Guidelines

Sep 14, 2025

GRG Documentation Guidelines

Describe the construction through its geometric definitions. Use plain language, explicit hypotheses, and reproducible evidence.

Current Formulas

Use geometric lengths and ratios. Keep circular constants outside the construction’s formulas and numerical kernel. Explicit computational comparisons may use reference constants, with their role clearly stated.

Claims and Evidence

Distinguish definitions, proved statements, numerical observations, proposals, and applications. Do not claim efficiency, numerical stability, exact hardware arithmetic, or originality without supporting evidence.

Versioning

Identify historical formulations and retain their provenance. The binary radial formulation is the current two-dimensional reference.

Presentation

Keep site content in English. Provide accessible diagram descriptions, readable mathematical displays, and verified links. Avoid polemical slogans and comparisons that imply an unmeasured advantage.

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.

Parameters

Parameters

Sep 12, 2025

Parameters and Notation

This page defines the current circle-kernel conventions. Historical methods may use other conventions; their symbols must be interpreted locally.

Geometric Scale

R is the positive radius. N is the number of complete sectors across one Turn. Binary depth b gives N = 2ᵇ; the nondegenerate chord–tangent kernels require b ≥ 2.

Angular Steps

The complete sector width is 1/N Turn, its half-width is 1/(2N), and the quarter-step parameter is 1/(4N). If M counts samples in a quarter of the full Turn, state explicitly how M relates to N; do not use the same symbol for both counts.

Local Geometry

a is the chord apothem, h the half-chord, and e the half-tangent side. The geometric ratios are c = a/R, s = h/R, and q = h/a.

Implementation Parameters

Specify arithmetic format, rounding, construction procedure, stopping rule, and any stored tables. Additional parameters for adaptive or noncircular methods require their own definitions.

Full definitions and formulas.