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.

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.
GRG Trapezoidal

GRG Trapezoidal

Sep 6, 2025

Earlier Formulation

This page preserves a historical GRG method. Its notation and formulas belong to that version. The current binary radial formulation and verification are available here. Historical computational claims require separate validation.

Δx Trapezoidal

This GRG block computes the circle’s area from the inside using uniform $\Delta x=r/N$ on the quarter. On the graph $y(x)=\sqrt{r^{2}-x^{2}}$, adjacent samples form trapezoids strictly under the arc; multiply by $4$ for the full circle. No angles are required and no external constants appear in the primary formulas.

Setup

$$ \Delta x \;=\; \frac{r}{N}, \qquad x_k \;=\; k\,\Delta x \quad (k=0,\dots,N), \qquad y_k \;=\; \sqrt{\,r^{2}-x_k^{2}\,}. $$
Quarter circle with Δx trapezoids (N=5) Axes, quarter arc of radius 100, and five Δx strips forming interior trapezoids strictly under the arc. x y
Fig. 1 — Δx trapezoids strictly inside the quarter arc (N=5).

Quarter-area sum (→ ×4 for the circle)

$$ A_{\text{quarter}}(N) \;=\; \sum_{k=0}^{N-1} \frac{\Delta x}{2}\,\bigl( y_k + y_{k+1} \bigr), \qquad A_{\text{circle}}(N) \;=\; 4\,A_{\text{quarter}}(N). $$
Single Δx trapezoid between x_k and x_{k+1} One strip showing the trapezoid under the quarter arc; bracket indicates Δx. x_k x_{k+1} y_k y_{k+1} Δx
Fig. 2 — One Δx trapezoid; the dashed arc shows the exact curve above it.

Why use this block

  • Monotone under-approximation (trapezoids are strictly inside the quarter arc).
  • Smooth convergence $\sim O(N^{-2})$ and numerically stable (simple sums and square roots).
  • No angle handling; easy to implement with fixed-point or vectorized code.

How to use

  • Choose $N$ (e.g., $200$–$1000$) to balance speed and accuracy.
  • Step-doubling check: compare results at $N$ and $2N$ to gauge the remaining gap.
  • Optional diagnostic: plot the trapezoids on the quarter to see the under-tiling.

Notes

  • The last strip degenerates to a triangle since $y_N=0$ at $x_N=r$.
  • Perimeter pairs naturally with equal-angle chords; see the Core Turns block for the chord formula.

See also: Core Turns · Original Projection · Applications / Circle

Δx Trapezoidal

Area from the inside: sample the quarter with uniform $\Delta x = r/N$, use trapezoids under the arc on $y(x)=\sqrt{r^{2}-x^{2}}$, then multiply by $4$ for the full circle. No angles required.

Formula box — Quarter setup

$$ \Delta x \;=\; \frac{r}{N}, \qquad x_k \;=\; k\,\Delta x \ \ (k=0,\dots,N), \qquad y_k \;=\; \sqrt{\,r^{2}-x_k^{2}\,}. $$

Formula box — Quarter area (→ ×4)

$$ A_{\text{quarter}}(N) \;=\; \sum_{k=0}^{N-1} \frac{\Delta x}{2}\,\bigl[\,y_k + y_{k+1}\,\bigr] $$
$$ A_{\text{circle}}(N) \;=\; 4\,A_{\text{quarter}}(N) $$

Properties

  • Monotone under-approximation: each trapezoid lies strictly inside the quarter arc.
  • Convergence $\sim O(N^{-2})$. Numerically stable: sums and square roots only.
  • The last strip degenerates to a triangle because $y_N=0$ at $x_N=r$.

How to use

  • Choose $N$ (e.g., $200$–$1000$) based on speed/accuracy needs.
  • Step-doubling check: compare results at $N$ and $2N$ to estimate the remaining gap.
  • Optional: plot the trapezoids on the quarter to visualize the under-tiling.
GRG Original Projection

GRG Original Projection

Sep 4, 2025

Earlier Formulation

This page preserves a historical GRG method. Its notation and formulas belong to that version. The current binary radial formulation and verification are available here. Historical computational claims require separate validation.

Original Projection — historical method This page presents the original GRG idea: the circle is generated from the inside. The circle area is computed on a quadrant by projecting heights onto the x-axis and using midpoint rectangles (discrete upper); a matching right-endpoint rule gives a discrete lower, forming an under/over corridor that shrinks as $N$ increases (the midpoint rule is second-order). The perimeter is obtained from the sum of equal-angle chords on the quadrant (then $\times 4$). This method remains alongside other GRG blocks and is kept intact for comparison and continued study. See also Core Turns and Δx Trapezoidal.

Original Projection (Quarter → ×4)

Work on the quarter $x\in[0,r]$ with the profile $y(x)=\sqrt{r^{2}-x^{2}}$. Use uniform $\Delta x=r/N$. Midpoint rectangles provide a discrete upper estimate; right-endpoint rectangles provide a discrete lower. Multiply by $4$ to obtain full-circle values.

