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

Chemistry

Chemistry

Sep 1, 2025

Generative Radial Geometry for Chemistry: From Conformers to MOFs and Solvent Shells

Angles in turns. We sample directions with turns \(u\in[0,1)\) at \(\Delta u=1/N\) (full turn). This keeps angular steps digital-native and π-free.
g(r)

Generative Radial Geometry (GRG) offers a center-out way to construct and search chemical structures. By organizing atoms, coordination sites, pores, or solvent molecules in radial shells with controlled distances and angles, GRG cuts down the combinatorial explosion and makes 3D models that are physically plausible and immediately ready for simulation.

Where GRG helps in chemistry

  • Molecule modeling & design: targeted conformer generation around rings and chiral centers; pre-screening of virtual libraries to propose sterically sensible radial scaffolds.
  • Crystallography & solids: crystal packing via radial neighborhoods of a repeat unit; localized models of defects and vacancies.
  • Molecular dynamics: fast computation of radial distribution functions (RDF) and balanced initialization of solvents/ions around a solute.
  • Porous materials (MOF/COF): radial placement of metal nodes around linkers; geometric control of free radii to tune pore accessibility.
  • Supramolecular self-assembly: prediction of micelles/liposomes and core–shell architectures by concentric, composition-aware layers.
  • Drug design & docking: radial abstractions of binding pockets and “radial grids” of electrostatic potential to guide polar/apolar moieties.
GRG Modes & Selection (Chemistry). We use three base modes: Δx Trapezoidal (area-like sums on a quarter, lower estimate), Projection Corridor (paired lower/upper bounds on a quarter), and Chord-Sum at Δu (loops on the full turn, lower; pair with tangent-sum for an upper bound). In 3D they extend naturally to spherical shells/belts/frusta.

Δx Trapezoidal (Cartesian)

Target: fast area/volume roll-ups • Estimate: lower • Domain: quarter (×4)

  • RDF histograms (shell counts), coordination numbers (baseline).
  • Solvent occupancy per shell; pore cross-section scans.

Projection Corridor

Target: certified bounds on integrals • Estimate: lower & upper • Domain: quarter (×4)

  • Bounds for \(N(R)=4\pi\rho\!\int_0^R\!g(r)\,r^2dr\) and shell occupancies.
  • Stop when gap ≤ tolerance; use to compare force fields robustly.

Chord-Sum at Δu (Radial)

Target: ring/loop sampling & perimeter • Estimate: lower (add tangent-sum ↑) • Domain: full turn

  • Uniform directions \(u\in[0,1)\) for conformers/solvent placement.
  • Pore apertures & ring tiling in MOFs/COFs; angular RDF slices.

Picker by task. Need speed → Trapezoid; need guarantees/threshold radii → Corridor; need angular uniformity/phase on rings → Chord-Sum (± tangent-sum); volumetrics → shell/belt extensions.

Core GRG idea for chemical systems

Place partners within radial bounds and enforce angular/steric constraints; then score candidates and iterate. Typical constraints:

$$ r_{\min} \;\le\; r \;\le\; r_{\max}, \qquad \angle(\text{A–C–B}) \;\ge\; \alpha_{\min}. $$

For distributions, the classic RDF (pair correlation) reads:

$$ g(r) \;=\; \frac{1}{4\pi r^{2}\,\rho\,N}\, \sum_{i\ne j}\, \delta\!\bigl(r-\lVert \mathbf{r}_i-\mathbf{r}_j \rVert \bigr). $$

GRG modes: Use Trapezoid to accumulate RDF shell counts quickly; switch to Corridor for certified bounds on coordination \(N(R)=4\pi\rho\!\int_0^R\!g(r)\,r^2dr\); use Chord-Sum at \(\Delta u=1/N\) to ensure uniform angular sampling when RDF is evaluated over directional sectors.

Worked examples

1) Conformer generation around a chiral center

Goal: explore the spatial arrangements around a chiral carbon and select the most stable isomer.

