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.

Roadmap

Roadmap

Sep 13, 2025

Research Roadmap

This page records work to be completed rather than retaining expired milestones or unsupported delivery promises.

Documentation

Align the current formulation, area-kernel article, notation, historical methods, and references. Consolidate duplicate content and preserve versions.

Implementation

Trace dependencies in every interactive tool. Specify which operations construct geometry, evaluate measures, compute comparison references, and draw the visualisation.

Verification

Review domains, normalisations, error proofs, rounding, cancellation, and stopping conditions. Validate general shapes and three-dimensional methods independently.

Benchmarks and Releases

Publish actual scripts, datasets, environments, licences, and measured results. Assess equal-accuracy performance before reporting a speed advantage.

Publication

List manuscripts and releases with verified status and persistent links. No release date, throughput target, or completed milestone is implied until evidence is available.

Reproducibility

Reproducibility

Sep 13, 2025

Reproducibility and Research Artifacts

This page defines the evidence required to reproduce a GRG result. It is a protocol, not a declaration that every artifact is already available.

Available Documentation

Required Record

Each numerical result should identify the construction version, input geometry, sector count, sampling convention, arithmetic format, rounding policy, software version, reference definition, and raw output.

Code and Environment

Provide executable source, a documented environment, test cases, and a release identifier. State the actual licence; do not imply an open-source release before one is published.

Measurement Protocol

Archive inputs, outputs, timing methodology, hardware details, memory measurements, and comparison methods. Include setup and precomputation costs. Compare complete implementations at equal achieved accuracy.

Verification Status

Previous completion checkmarks and sample commit identifiers were templates. They have been removed. Repository, continuous-integration, dataset, and DOI availability must be verified individually before being marked complete.

Use Publications and Research References for documented external records.

Advanced Applications

Advanced Applications

Sep 13, 2025

Advanced Applications

Category: Generative Radial Geometry (GRG)

GRG is a center-seeded, turns-based framework where the step on the full turn is dyadic Δu = 2^{-n} and a quarter has M = 2^{\,n-2} samples. Beyond didactic examples, GRG scales to rich, real-world use-cases across design, science, and engineering.

$$ n\in\mathbb{N}_{\ge 0},\quad \Delta u=2^{-n}\ \text{(full turn)},\quad M=2^{\,n-2}\ \text{(samples per quarter)}. $$

Metamaterials & Radial Lattices

  • Cell parameterization: drive lattice thickness and porosity with radial weights w(u;\theta) aligned to sectors.
  • Gradient structures: encode density transitions by modulating n and w(u) per quadrant.
  • Export: polylines for slicing; surface thickening downstream; per-sector tags in JSON for auditability.

Bio-inspired Morphologies

  • Venation & ribbing: use adaptive refinement where the sagitta/chord proxy $\,\rho(\Delta u)=\dfrac{1-c(\Delta u/2)}{2\,s(\Delta u/2)}\,$ is high to mimic leaf veins or shell ribs.
  • Seed layouts: multi-seed blends reproduce spoke-and-ring organizations seen in echinoderms and corals.

Procedural Graphics & VFX

  • Shader-ready primitives: chords and strips form compact GPU-friendly loops; dyadic refinement maps to MIP-like levels.
  • Stylization: directional weights create radial sunbursts, rosettes, and moiré effects with deterministic control.

Architecture, CAD/BIM & Fabrication

  • Toolpaths: quarter aggregation supports symmetry-aware infill and rib layouts for AM/robotic fabrication.
  • Interoperability: export nodes/strips (.OBJ polylines) and parameter logs (JSON). Downstream conversion to .STEP for BIM/CAD workflows.
  • Tolerances: pick n from budgeted error using second-order behavior on smooth parts.

Scientific & Medical

  • Vascular/airway sketches: multi-seed radial fields provide reproducible centerlines for exploratory models.
  • Acoustics/CFD pre-meshing: use GRG sectors to place probe rings and boundary samples consistently.

Design Patterns (Recipes)

GoalSetupParametersNotes
Lightweight radial shell Chord—Midpoint (perimeter) + Δx Trapezoidal (area) n=11, adaptive on, β=0.3, m=2 Uniform silhouette, denser ribs near high curvature
Bio-venation panel Multi-seed with curved blending n=12, τ=0.05, β=0.5, m=1 Leaf-like strands; ensure hole guards on internal voids
Procedural rosette (VFX) Directional weights only n=10, β=0.6, m=6, \phi=0 Deterministic radial motifs, shader-friendly

Export Checklist

  1. Log seed, n, rule, adaptive thresholds, and weights in JSON.
  2. Export quarter nodes; reconstruct full geometry via symmetry.
  3. Convert to surfaces/solids downstream only after verification of P(n), A(n) stabilization.