Setup

$$ \Delta x \;=\; \frac{r}{N}, \qquad x_k \;=\; k\,\Delta x \ (k=0,\dots,N), \qquad x_{k+\tfrac12} \;=\; \Bigl(k+\tfrac12\Bigr)\Delta x, \qquad y(x) \;=\; \sqrt{\,r^{2}-x^{2}\,}. $$

Quarter area: discrete corridor (upper vs lower)

Midpoint (discrete upper, second-order)

$$ A_{\text{quarter}}^{\text{mid}}(N) \;=\; \sum_{k=0}^{N-1} \Delta x \; y\!\bigl(x_{k+\tfrac12}\bigr), \qquad A_{\text{circle}}^{\text{mid}}(N) \;=\; 4\,A_{\text{quarter}}^{\text{mid}}(N). $$

Right-endpoint (discrete lower)

$$ A_{\text{quarter}}^{\text{right}}(N) \;=\; \sum_{k=0}^{N-1} \Delta x \; y\!\bigl(x_{k+1}\bigr), \qquad A_{\text{circle}}^{\text{right}}(N) \;=\; 4\,A_{\text{quarter}}^{\text{right}}(N). $$

Corridor: $A_{\text{circle}}^{\text{right}}(N) \;\le\; \text{Area(circle)} \;\le\; A_{\text{circle}}^{\text{mid}}(N)$, with gap shrinking like $O(N^{-2})$ under smoothness.

Perimeter via equal-angle chords (quadrant → ×4)

Partition the quarter by equal angular steps in turns. Let the quarter have $Q$ equal-angle chords of step $\Delta u=\tfrac{1}{4Q}$. The sum over the quarter is

$$ C_{\text{quarter}}(Q) \;=\; 2\,Q\,r\,s\!\Bigl(\tfrac{\Delta u}{2}\Bigr) \;=\; 2\,Q\,r\,s\!\Bigl(\tfrac{1}{8Q}\Bigr), \qquad C_{\text{circle}} \;\approx\; 4\,C_{\text{quarter}}(Q) \;=\; 8\,Q\,r\,s\!\Bigl(\tfrac{1}{8Q}\Bigr). $$

Equivalently, with $N=4Q$ steps on the full turn: $C_{N}=2\,N\,r\,s\!\bigl(\tfrac{1}{2N}\bigr)$ (see Core Turns).

Figure — Quarter with midpoint and right rectangles

Quarter: midpoint (upper) vs right-endpoint (lower) Quarter arc of a circle (r=100) with Δx strips. Green: midpoint rectangles; orange: right-endpoint rectangles. Both lie inside the quarter; midpoint sits above right. x y midpoint (upper) right (lower)
Midpoint vs right-endpoint corridor on the quarter.

Why use this block

  • Explicit under/over corridor with midpoint/right rules on the quarter.
  • Second-order convergence for the midpoint rule ($O(N^{-2})$) with simple operations.
  • No angle handling; complements chord-based perimeter via equal-angle steps.

How to use

  • Choose $N$ (e.g., $200$–$1000$) for the quarter; multiply results by $4$ for the circle.
  • Step-doubling check: compare $N$ vs $2N$ to estimate the residual gap between upper and lower.
  • Pair with equal-angle chords on the quarter to report both area and perimeter consistently.

See also

Core Turns · Δx Trapezoidal · Applications / Circle

Original Projection

The original GRG area method: sample the quarter along the x-axis with uniform step $\Delta x=r/N$. At each sample, project the height on the circle graph $y(x)=\sqrt{r^{2}-x^{2}}$ and build discrete rectangles. Two rules create a shrinking corridor: midpoint rectangles (upper) and right-endpoint rectangles (lower). Multiply by $4$ for the full circle. No angles are required in this block.

Setup

$$ \Delta x \;=\; \frac{r}{N}, \qquad x_k \;=\; k\,\Delta x \quad (k=0,\dots,N), \qquad x_{k+\tfrac12} \;=\; \Bigl(k+\tfrac12\Bigr)\Delta x, \qquad y(x) \;=\; \sqrt{\,r^{2}-x^{2}\,}. $$

Quarter-area sums (→ ×4 for the circle)

$$ \textbf{Midpoint (upper):}\quad A_{\mathrm{mid}}^{\mathrm{q}}(N) \;=\; \sum_{k=0}^{N-1} y\!\bigl(x_{k+\tfrac12}\bigr)\,\Delta x $$
$$ \textbf{Right-end (lower):}\quad A_{\mathrm{right}}^{\mathrm{q}}(N) \;=\; \sum_{k=0}^{N-1} y\!\bigl(x_{k+1}\bigr)\,\Delta x $$
$$ \textbf{Circle totals:}\quad A_{\mathrm{mid}} \;=\; 4\,A_{\mathrm{mid}}^{\mathrm{q}}, \qquad A_{\mathrm{right}} \;=\; 4\,A_{\mathrm{right}}^{\mathrm{q}} $$
$$ \textbf{Corridor:}\quad A_{\mathrm{right}} \;\le\; A_{\mathrm{true}} \;\le\; A_{\mathrm{mid}} $$
$$ \textbf{Estimate (optional):}\quad A_{\mathrm{avg}} \;=\; \frac{A_{\mathrm{mid}}+A_{\mathrm{right}}}{2} \quad\text{(use half-gap as an error bar).} $$

Notes

  • $y(x)=\sqrt{r^{2}-x^{2}}$ is concave on $[0,r]$; hence midpoint gives an upper estimate and right-endpoint a lower one.
  • Increase $N$ to tighten the corridor; publish $A_{\mathrm{avg}}$ with its half-gap if you need an error bar.

See also: Δx Trapezoidal · Core Turns · Applications / Circle