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.

Interactive Lab + FAQ

Interactive Lab + FAQ

Oct 13, 2025

Generative Radial Geometry (GRG)

A discrete–exact grammar for curves and surfaces using turns, chords, and radial projections. π is not imported as a magic constant; it emerges from dyadic steps and converges in the classical limit.

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Turns-first definitions

We discretize the unit turn; dyadic refinement keeps operations exact on the mesh and drives accuracy.

$$ b \in \mathbb{N}_{\geq 0}, \qquad N = 2^{b} $$
$$ \Delta u = 2^{-(b+2)} \quad \text{(quarter sampling)} \qquad \Rightarrow \qquad \Delta u = \frac{1}{4N}. $$

For a circle of radius \(R\), the inscribed polygon at \(N\) turns has:

$$ P_{N} = 2 N R \sin\!\left(\frac{\pi}{N}\right), \qquad A_{N} = \frac{1}{2} N R^{2} \sin\!\left(\frac{2\pi}{N}\right). $$
Dyadic polygon and chords Circle with an N=16 inscribed polygon, dyadic rays, and a highlighted chord.
Turns, chords, and dyadic rays. Crisp vectors at any zoom.

Interactive Lab

Switch presets and adjust the dyadic exponent \(b\). For the star case \(r(\theta)=1+a\cos(k\theta)\), the exact area is \(A=\pi\!\left(1+\frac{a^{2}}{2}\right)\).

N (turns)
—
Perimeter (approx)
—
Area (approx)
—
Perimeter true
—
Area true
—
Errors (|P|, |A|)
—

Notes. Circle & atlas use exact finite-step formulas (chords) and converge monotonically. Star-shape uses inscribed polygons; C-shape uses odd–even radial sweep for area (perimeter shown exact for reference).

Critical FAQ (condensed)

Q: Is GRG just numerical integration with a trick?
A: For circles with dyadic bisections steps are exact at finite \(N\) (chords & projections). The limit recovers classical values; nothing is imported.

Q: Where does \(\pi\) come from?
A: From the discrete perimeter/area growth of dyadic polygons. \(\pi\) appears as a rotational-symmetry shadow as \(N\to\infty\).

Q: Non-convex shapes “jump across the outside”.
A: We prevent that with visibility-aware schemes: (i) odd–even radial sweep; (ii) radial atlas (locally star-shaped patches with ± signs for outer components/holes).

Q: Convergence?
A: \(O(N^{-2})\) for \(C^{2}\) boundaries; \(O(N^{-1})\) near corners (mitigated via adaptive dyadics). Inscribed perimeter/area are monotone.

Q: Why is GRG pedagogically useful?
A: You can teach Euclidean geometry from bisections and chords; radians & trig tables emerge rather than being prerequisites.

Technical Appendix — Non-Convex & Complex Parametrizations

  1. Star-Check. If a center sees the whole boundary (star-shaped), sample \(\theta\) in dyadic turns and use \(r(\theta)\). Inscribed area/perimeter are monotone \(\uparrow\) with \(O(N^{-2})\).
  2. Odd–Even radial sweep. For each \(\theta_{k}\), intersect the ray with the boundary, sort radii \(r_{k,j}\), and add inside segments \([r_{k,1},\, r_{k,2}],\ [r_{k,3},\, r_{k,4}], \dots\).
  3. Radial atlas (± signs). Decompose into locally star-shaped patches; outer components \(+\), holes \(−\). Sum areas with signs and perimeters by absolute length. Same \(O(N^{-2})\) patch-wise.
  4. Robustness. \(C^{2} \Rightarrow O(N^{-2})\); corners \(\Rightarrow O(N^{-1})\) unless adaptively refined. Self-intersections: use winding rule instead of parity.

All computations use turns, chords, projections; operations are exact at finite dyadic steps for circular primitives and exact on the mesh for polygons.

Team

Roles are presented by responsibility area. Individual bios can be added later.

Team roles pictogram Minimal icons representing research, algorithms, systems, and editorial review.
Role-based structure. Short scope, clear ownership.

Leadership & Research

  • Sets the research agenda and publishes specifications with reproducible examples.
  • Maintains the “turns-first” glossary and consistency across demos and notes.
  • Approves public statements and abstracts before release.

Methods & Algorithms

  • Designs visibility-aware procedures (odd–even sweep, radial atlas with ± signs).
  • Owns convergence studies and baselines for circles, ellipses, and polygons.
  • Runs complexity/accuracy benchmarks; documents limits and mitigations.

Systems & Frontend

  • Implements interactive canvases; keeps demos accessible and performant.
  • Maintains build tooling and test fixtures for numerical correctness.
  • Coordinates localization and analytics (non-invasive, anonymized).

Editorial Review & QA

  • Checks clarity, tone, and scope; enforces the evidence-led style.
  • Validates formulas and captions (MathJax + SVG semantics and accessibility).
  • Controls versioning of figures; ensures non-repetitive hero imagery.

Collaborations

We welcome collaborations with educators and researchers. A public contact form will be added once finalized.

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