Roadmap

Roadmap

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.

Interoperability

Interoperability

Interoperability & Export

Category: Generative Radial Geometry (GRG)

GRG exports reproducible geometry and metadata for CAD/BIM, DCC, and scripting pipelines. Angles use turns with dyadic step Δu=2^{-n} (full turn); a quarter has M=2^{\,n-2} samples.

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

GRG Export Pack (recommended)

  • config.json — seed, n, rule, adaptive thresholds, weights, commit, timestamps.
  • nodes_quarter.obj — quarter polyline (units: meters; Z=0).
  • nodes_full.obj — reconstructed full polyline via ×4 symmetry.
  • strips.csv — per-step area strips (Δu, chord, sagitta).
  • metrics.json — P(n), A(n), ρ(Δu) stats, runtime, memory.
  • LICENSE — permissive license (MIT/Apache-2.0).
  • README.md — how to import into common tools.

Formats & Usage

Recommended export formats with contents and usage notes
FormatContentNotes
.OBJPolylinesLightweight; ideal for downstream thickening or meshing
.SVG2D vectorsGreat for print/CNC; preserves quarters and layers
.glTFRuntime-friendlyEmbed as lines/triangles for engines and viewers
.STEPCAD solids/surfacesProduce via downstream thickening/loft from polylines
.CSV/.JSONMetrics & paramsReproducibility and scripts; easy diffs

Units & Coordinate Frame

  • Units: meters by default; set units in config.json.
  • Frame: origin at seed; X to the right, Y up; quarter in first quadrant by convention, Z=0.
  • Winding: counter-clockwise sampling by increasing u.

Naming Conventions

  • grg_{project}_{yyyy-mm-dd}_{commit}/ root folder.
  • quarter_M{M}_n{n}.obj and full_M{M}_n{n}.obj for geometry.
  • metrics_P_A_{n}.json, strips_{n}.csv for analytics.

Export Pseudocode

GRG_Export(seed, n, rule, weights, adaptive):
  Δu ← 2^{-n}; M ← 2^{n-2}
  nodes_q ← BuildQuarter(seed, Δu, rule, weights, adaptive)
  nodes_f ← Reflect4(nodes_q)
  P, A  ← Aggregate(nodes_q)  # chords & strips
  write_obj("nodes_quarter.obj", nodes_q)
  write_obj("nodes_full.obj", nodes_f)
  write_csv("strips.csv", per_step_values)
  write_json("metrics.json", {P, A, rho_stats, runtime, mem})
  write_json("config.json", {seed, n, rule, weights, adaptive, units:"m"})

Round-tripping

  • To CAD: loft/thicken polylines; convert to NURBS; export STEP.
  • To DCC/Engines: glTF with line primitives or triangulated ribbons.
  • Back to GRG: keep config.json as the single source of truth.

Checks Before Shipping

  • Verify P(n), A(n) stabilization across dyadic refinements.
  • Ensure units and frame documented; include LICENSE.
  • Attach README with minimal import steps for target tools.
Advanced Applications

Advanced Applications

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)

Ready-to-use GRG design 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

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.