Wedge Visualization (Interactive)

Wedge Visualization (Interactive)

Sep 21, 2025

GRG — Wedge Visualization: Interactive Uniform vs Adaptive Discretization

This mini-demo provides concrete, reproducible numbers for wedge-stratified GRG. Adjust the controls to compare a uniform wedge count against an adaptive profile on the outer annulus, and see how certified corridor half-widths contract roughly like 1/\sqrt{W_k} for comparable ring budgets.

How the Demo Works (plain language)

  • Dyadic edges. Rings follow turns-first dyadic refinement with b∈ℕ≥0, N=2^b, and quarter sampling Δu=2^{-b-2}.
  • Corridor half-width. Each sector’s estimate has a certified interval [E^- , E^+]. The half-width is (E^+−E^-)/2; decisions stop when this width ≤ tolerance τ.
  • Field model. We synthesize an anisotropic field with mixed angular orders up to fourth order to stress the wedges; values are computed in-page so the numbers are concrete and repeatable.
$$ F(\rho,\theta)=A(\rho)\Big[1+\epsilon_2\cos 2(\theta-\theta_2)+\epsilon_3\cos 3(\theta-\theta_3)+\epsilon_4\cos 4(\theta-\theta_4)\Big], $$
$$ \textbf{Stop when}\quad \widehat{E}_b^{+}-\widehat{E}_b^{-}\le \tau,\qquad \text{with}\ \ \Delta u=2^{-b-2}. $$

Controls

Results — Per-Sector Estimates on the Outer Annulus

Hover or tap a bar to read exact values (estimate and corridor half-width). The polar plots show the directional pattern and corridor bands.

Uniform W — outer annulus sectors (bars = estimate, whiskers = corridor half-width).
Adaptive outer Wk — tighter corridors where anisotropy is richer.
Polar — Uniform W.
Polar — Adaptive Wk.

Concrete Numbers (auto-computed)

The table aggregates sector values (outer annulus) for the current settings. Replace later with CSVs from your scripts; the structure is the same.

Notes & Practical Guidance

  • When to prefer GRG. As angular richness grows or is localized, adaptive W_k exposes structure with linear O(K·W) cost and certified decisions via τ.
  • When low-order bases suffice. Near-radial scenes with mild 2-fold variation are well captured by a short angular basis; choose GRG when auditability (corridors) matters.
  • Next steps. Publish side-by-side benchmarks (Zernike, B-splines, Fourier–Bessel) with the same corridor-aware stopping rule and public scripts/datasets.
Residual Critiques v2

Residual Critiques v2

Sep 20, 2025

GRG — Residual Critiques: Demo Status, Benchmarks-in-Progress, Complex Anisotropy, and Cost Comparison

This page acknowledges what is still pending (public demo and published benchmarks) and adds a more demanding wedge-stratified example (freeform-like anisotropy). We also outline when GRG is preferable versus angular bases (e.g., Zernike) as W grows.

1) Mini-App Demo & Screencast — Current Status

  • Badge: Demo forthcoming — the public link and 2–3 min screencast will appear in the site header Community → Mini-App and in this page’s hero as a primary CTA.
  • Presets: Optics (MTF/PSF), Chemistry (RDF). Each preset provides CSV with estimate and certified corridors.
  • Transparency: until the link is live, this page keeps the badge above the fold, so the gap is explicit.

2) Benchmarks — What Will Be Published

Side-by-side, corridor-aware comparisons vs Zernike, polar B-splines, Fourier–Bessel, and pixel discretizations. Datasets and scripts are public; winner is declared only when corridor half-width is < task tolerance \tau and intervals do not overlap.

$$ \text{Report}\quad \|T_{\text{GRG}} - T_{\text{ref}}\|_2,\ \text{runtime},\ \text{memory}\quad\text{vs}\quad b,\ N=2^b,\ \text{problem size}. $$

3) Complex Anisotropy — Freeform-Like Example (Concrete K, Adaptive W)

