Interoperability

Interoperability

Sep 13, 2025

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

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

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.
Benchmarks

Benchmarks

Sep 13, 2025

Validation and Benchmark Protocol

This page specifies planned comparisons. It does not present a completed benchmark or a demonstrated performance advantage.

Separate Tasks

Measure circumference and area evaluation separately from boundary generation, rasterisation, and three-dimensional modelling. Equal sector counts alone do not make different tasks comparable.

Accuracy and Cost

Compare runtime at the same achieved error, and error at the same runtime. Report arithmetic format, rounding, memory, hardware, software versions, warm-up, repetitions, timing variability, setup cost, and precomputation.

Baselines

Include geometric bisection with inner–outer bounds, comparable weighted or extrapolated kernels, and direct analytical evaluation where appropriate. Give every method the same target and include its full computational cost.

Data and References

Begin with the circle kernels whose hypotheses are documented in the current formulation. Validate other shapes and three-dimensional methods separately. Publish reference precision and how it is obtained.

Convergence

The current regular-sector proofs distinguish second-order arithmetic kernels from fourth-order weighted kernels. These rates must not be transferred automatically to Cartesian trapezoids, endpoint-singular integrands, general shapes, or finite-precision implementations.

Reporting

Publish executable scripts, inputs, raw CSV or JSON, version identifiers, plots, and the conditions of every comparison. Former sample timing rows and schematic plots are not retained as measured results.