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)
| 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.