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.
GRG Mini app

GRG Mini app

Aug 24, 2025

GRG Mini-App

Explore Generative Radial Geometry (GRG): figures are generated from the center at a chosen resolution, without assuming π a priori. Angles are measured in turns \(u\in[0,1)\) with period-1 trig \(c(u)=\cos(2\pi u),\ s(u)=\sin(2\pi u)\).

Controls

N = 24
Circumference is always from equal-angle chords in turns.
Sphere uses octant symmetry (×8) and π-free operators; volume is O(N²).
%
“Internal dominance” compares N to 2N (self-consistency). “Victory” compares to the π-based value (external).

Circle (2D)

Area estimate
—
Area corridor (if applicable)
—
Circumference (chords)
—
Internal Δ (N → 2N), area
—
Internal Δ (N → 2N), circumf.
—
External error vs π-based, area
—
External error vs π-based, circumf.
—
area: — circ: —

Sphere (3D)

Surface (octant fusion ×8)
—
Volume (stacked quarters ×8)
—
Internal Δ (N → 2N), surface
—
Internal Δ (N → 2N), volume
—
External error vs 4πr²
—
External error vs (4/3)πr³
—
surface: — volume: —

How the GRG Mini-App works — Circle & Sphere

What you control

Inputs. Two modules:

  • Circle (2D): choose radius r, subdivisions N and the Area method:
    • Δx Trapezoidal — quarter → ×4 (concave arc, under-tiling, fast O(N⁻²)).
    • Original Projection — midpoint (upper) / right (lower) corridor, quarter → ×4.
    • Radial Sectors (turns) — full turn, equal-angle sectors (inscribed, product form).
    Note: for Δx/Original the slider N refers to the quarter; for Radial it refers to the full turn. Circumference always uses turns with equal-angle chords.
  • Sphere (3D): choose radius R, vertical slices Nz, radial trapezoids for quarter-areas Mr, and quarter-circumference subdivisions Mc.
  • Show traditional comparison toggles all external (vs π) quantities and the grey series in charts.
  • Show π estimates reveals π computed internally (from A/C in 2D; S/V in 3D).
  • Download CSV exports the full range (1…Nmax); Copy link preserves the current settings.

Status badges & thresholds

🟢 Internal dominance The internal change between the current resolution and the doubled one is small relative to the value:

\(\displaystyle \frac{\Delta_{\mathrm{int}}}{\text{value}} \le 0.15\%\).

🏆 Victory vs classical The external error is also small:

\(\displaystyle \frac{\Delta_{\mathrm{ext}}}{\text{value}_\pi} \le 0.10\%\).

Formulas used — Circle (2D)

  • Area (Δx Trapezoidal): quarter sum of trapezoids with uniform \(\Delta x=r/N\) → ×4 (Kahan summation for stability).
  • Area (Original Projection): midpoint (upper) / right (lower) corridor on the quarter → ×4 (Kahan).
  • Area (Radial Sectors, turns): \(A_N=\tfrac{N}{2}r^2\,s(\tfrac{1}{N})\) with \(s(u)=\sin(2\pi u)\).
  • Circumference (turns): equal-angle chords on the full turn: \(C_{N_{\text{full}}}=2N_{\text{full}}\,r\,s(\tfrac{1}{2N_{\text{full}}})\). For Δx/Original we use \(N_{\text{full}}=4N\); for Radial, \(N_{\text{full}}=N\).

Formulas used — Sphere (3D)

  • Surface: octant fusion of quarter-circumferences along meridian (\(\Delta s\)), then ×8.
  • Volume: octant stacking of quarter-areas along \(z\) with trapezoid in \(\Delta z\), then ×8.

What this app demonstrates

GRG reconstructs circle and sphere without assuming π. With angles in turns and quarter/octant symmetries, internal stabilization (dominance) precedes external agreement (victory). Construct first, constants emerge.

GRG Mini-App — 3 righe

GRG Mini-App — 3 righe

Layout a righe: Controls → Circle (2D) → Sphere (3D). Tutti gli slider sono two-way con i numeri; i turns (U) hanno preview live (corde uguali).

Controls

Circle

Angular (turns)

Δu = 1/48 · Δθ = 0.130899694 rad
U multiple of 4 ✓; U = 4·N·k ✓

Sphere

Slider (mirrors)

Circle (2D)

Area (GRG)
0
Corridor width (mid−right)
—
Circumference (chords)
0
Δint A (2N vs N)
—
Δint C (2U vs U)
—
Δext A vs πr²
—
Δext C vs 2πr
—
area: — circ: —
Anteprima: poligono inscritto con U settori uguali (corde uguali).

Sphere (3D)

Surface S (octant fusion ×8)
0
Volume V (stacked quarters ×8)
0
Δint S
—
Δint V
—
Δext S vs 4πR²
—
Δext V vs (4/3)πR³
—
surface: — volume: —
GRG for ellipses

GRG for ellipses

Aug 22, 2025

GRG for Ellipses — Constructive Area, Perimeter, and Equal-Area Sectors

The “Variable Radius” in GRG

The effective radius depends on the chosen direction.

