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.

Selector

Selector

Sep 12, 2025

Choosing a GRG Method

Choose a method by its documented geometry, required output, error model, and implementation evidence.

Current Circle Measures

For regular binary sectors, the current formulation supplies arithmetic and weighted kernels, explicit area bounds, and exact-arithmetic convergence proofs. Use the stated domains and preserve the distinction between geometric approximation and rounding.

Historical Cartesian Methods

Projection and trapezoidal constructions are separate methods. Their endpoint behaviour, stopping rules, and error rates require independent validation.

General Shapes and Adaptivity

A circular chord–sagitta ratio alone is not a demonstrated universal curvature or stability selector. An adaptive rule must define the boundary, local error indicator, limits, and termination behaviour before it is recommended.

Performance

No automatic selector is claimed as validated here. Compare actual implementations under the benchmark 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.

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.