Scientific correction — 13 October 2026

The application intervals and charts on this historical page are proposals or illustrations, not certified bounds for arbitrary fields or validated performance results. Certification requires assumptions on each integrand, a proved quadrature residual, rounding control, and propagation through subsequent transformations. Regular circular-sector bounds do not automatically bound optical, chemical, or anisotropic quantities.

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 illustrative corridor half-widths (not certified error bounds) 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 proposed (not independently 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

Rings K: Uniform W: Adaptive outer W_k: Tolerance τ:

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 W_k — tighter corridors where anisotropy is richer.

Polar — Uniform W.

Polar — Adaptive W_k.

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 proposed (not independently 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.