  • From the center (angle \(\phi\) from the major axis):

    $$ r_c(\phi)=\frac{ab}{\sqrt{(b\cos\phi)^2+(a\sin\phi)^2}}. $$

  • From the focus (true anomaly \(\theta\)):

    $$ r_f(\theta)=\frac{p}{1+e\cos\theta},\qquad e=\sqrt{1-\frac{b^2}{a^2}},\qquad p=\frac{b^2}{a}. $$

  • Without trigonometry (slope \(m=\tan\phi\)):

    $$ r_c(m)=\frac{ab\,\sqrt{1+m^2}}{\sqrt{\,b^2+a^2m^2\,}}. $$

Ellipse Area in GRG

A1) Cartesian strips in the first quadrant

Split the semi-axis \(a\) into \(N\) parts: \(x_i=i\,a/N\) with \(\Delta x=a/N\). Heights \(y_i=b\sqrt{1-x_i^2/a^2}\).

$$ A_{1/4}^{(N)}=\sum_{i=1}^{N} y_i\,\Delta x,\qquad A^{(N)}=4\,A_{1/4}^{(N)}. $$
A1 — Cartesian strips in the first quadrant x–y axes, ellipse centered at (400,250) with a=240, b=160; shaded vertical strip inside Q1. +y +x Cartesian strips in Q1; then ×4
Quarter-by-strips: sum \(y_i\,\Delta x\) in Q1 and multiply by 4 for the ellipse area.

A2) Radial wedges from the center (by quarters)

With \(\Delta\phi=90^\circ/N\) and \(\phi_i=i\,\Delta\phi\):

$$ A_{1/4}^{(N)}=\sum_{i=0}^{N-1}\tfrac12\,r_c(\phi_i)\,r_c(\phi_{i+1})\,\sin\Delta\phi, \qquad A^{(N)}=4\,A_{1/4}^{(N)}. $$
A2 — Center-based wedges (quarter symmetry) Rays from center to ellipse boundary; the wedge between two rays approximates an area piece. C +x +y Center-based wedges (quarter symmetry)
Sum center-based wedges over a quarter (then ×4). Summing the chords between successive boundary points gives the corresponding perimeter.

A3) Radial wedges from the focus (natural for orbits)

With \(\Delta\theta=360^\circ/M\) and \(\theta_i=i\,\Delta\theta\):

$$ A^{(M)}=\sum_{i=0}^{M-1}\tfrac12\,r_f(\theta_i)\,r_f(\theta_{i+1})\,\sin\Delta\theta. $$
A3 — Focus-based sectors (equal areas → equal times) Sectors with apex at a focus; each filled sector has the same area increment ΔA (Kepler-2). F F′ periapsis → Equal-area sectors from the focus (constant ΔA; angle widths vary)
Equal-area stepping from the focus: constant \(\Delta A\) gives uniform time steps (Kepler’s 2nd law). Near periapsis the angular gaps shrink; away they widen.

Perimeter (sum of chords)

P1) Chords from the center (by quarters)

With \(\Delta\phi=90^\circ/N\):

$$ P_{1/4}^{(N)}=\sum_{i=0}^{N-1}\sqrt{\,r_c(\phi_i)^2+r_c(\phi_{i+1})^2-2\,r_c(\phi_i)\,r_c(\phi_{i+1})\cos\Delta\phi\,}, \qquad P^{(N)}=4\,P_{1/4}^{(N)}. $$

P2) Chords from the focus (full ellipse)

$$ P^{(M)}=\sum_{i=0}^{M-1}\sqrt{\,r_f(\theta_i)^2+r_f(\theta_{i+1})^2-2\,r_f(\theta_i)\,r_f(\theta_{i+1})\cos\Delta\theta\,}. $$

GRG Error Estimate (after-the-fact)

With step doubling the error typically decays ~quadratically. Method-agnostic (π-free):

  • Compute \(S_N\) and \(S_{2N}\) with the same operator (A1–A3, P1–P2).

  • Accelerated approximation:

    $$ S_\star \approx S_{2N}+\frac{S_{2N}-S_N}{3}. $$

  • Error estimate:

    $$ \varepsilon \approx \frac{|S_{2N}-S_N|}{3}. $$

Practical Pipeline (geometry → orbits)

  • Geometry: use A1 or A2 for area and P1 for perimeter with quarter symmetry.

  • Orbits: use A3. For uniform time steps, enforce constant-area sectors:

    $$ \tfrac12\,r_f(\theta_k)\,r_f(\theta_{k+1})\,\sin(\theta_{k+1}-\theta_k)=\Delta A. $$

  • The arc length per step is the P2 chord between successive points.