  1. Define the chiral center C and the four substituents (A, B, C, D).
  2. Choose distance bounds for C–X: e.g., 1.45–1.55 Å.
  3. Partition the interval into k concentric shells (e.g., 5).
  4. On each shell, sample directions with turns \(u\in[0,1)\) at \(\Delta u=1/N\), enforcing tetrahedral angles ≈\(109.5^\circ\) and min separations.
  5. Score with a light force field (e.g., UFF) and retain the lowest-energy 10 conformers.
C* A B C D Constraints: 1.45 ≤ r(C–X) ≤ 1.55 Å ∠ ≈ 109.5° (tetrahedral) Shells: 5

GRG modes: Chord-Sum (full turn, turns \(u\)) for uniform angular sampling and ring-wise sterics; Trapezoid to tally shell feasibility quickly; Corridor to certify bounds on steric/energy scores over shells before QM refinement.

2) Crystal packing of a simple MOF

Goal: arrange metal nodes and organic linkers in a cubic unit cell using radial symmetry.

  1. Place a metal node M (e.g., Zn2+) at the cell center; four linkers L at the corners of a radial tetrahedron.
  2. Set coordination bounds (e.g., M–O = 1.9–2.2 Å).
  3. For each L site, sample positions on a spherical shell about M, keeping \(\angle\text{L–M–L}\approx 109.5^\circ\).
  4. Tile the cell periodically with vectors \((a,0,0)\), \((0,a,0)\), \((0,0,a)\).
  5. Run a short geometry minimization (e.g., Powell) to relieve strain.
M L L L L

GRG modes: Chord-Sum for ring apertures and angular registry of linkers (periodic tiling over full turn); Trapezoid for cross-sectional pore area scans (quarter ×4); Corridor to bound pore-limiting/free-sphere radii derived from those scans.

3) Solvent bubble for MD around a solute

Goal: generate a balanced water sphere around a charged solute without overcrowding.

  1. Locate the solute center of mass and choose \(R_1\) (solute + first shell) and \(R_2\) (outer boundary).
  2. Split \([R_1,R_2]\) into \(N\) shells (typically 3–5).
  3. For each shell, place water molecules by sampling directions uniformly with turns \(u\) and \(\Delta u=1/N\); set distances \(r\) within that shell and enforce \(d_{\min}(\mathrm{O}, \text{solute}) \ge 2.5\ \text{Å}\).
  4. Assign water orientations (H–O–H) with small random perturbations.
  5. Export to PDB and run a short heating/minimization to relax the sphere.
solute

GRG modes: Chord-Sum for uniform angular placement/orientations; Trapezoid for quick shell occupancy and radial density; Corridor to bracket hydration numbers \(N(R)\) and to stop when bounds meet your tolerance.

4) Docking on radial electrostatic grids (quick setup)

Goal: approximate binding pocket neighborhoods with radial shells and guide pose sampling.

  1. Center a pocket reference and define shells \([r_k,r_{k+1}]\) plus ring samples at turns \(u_j=j/N\).
  2. Precompute coarse electrostatic potential on these cells; score ligand fragments by shell and ring.
  3. Promote top cells to fine sampling; enforce sterics and ring-wise clashes; export poses to your docking engine.

GRG modes: Chord-Sum for ring coverage of orientations; Trapezoid to roll up cell scores radially; Corridor to keep upper/lower envelopes on pocket accessibility/energy as you refine.

From generation to computation

  • Search space reduction: symmetries and radial bounds focus the exploration on chemically sensible regions.
  • 3D realism with control: enforce minimum/maximum inter-center distances and angles while sampling.
  • Drop-in to pipelines: GRG output meshes with QM, MD, and docking workflows (e.g., UFF pre-screen → PDB export → MD heating → production run).

GRG modes (pipeline): Use Trapezoid during fast scans; switch to Corridor to certify thresholds (coordination/hydration/pore radii); validate angular uniformity/perimeters with Chord-Sum (± tangent-sum) before committing to costly runs.