Scenario. A weakly freeform optical surface induces an intensity field with mixed angular content up to fourth order. We integrate over dyadic rings and adaptive wedges to capture anisotropy where gradients are steep.

$$ F(\rho,\theta) \;=\; A(\rho)\,\bigl[\,1+\epsilon_2\cos 2(\theta-\theta_2)+\epsilon_3\cos 3(\theta-\theta_3)+\epsilon_4\cos 4(\theta-\theta_4)\,\bigr] $$
  • Rings: K=32 (dyadic edges; quarter sampling uses \Delta u=2^{-b-2} with b=5).
  • Adaptive wedges: inner rings (\rho<0.4) use W_k=8; mid (0.4\le \rho<0.7) use W_k=12; outer (\rho\ge 0.7) use W_k=16.
  • Parameters: A(\rho)=\rho^{0.6}, \epsilon_2=0.14, \epsilon_3=0.07, \epsilon_4=0.05, phases \theta_2=15^\circ, \theta_3=50^\circ, \theta_4=100^\circ.

Per-sector results (outer annulus, \rho\in[0.7,1))

Values below are illustrative and show how corridors tighten with higher W_k where angular content is richer.

Uniform W = 8

Sector wEstimateCorridor ±
10.121±0.0070
20.128±0.0072
30.136±0.0074
40.142±0.0075
50.141±0.0075
60.134±0.0073
70.126±0.0071
80.121±0.0070

Adaptive Wk (8→12→16)

Sector wEstimateCorridor ±
10.121±0.0046
20.129±0.0048
30.138±0.0049
40.144±0.0050
50.143±0.0050
60.135±0.0048
70.127±0.0047
80.121±0.0046

Takeaway. Adaptive wedges localize work where anisotropy is strongest; corridor half-widths fall ≈ proportional to 1/\sqrt{W_k} for comparable ring budgets, exposing higher-order angular content without oversampling isotropic regions.

4) Cost for Large W — When GRG Beats/Matches Angular Bases

With K rings and W wedges, work and memory scale as:

$$ \text{GRG (wedge)}:\ \ \text{work}\sim O(K\cdot W),\quad \text{memory}\sim O(K\cdot W). $$

Rule of thumb. If anisotropy is low (energy mostly in orders ≤ 2), a short Zernike truncation can be competitive. As angular richness grows or is localized, GRG with adaptive W_k becomes preferable—no global orthogonalization, easy corridor control, and targeted refinement.

Scenario Angular content Suggested method Rationale
Near-radial optics Dominant 2-fold Zernike (low order) or GRG with W=8 Few modes suffice; GRG adds corridors if decisions must be certified.
Freeform moderate Up to 4-fold, localized GRG adaptive (W_k=8→16) Targeted wedges reduce cost vs global mode expansions.
Highly anisotropic Mixed orders, sharp sectors GRG adaptive with corridor-based stopping Local control; corridors guide where to refine.

5) Corridor-Based Stopping (unchanged)

Independently of W, choose the smallest dyadic level b such that corridor half-width ≤ tolerance \tau.

$$ \textbf{Stop when}\quad \widehat{E}_b^{+}-\widehat{E}_b^{-}\le \tau. $$

What Changes with This Update

  • Honest status: demo still pending (badge visible); screencast to ship with the demo.
  • Benchmarks: publication plan remains; results will be linked into this page once released.
  • Complex example: freeform-like field with higher-order anisotropy and adaptive W_k, including per-sector corridors.
  • Cost clarity: when to prefer GRG vs short angular bases as W grows.
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.
The Circle

The Circle

Aug 18, 2025

The Circle — Radius-First (Three Methods)

Principle

The circle is generated from the center. The radius is the only primitive; the perimeter is the path traced by the radius, and the area is the surface swept by its rotation. We work in turns (fractions of a full rotation), not in sexagesimal degrees.

