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
Format	Content	Notes
.OBJ	Polylines	Lightweight; ideal for downstream thickening or meshing
.SVG	2D vectors	Great for print/CNC; preserves quarters and layers
.glTF	Runtime-friendly	Embed as lines/triangles for engines and viewers
.STEP	CAD solids/surfaces	Produce via downstream thickening/loft from polylines
.CSV/.JSON	Metrics & params	Reproducibility 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

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
Goal	Setup	Parameters	Notes
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

Log seed, n, rule, adaptive thresholds, and weights in JSON.
Export quarter nodes; reconstruct full geometry via symmetry.
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.

Left: sector-aligned lattice cells driven by w(u;\theta). Center: venation-like refinement where $\rho(\Delta u)$ is large. Right: stylized rosette via directional weights and dyadic levels.