Conclusion

Across molecular design, crystal engineering, porous frameworks, self-assembly, and simulation setup, GRG brings a clear center-out logic: define radial bounds, discretize into shells, generate angles under constraints, and score. Choosing among Δx Trapezoidal, Projection Corridor, and Chord-Sum at Δu—or combining them—yields fast baselines, certified bounds, and robust angular control for meaningful chemistry.

RDF & Coordination via GRG Corridor — bounds on N(R) and Rn

Definitions. With bulk number density \(\rho\) and radial distribution \(g(r)\), the coordination number inside radius \(R\) is

$$ N(R)=4\pi\rho\int_0^{R} g(r)\,r^2\,dr. $$

Discrete GRG shells. Choose radii \(0=r_0turns \(u\in[0,1)\)) and use min/max over the sector.

Corridor bounds (cumulative up to \(R\)).

$$ N^{\downarrow}(R)=\rho\!\!\sum_{k:\,r_{k+1}\le R}\! V_k\,g_k^{\min},\qquad N^{\uparrow}(R)=\rho\!\!\sum_{k:\,r_{k+1}\le R}\! V_k\,g_k^{\max}, $$
where \(g_k^{\min},g_k^{\max}\in\{g_k^{\mathrm{in}},g_k^{\mathrm{out}}\}\) for smooth/monotone segments, or the angular min/max in that shell otherwise. Always \(N^{\downarrow}(R)\le N(R)\le N^{\uparrow}(R)\).

Bracketing a coordination radius. For a target coordination \(n\) (e.g., “first shell”):

  1. \(R_{n}^{-}=\min\{R:\,N^{\uparrow}(R)\ge n\}\) is a lower bound of the true \(R_n\).
  2. \(R_{n}^{+}=\min\{R:\,N^{\downarrow}(R)\ge n\}\) is an upper bound of the true \(R_n\).
  3. Refine by halving shell width \(\Delta r\) near the bracket or by bisecting within the shell until \(R_{n}^{+}-R_{n}^{-}\le\) tolerance.

Method picker. Isotropic liquids: use shells only (fast). Structured/anisotropic systems or near interfaces: split shells into polar cells; sample azimuth with Chord-Sum (\(\Delta u=1/N\)) to keep directional coverage, then apply the corridor per cell. Use Trapezoid for quick scans of \(N(R)\); switch to Corridor when you need guarantees.

N_up above true N, N_down below, horizontal n line; bracket on R. R N n R− R+

Practical notes. Use \(\Delta r\) small enough to resolve the first peak of \(g(r)\) (typ. 0.02–0.05 nm); respect PBC and the minimum-image convention (limit \(R\le L/2\) in a cubic box). For noisy \(g(r)\), smooth lightly after computing corridor bounds, not before.

Optics

Optics

Sep 1, 2025

Generative Radial Geometry for Optics: From Aspheres to Metasurfaces

Generative Radial Geometry (GRG) treats optical elements as radially generated structures optimized from the center outward. This viewpoint enables targeted aberration correction, compact/lightweight systems, and fast iteration via simulation—spanning refractive aspheres, GRIN profiles, diffractive zone plates, and radially programmed metasurfaces.

What GRG means in optics

In GRG, a surface or refractive-index profile is parameterized radially and then optimized against a merit function (e.g., wavefront error, Strehl, or MTF) over the full pupil. Rather than restricting to a low-order conic/asphere, GRG explores richer radial degrees of freedom that better align the phase over wide apertures and fields.

GRG Modes & Selection for Optics. We use three base modes: Δx Trapezoidal (fast area-like radial sums, lower estimate), Projection Corridor (paired lower/upper area bounds via right/midpoint sampling), and Chord-Sum at Δu (circumference/loops on the full turn, lower; pair with tangent-sum for an upper bound). In optics they map to pupil integrals (energy/WFE), certified bounds on merit terms, and phase/loop controls on rings/zones.

Δx Trapezoidal (Cartesian)

Target: area-weighted radial sums (pupil cross-sections) • Estimate: lower

  • Encircled energy, WFE/OPD RMS, throughput vs radius.
  • Baseline fast integration in design loops.