Angles in turns & radial node

Let \(u\in[0,1)\) be an angle in turns (period \(1\)). We use period-1 trigonometric functions \(c(u), s(u)\) with \(c(0)=1, s(0)=0\) and \((c(\tfrac14), s(\tfrac14))=(0,1)\). The exact radial node is:

$$ P(u)=\bigl(r\,c(u),\; r\,s(u)\bigr). $$

Bit-friendly turns

Use fixed-point angles \(u = k/2^B\). Picking \(N\) that divides \(2^B\) makes equal-angle steps exact (\(\Delta u=1/N\)).
Optional: LUT + interpolation for \(c(u), s(u)\).

Method scope & scaling (quick key)

  • Radial Sectors (equal-angle, full turn) → no ×4.
  • Δx Trapezoidal (quarter tiling) → ×4 to get the full circle.
  • Original Projection (midpoint/right on a quadrant) → ×4.
Perimeter (equal-angle chords) is naturally full-turn (no ×4) and shared across methods.

Perimeter (shared across methods)

Equal-angle chords (full turn)

$$ \boxed{\, C_N \;=\; 2\,N\,r\,s\!\Bigl(\tfrac{1}{2N}\Bigr) \,} $$
Inscribed regular \(N\)-gon in turns. Monotone ↑ in \(N\). No quadrant scaling.
Perimeter by equal-angle chords (Δu = 1/N turns) Cartesian axes; circle r=100; sample radii and dashed chords (example N=12).
Perimeter by equal-angle chords: \(C_N = 2N r\, s(1/(2N))\) (full turn).

Area — three interchangeable builds (pick one)

Δx Trapezoidal (quarter → ×4)

$$ \Delta x=\frac{r}{N},\quad x_k=k\,\Delta x,\quad y_k=\sqrt{r^2-x_k^2} $$
$$ A_{\text{quarter}}(N)=\frac{\Delta x}{2}\!\left(y_0+2\sum_{k=1}^{N-1}y_k+y_N\right),\qquad A(N)=4\,A_{\text{quarter}}(N). $$
Under-approx on a concave arc; smooth convergence \(O(N^{-2})\).

Original Projection (midpoint/right corridor, quarter → ×4)

$$ \Delta x=\frac{r}{N},\ \ x_{k+\tfrac12}=(k+\tfrac12)\Delta x,\ \ y(x)=\sqrt{r^2-x^2} $$
$$ A^{\text{quad}}_{\text{mid}}=\sum_{k=0}^{N-1} y\!\bigl(x_{k+\tfrac12}\bigr)\,\Delta x,\qquad A^{\text{quad}}_{\text{right}}=\sum_{k=0}^{N-1} y\!\bigl(x_{k+1}\bigr)\,\Delta x, $$
$$ A_{\text{mid/right}}=4\,A^{\text{quad}}_{\text{mid/right}},\qquad A_{\text{right}}\le A_{\text{true}}\le A_{\text{mid}}. $$
Midpoint is second-order; corridor shrinks with \(N\).

Radial Sectors (equal-angle sectors, full turn)

$$ \boxed{\, A_N = \tfrac{N}{2}\,r^2\,s\!\Bigl(\tfrac{1}{N}\Bigr) \,} $$
Sum of \(N\) congruent isosceles triangles with included angle \(1/N\) turns (inscribed lower; monotone ↑ in \(N\)). No ×4.
Quarter area by Δx trapezoids (under the arc) Cartesian x–y axes; quarter arc with Δx verticals and interior trapezoids. Δx trapezoids
Quarter area by Δx trapezoids; multiply by 4 for the full circle.
Radial sector tiling (equal-angle sectors, full turn) Two example sectors on a unit circle (illustrative N=8).
Radial sectors: \(A_N=\tfrac{N}{2}r^2\,s(1/N)\) (full turn, no ×4).

Practical notes

  • Pick \(N\). If angles are \(k/2^B\) turns, choose \(N\mid 2^B\). Use step-doubling (\(N\!\to\!2N\)) to gauge the gap.
  • Monotonicity. \(C_N\) ↑ with \(N\). Δx trapezoids ↑ toward area (under-tiling). Original corridor shrinks. Radial sectors ↑ (inscribed).
  • All in turns. No degrees, no π in the constructive steps.