  • Bit-shift counts: N = 1 << b replaces divisions with a native shift.
  • Mirrored quarters: memory and symmetry benefits; identical update sequences aid reproducibility.
Results

Results

Sep 13, 2025

Results and Verification

This page separates analytical results from numerical illustrations and performance measurements.

Analytical Results

The current binary radial formulation defines regular chord–tangent sectors and gives exact-arithmetic proofs of second-order error for arithmetic kernels and fourth-order error for the weighted circumference and area kernels. It also proves an area bracket for the defined regular sectors.

These statements have specific geometric hypotheses. They do not establish numerical exactness at finite resolution, hardware stability, originality, or superiority over other methods.

Numerical Checks

Numerical evaluations may check consistency with the analytical results. A publishable record must include the generating procedure, radius, sector count, precision, reference definition, raw output, and executable code.

Performance Results

The former gallery mixed example cards with timing figures and incomplete dataset records. Those figures are not retained as validated performance results. No measured efficiency advantage is established by this page.

Performance results will be added when complete reproducible evidence is available under the comparison protocol.

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.