Projection Corridor

Target: same as above • Estimate: lower & upper (certified)

  • Bounds for energy, WFE, Strehl/MTF contributions.
  • Use to stop/refine when gap ≤ tolerance.

Chord-Sum at Δu (Radial)

Target: loops on rings (full turn) • Estimate: lower (pair with tangent-sum ↑)

  • Zone edges, phase wraps (2π), vortex/Bessel rings.
  • Perimeter/phase uniformity & ring tiling quality.

Picker by task. Quick merit updates → Trapezoid; need guarantees → Corridor; ring/zone phase & perimeter → Chord-Sum (± tangent-sum); 3D volume/weight/thermal → shell/frusta extensions.

Why GRG improves optical design

  • Targeted aberration correction: spherical aberration, coma, and astigmatism can be minimized across large apertures by flattening phase error over the beam.
  • Compact, lightweight systems: high-performance single elements can replace multi-lens groups; GRG also supports radial GRIN to combine focus and correction in one body.
  • Advanced diffractive elements: optimized radial zone partitioning for resolution, efficiency, and robustness—supporting multi-order/multi-focal designs.
  • Metasurfaces & LIDAR: radial placement/shaping of nano-scatterers for phase/amplitude control (beam shaping, Bessel/vortex), wide-FOV collimation, compact 360° sensing.
  • Rapid simulation-to-prototype: GRG loops tightly with FEM/FDTD engines to explore candidate radial profiles quickly.

Design workflow (radial parametrization → merit function → solver)

A typical GRG workflow parameterizes the sag or index as a radial series; samples the pupil; defines a merit function over wavefront or image metrics; and solves for the radial coefficients (often via linearization around a base conic).

Radial surface model

We model the sag z(r) as a base conic plus a radial polynomial tail:

$$ z(r) \;=\; \frac{c\,r^{2}}{\,1+\sqrt{\,1-(1+k)c^{2}r^{2}\,}\,} \;+\; a_{4}r^{4} \;+\; a_{6}r^{6} \;+\; a_{8}r^{8}. $$
Axis r vs z(r); blue curve shows base conic; accent curve shows GRG-corrected sag. r z base conic GRG tail

Sampling and merit function

Sample radii across the clear aperture and minimize wavefront residual at wavelength λ for the target focal length:

$$ \min_{a_4,a_6,a_8}\;\sum_{i} \bigl[\Phi\bigl(z(r_i)\bigr)\bigr]^2, $$
Circle aperture with 5 concentric rings and sample points.

Linearized solve

Linearizing in the coefficients leads to a system

$$ M \begin{pmatrix} a_{4}\\[2pt] a_{6}\\[2pt] a_{8} \end{pmatrix} \;=\; b. $$

GRG modes: Use Trapezoid for fast radial contributions to WFE/energy; switch to Corridor when you need a guaranteed bound on residuals; use Chord-Sum on rings to enforce phase consistency (2π wraps) and to check zone/perimeter uniformity; add tangent-sum to bracket loop metrics.

Worked example: aspheric–radial singlet at λ = 550 nm

For a collimated beam at λ=550 nm focused to f=100 mm with D=25 mm, take a base paraboloid (k=–1) with curvature c=0.010 mm⁻¹ (R=100 mm). Solving the linearized system yields:

$$ \begin{aligned} a_{4} &= 2.50\times 10^{-11}\,\mathrm{mm}^{-3},\\ a_{6} &= -1.20\times 10^{-16}\,\mathrm{mm}^{-5},\\ a_{8} &= 3.00\times 10^{-22}\,\mathrm{mm}^{-7}. \end{aligned} $$