Good Defaults

  • Step: Δu=2^{-10} to start; refine dyadically where $\rho(\Delta u)$ exceeds τ.
  • Rules: midpoint for perimeter, Δx trapezoidal for area; averaged chords for mixed targets.
  • Blending: keep $\sum w_i=1$, $w_i\ge 0$; make weights vanish across cut-loci around holes.
Limitations

Limitations

Sep 13, 2025

Scope and Limitations

What the Current Proofs Cover

The current circle kernels use regular symmetric sectors and specific internal and external triangles. Their convergence results assume exact arithmetic and fixed radius.

What Requires Separate Verification

Floating-point and fixed-point stability, computational efficiency, general curves, nonconvex geometry, holes, adaptive sampling, and three-dimensional constructions require their own specifications and evidence.

Refinement

Increasing geometric resolution does not guarantee increasing numerical accuracy indefinitely. Cancellation, quantisation, and underflow or overflow may limit an implementation.

Historical Methods

Cartesian projection and trapezoidal methods have different sampling and endpoint behaviour. Their error rates must not be inferred from the angular kernel proofs.

Research Claims

The existence of a coherent formulation and a valid error proof does not establish original authorship of a local weighting or superiority over alternative implementations.

Current definitions and verification.

Integration

Integration

Sep 13, 2025

Integration: Structural Optimization & Machine Learning

Category: Generative Radial Geometry (GRG)

GRG integrates with structural optimization (SIMP/level set) and with machine learning for parameter calibration. Angles are measured in turns with dyadic step Δu = 2^{-n} on the full turn; a quarter has M = 2^{\,n-2} steps. Formulas remain π-free at finite resolution.

$$ n \in \mathbb{N}_{\ge 0},\qquad \Delta u = 2^{-n},\qquad M = 2^{\,n-2}\ \text{(steps per quarter)}. $$

Topology Optimization Bridge

We expose GRG’s radial density as design variables for SIMP/level-set frameworks. Let x denote per-sector densities, mapped from radial weights:

$$ x(u_k)\;=\;\sigma\!\big(w(u_k;\,\theta)\big),\qquad u_k = k\,2^{-n},\ \ k=0,\ldots,2^{n}-1, $$

with a smooth logistic σ and parameters θ controlling w(u). A standard compliance-minimization model uses:

$$ \min_{x}\; C(x)=\mathbf f^\top \mathbf u \quad\text{s.t.}\quad \mathbf K(x)\,\mathbf u=\mathbf f,\;\; \sum_i x_i \le \bar{v}\,m,\;\; x_{\min}\le x_i\le 1. $$

By parameterizing x via GRG weights, the optimizer searches a low-dimensional, center-seeded space that preserves radial structure while meeting volume and stiffness targets.

Level-Set Coupling

Alternatively, drive a level-set ϕ by GRG fields: initialize ϕ from radial strips and evolve with curvature-regularized motion under structural sensitivities. GRG provides stable seeding and symmetric updates across quarters.

ML-Assisted Calibration

Given exemplars (e.g., bones, shells, corals), regress GRG parameters θ to match silhouettes or performance. Use mixed geometric + physical loss:

$$ \mathcal L(\theta)\;=\;\lambda_{\text{geom}}\,d_{\text{Chamfer}}\!\big(\partial S_{\text{GRG}}(\theta),\,\partial S^\*\big) \;+\; \lambda_{\text{phys}}\!\left(\alpha\,\big|A(\theta)-A^\*\big|+\beta\,\big|P(\theta)-P^\*\big|\right) \;+\;\lambda_{\text{reg}}\|\theta\|_2^2. $$

Perimeter and area use GRG quarter aggregation with Δu=2^{-n}:

$$ P(n)\;=\;4\sum_{k=1}^{M} 2\,s\!\left(2^{-(n+1)}\right),\qquad A(n)\;=\;4\sum_{k=1}^{M}\tfrac{1}{2}\,s\!\left(2^{-n}\right), \quad M=2^{\,n-2}. $$

Pipeline

  1. Radial seed & step: pick n so Δu=2^{-n}; choose rule for chords/strips (midpoint, averaged, Δx trapezoidal).
  2. Design variables: set θ for w(u) or direct densities x(u_k).
  3. Solver coupling: run SIMP/level-set; back-propagate sensitivities to θ.
  4. Calibration (optional): fit θ to exemplars using the mixed loss.
  5. Export: reconstruct full shape from quarter; output polylines/meshes.

Safety & Regularization

  • Step cap: enforce a minimum Δu_{\min} to prevent local collapse.
  • Curvature damping: penalize large changes of w(u) across adjacent sectors.
  • Volume guard: soft penalty if \sum x_i drifts from target fraction \bar v.

Interoperability

  • Inputs: seed, n, rule, parameters θ or densities x.
  • Outputs: perimeter P, area A, nodes/strips, optional field samples.
  • Formats: .OBJ polylines, .STEP (via downstream CAD), JSON config for reproducibility.

Defaults

  • Rule: chord — midpoint for perimeter; Δx trapezoidal for area.
  • n: start with n=10 (Δu=2^{-10}) and refine dyadically.
  • ML loss: set \lambda_{\text{geom}}=1, \lambda_{\text{phys}}=0.3, \lambda_{\text{reg}}=10^{-3} as safe starters.