Benchmarks

Benchmarks

Sep 13, 2025

Validation and Benchmark Protocol

This page specifies planned comparisons. It does not present a completed benchmark or a demonstrated performance advantage.

Separate Tasks

Measure circumference and area evaluation separately from boundary generation, rasterisation, and three-dimensional modelling. Equal sector counts alone do not make different tasks comparable.

Accuracy and Cost

Compare runtime at the same achieved error, and error at the same runtime. Report arithmetic format, rounding, memory, hardware, software versions, warm-up, repetitions, timing variability, setup cost, and precomputation.

Baselines

Include geometric bisection with inner–outer bounds, comparable weighted or extrapolated kernels, and direct analytical evaluation where appropriate. Give every method the same target and include its full computational cost.

Data and References

Begin with the circle kernels whose hypotheses are documented in the current formulation. Validate other shapes and three-dimensional methods separately. Publish reference precision and how it is obtained.

Convergence

The current regular-sector proofs distinguish second-order arithmetic kernels from fourth-order weighted kernels. These rates must not be transferred automatically to Cartesian trapezoids, endpoint-singular integrands, general shapes, or finite-precision implementations.

Reporting

Publish executable scripts, inputs, raw CSV or JSON, version identifiers, plots, and the conditions of every comparison. Former sample timing rows and schematic plots are not retained as measured results.

Complexity

Complexity

Sep 12, 2025

Computational Complexity

Complexity must identify the task, algorithm, arithmetic precision, and output representation. This page distinguishes costs rather than assigning a single complexity to all GRG methods.

Scalar Kernel Evaluation

Given local geometric ratios, the current circle kernels require a fixed number of arithmetic operations. Obtaining the ratios has its own cost. A binary-bisection evaluation performs a number of refinement stages proportional to the binary depth; square-root cost depends on precision and implementation.

Explicit Geometry

Generating and storing all sector nodes has a cost proportional to the output count. Streaming may reduce storage, but does not remove output work.

Raster and Volume Tasks

Boundary drawing, disk filling, surface generation, and voxel filling have different output sizes. Their cost statements must specify whether every output element is generated.

Comparisons

Report setup, precomputation, precision, memory, and runtime at equal achieved accuracy under the benchmark protocol. Earlier repeated cost models have been consolidated here.

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.

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.
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.

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.