With these coefficients, the residual wavefront error across the pupil falls below λ/20 and the MTF at 50 lp/mm improves by ~30% relative to the spherical counterpart:

$$ \begin{aligned} \mathrm{WFE}_{\mathrm{rms}} &\le \frac{\lambda}{20},\\ \mathrm{MTF}\!\left(50\,\mathrm{lp/mm}\right) &\approx 1.30\,\times\,\mathrm{MTF}_{\text{spherical}}. \end{aligned} $$
MTF curves vs spatial frequency; GRG shows higher contrast near 50 lp/mm. spatial freq (lp/mm) MTF 50 ~30% gain @ 50 lp/mm (GRG) spherical GRG Two small fan plots suggesting reduced aberrations after correction. Ray fan (tangential) Ray fan (sagittal)

GRG modes: Trapezoid for quick MTF/Strehl-related radial sums; Corridor to certify WFE/energy targets before freezing coefficients; Chord-Sum (+ tangent-sum) to monitor ring-wise phase/perimeter and edge artifacts.

Diffractive optics and zone plates

GRG optimizes the radial subdivision and phase steps of diffractive optical elements (DOEs) to jointly maximize resolution, first-order efficiency, and fabrication tolerance, and can be extended to multi-order/multi-focal targets.

Concentric rings with varying widths representing an optimized radial phase profile.

GRG modes: Chord-Sum to place ring edges by phase advance along the circumference (2π steps, vortex charge checking); Corridor to bound first-order efficiency/EE across radii; Trapezoid for throughput and fill-factor roll-ups.

Metasurfaces and 360° sensing (LIDAR)

For metasurfaces, GRG prescribes the radial distribution and geometry of nano-scatterers (meta-atoms) to deliver the desired phase/amp profile—enabling beam shaping (Bessel, vortex), wide-FOV collimation, or compact 360° sensing modules.

Dots arranged in concentric rings with varying densities and azimuthal offsets.

GRG modes: Chord-Sum for ringwise phase conformity and azimuthal uniformity (Bessel/vortex); Corridor to bracket efficiency across pupil zones under scatterer quantization; Trapezoid for fast aggregate metrics during layout sweeps.

GRIN materials and fabrication

Beyond pure surface shaping, GRG covers radial gradient-index (GRIN) profiles co-optimized with geometry, compatible with precision 3D printing, advanced lithography, and layered deposition processes.

Circle with radial gradient indicating higher index in the center. n(r)

GRG modes: Trapezoid/Corridor for OPD and energy integrals through shells (robustness to dispersion/tolerances); Chord-Sum to verify iso-phase circles after co-optimizing n(r) + sag.

Simulation loop and time-to-prototype

A GRG toolchain linked to wave/EM solvers rapidly screens candidate radial profiles, reducing the number of physical iterations needed before fabrication.

Blocks: Parametrization → Solver → EM Simulation → Tolerance → Prototype. Radial parametrization Solver EM simulation Tolerance Prototype

GRG modes: Loop fast with Trapezoid; certify with Corridor (stop when the bound gap is within tolerance); validate ring/zone behavior with Chord-Sum (+ tangent-sum) before tape-out.

Practical notes: tolerances, costs, and when GRG is worth it

  • Tolerancing: include manufacturing/assembly error budgets in the merit function so the solved profile remains robust after polishing, litho rounding, or index dispersion.
  • Cost trade-offs: in commodity systems, classic low-order aspheres may stay cheaper; GRG shines when performance/weight constraints justify advanced profiles.
  • System impact: a single GRG element can replace a small lens group—simplifying alignment and improving thermal stability—especially with GRIN or metasurface layers.

Conclusion

GRG augments—not replaces—classical optical design. By combining Δx Trapezoidal, Projection Corridor, and Chord-Sum at Δu (with tangent-sum for upper loop bounds), you get fast merit updates, certified guarantees, and precise ring/zone control across refractive, diffractive, GRIN, and metasurface optics.

Encircled Energy (EE) via GRG Corridor — bounds on Rτ

Definition. For a pupil-normalized PSF intensity \(I\), the encircled energy at radius \(R\) is

$$ E(R)=\frac{\iint_{x^2+y^2\le R^2}\! I(x,y)\,dA}{\iint_{x^2+y^2\le R_{\max}^2}\! I(x,y)\,dA}\,. $$

Axisymmetric shortcut. If \(I(x,y)=I(r)\) then

$$ E(R)=\frac{2\pi\int_0^{R} r\,I(r)\,dr}{2\pi\int_0^{R_{\max}} r\,I(r)\,dr}\,. $$

GRG corridor (discrete). Choose radii \(0=r_0

Lower/upper cumulative EE (up to the smallest index with \(r_{k+1}\le R\)):

$$ E^{\downarrow}(R)=\frac{\sum_{k\,:\,r_{k+1}\le R} A_k\,I_{k}^{\min}}{\sum_{k=0}^{K-1} A_k\,I_{k}^{\mathrm{mid}}},\qquad E^{\uparrow}(R)=\frac{\sum_{k\,:\,r_{k+1}\le R} A_k\,I_{k}^{\max}}{\sum_{k=0}^{K-1} A_k\,I_{k}^{\mathrm{mid}}}, $$
where \(I_k^{\min}, I_k^{\max}\) can be chosen as \(\{I_k^{\mathrm{out}}, I_k^{\mathrm{in}}\}\) for monotone-decreasing \(I(r)\), or via angular min/max for general PSFs. Always \(E^{\downarrow}(R)\le E(R)\le E^{\uparrow}(R)\).

Bracketing the EE radius. For a target \(\tau\) (e.g., 0.80):

  1. \(R_{\tau}^{-}=\min\{R:\,E^{\uparrow}(R)\ge \tau\}\) is a lower bound on the true \(R_{\tau}\).
  2. \(R_{\tau}^{+}=\min\{R:\,E^{\downarrow}(R)\ge \tau\}\) is an upper bound on the true \(R_{\tau}\).
  3. Refine by increasing annuli (or bisect within the bracket). Stop when \(R_{\tau}^{+}-R_{\tau}^{-}\le\) tolerance.

Method picker. Axisymmetric/on-axis: use annular corridor above (fast). General PSF or field points: use GRG Projection Corridor on a quarter (area bounds), or polar cells with per-cell min/max. Use Chord-Sum (+ tangent-sum) to validate ring perimeter/phase uniformity when EE is tied to zone edges.

E_up above true E, E_down below, horizontal tau line; bracket on R. R E τ Rτ− Rτ+
Astronomy & Cosmology

Astronomy & Cosmology

Aug 31, 2025

Generative Radial Geometry in Astronomy: A New Paradigm for Radial Modeling

Category: Generative Radial Geometry (GRG)
Tags: astronomy, cosmology, radial models, gravitational lensing, data visualization

Abstract. Generative Radial Geometry (GRG) is a discrete, π-free approach that models shapes from the inside-out via radial generation. Because many cosmic phenomena are radially organized—from galactic halos to gravitational lenses—GRG offers natural, efficient tools to simulate, analyze, and visualize the universe.

GRG notation. Angles are fractions of a full turn: u∈[0,1). A turn-based logarithmic spiral is written as r = a·e^{βu}, where β controls growth per turn. This avoids π and keeps the representation digital-native.

Hero — Radial Layers with Logarithmic Arms (turn-based)

Cosmic radial layers with logarithmic spiral arms Concentric radial layers and two opposite turn-based logarithmic spiral arms r=a·e^{βu}.
Figure 0 — Radial layers with two opposite logarithmic spiral arms r=a\,e^{\beta u}.

Radial Mapping of Stellar and Galactic Systems

Galaxies and clusters frequently exhibit radial or pseudo-radial structure. GRG models such systems as cumulative radial layers:

$$ \rho(r)=\sum_{k=1}^{N} f_k(r) $$

This representation captures luminosity gradients and supports multi-scale “zoom” without discontinuities.

Dark Matter Distribution

Halo profiles are commonly modeled by radial functions such as the Navarro–Frenk–White (NFW) profile:

$$ \rho_{\mathrm{NFW}}(r)\;=\; \frac{\rho_0}{\big(\tfrac{r}{r_s}\big)\,\big(1+\tfrac{r}{r_s}\big)^2} $$

In GRG, synthetic halos arise from discrete radial accumulation, enabling efficient comparisons with survey data.

Gravitational Lensing Simulations

Deflection stems from radial potentials. Under GRG, a quick discretized approximation is:

$$ \vec{\alpha}(r)\;\propto\;\frac{M(r)}{r^2}\,\vec{r} $$

where M(r) is the generative radial mass. The map below shows curved light-ray paths and equipotential-like curves around a central mass.

Gravitational lensing map (radial potential) A central mass with families of equipotential-like curves and bent light paths.
Figure 2 — Radial potential and light deflection.

Spherical Surveys and Data Navigation

Large surveys (e.g., GAIA) live in spherical coordinates. GRG promotes seamless inside-out navigation using radial and angular indexing:

$$ \begin{aligned} \text{radial index:}\quad& z_k = k\,\Delta z,\; k=0,\dots,N_z \\ \text{angular index:}\quad& u_j = j\,\Delta u,\; \Delta u=\tfrac{1}{N_u} \\ \text{radial bound:}\quad& \rho_k=\sqrt{R^2-z_k^2} \end{aligned} $$

This allows continuous navigation across scales and directions without discontinuities.

Radial and angular indexing for spherical surveys Concentric rings and angular sectors; sample stars mark indexed bins.
Figure 3 — Rings + angular sectors for within-r queries and multi-resolution bins.

Computational Benefits

  • Multi-scale coherence: continuous zoom from core to periphery
  • Simplified algorithms in radial/turn coordinates
  • Natural fit for circular or spherical symmetries
  • Efficient indexing and clustering (e.g., “find all stars within r = 5 pc”)
  • Digital-native arithmetic: sums and square roots, π-free

Conclusion

Applying GRG to astronomy yields natural models of cosmic structure, high-fidelity simulations, and streamlined analysis pipelines. It couples geometric intuition with computational efficiency and offers a practical language for radial phenomena.

π (GRG)

π (GRG)

Aug 25, 2025

Historical Comparison Page

This page preserves earlier experiments involving a circular constant. It is not the current GRG construction specification and does not establish a computational advantage. Read the current geometric formulation. The calculators and efficiency claims on this historical page require separate validation.

What this page is about

Here we determine π by looking at the circle from the inside. We rotate a central ray with equal step and sum only internal chords. If you “really want π”, this is the simplest way I know: increase only N. Everything else follows from the circle you already hold in your hands.

Angles are measured in turns \(u\in[0,1)\) with period-1 trigonometry: \( c(u):=\cos(2\pi u),\; s(u):=\sin(2\pi u) \). (For degree readers: \( \Delta u = \text{degrees}/360 \).)

Circle — three GRG options: overview · Δx trapezoids · Original corridor · Radial sectors.

The definition is exact and π is not assumed in advance. In turns (period-1 sine s(u)=\sin(2\pi u)) it is fixed by the limit

$$ \pi \;=\; \lim_{N\to\infty}\; 4N\, s\!\Big(\tfrac{1}{8N}\Big) \;=\; \lim_{N\to\infty}\; 2N\, s\!\Big(\tfrac{1}{4N}\Big). $$

For practical work we use the finite version

$$ \pi_N \;=\; 4N\, s\!\Big(\tfrac{1}{8N}\Big), $$

which increases from below and converges to π as N grows. A useful accuracy hint is

$$ \pi - \pi_N \;\approx\; \frac{\pi^3}{96\,N^2}. $$

Consequences you can feel while using the page:

  • Only one knob: raise N for more precision; the formatting “decimals to display” is just how many digits you show.
  • Powers of two shine: when N = 2^k we can compute by pure halving (only square-roots), no trigonometry, no π inside.
  • Predictable growth: multiplying N by 10 adds about 2 correct digits; doubling N adds roughly 0.6 digits.

Angles are measured in turns \(u\in[0,1)\) with period-1 trigonometry: \( c(u):=\cos(2\pi u),\; s(u):=\sin(2\pi u) \). (For degree readers: \( \Delta u = \text{degrees}/360 \).)

Circle — three GRG options: overview · Δx trapezoids · Original corridor · Radial sectors.

Interactive: increase only N

Choose N and how many decimals to display. For powers of two, the calculator uses pure halving (no trig). For general N it uses a high-precision series. Output shows the value, Δ to \( \pi_{2N} \), an error estimate, and detects when numerical verification becomes the limit (“inversion”).

Mode: —

—

—

How this π calculator works

Formula (TURNS): \( \pi_N = 4N\, s(\tfrac{1}{8N}) \) with \(s(u)=\sin(2\pi u)\). Increase only N; the value grows from below and converges to π.

Two modes

Halving (no trig): if N is a power of two, the chord angle is halved repeatedly starting from the quarter-turn \(u=\tfrac{1}{4}\). This uses only square-roots; no trig, no π inside.

Series (turns → radians): for general N, it evaluates \(s(u)\) at \(u=\tfrac{1}{8N}\) via a high-precision series (internally converts to radians).

Precision & inversion

The line shows Δ(N→2N), an internal error estimate \( \approx \tfrac{4}{3}\Delta_N \), and a Richardson “verified digits” check. When internal ratios drift far from the expected \(1/4\) or verified digits stop growing at high precision, the panel turns amber: GRG may be more precise than the numerical checker (inversion).

Concrete examples (GRG)

Here is how the value improves as you raise N (rounded):

  • N = 16 → πN ≈ 3.140331156954753 (error ≈ 1.261×10⁻³)
  • N = 128 → πN ≈ 3.141572940367091 (error ≈ 1.971×10⁻⁵)
  • N = 1024 → πN ≈ 3.141592345570118 (error ≈ 3.080×10⁻⁷)

These numbers come straight from the turn-native formula

$$ \pi_N \;=\; 4N\, s\!\Big(\tfrac{1}{8N}\Big) \quad\text{and}\quad \pi - \pi_N \;\approx\; \frac{\pi^3}{96\,N^2}. $$

When verification becomes the limit (“inversion”)

As you increase N, GRG’s error drops predictably—until a point where the numerical tools used to check your result (floating-point, auxiliary series, etc.) become the real bottleneck. From there on, GRG may be more precise than your verifier: that is the “inversion”.

Angles are measured in turns \(u\in[0,1)\) with period-1 trigonometry:

$$ s(u) := \sin(2\pi u), \qquad c(u) := \cos(2\pi u). $$

What really happens — three regimes

  • Model error dominates. The inside chord-sum approximation $$ \pi_N \;=\; 4N\,s\!\Big(\tfrac{1}{8N}\Big) $$ satisfies the classical asymptotic (from expanding \(s(u)\) at \(u=0\)): $$ \pi - \pi_N \;\approx\; \frac{\pi^{3}}{96\,N^{2}}. $$ Doubling N quarters the error.
  • Transition zone. The theoretical error keeps falling, but rounding and finite working precision begin to matter. If “shown decimals” or internal precision do not increase accordingly, visible gains slow down.
  • Inversion. Numerical/verification error exceeds the GRG error. Philosophically: the method is now better than your ruler.

Self-checking from the inside (no external \( \pi \))

  1. \(\Delta\)–ratio test (asymptotics). Define $$ \Delta_N := \pi_{2N}-\pi_N, \qquad R_N := \frac{\pi_{4N}-\pi_{2N}}{\pi_{2N}-\pi_N}. $$ For clean numerics, \( R_N \to \tfrac{1}{4} \). If you see ≈ 0.25, you are still in the \(1/N^{2}\) regime (all good). If it wanders far from 0.25, numerics have taken over ⇒ inversion.
  2. Internal error estimate (no \( \pi \)). From the series behavior, $$ \pi - \pi_N \;\approx\; \frac{4}{3}\,\Delta_N. $$ The constant \( \tfrac{4}{3} \) comes from analysis; the factors \( \pi^{3} \) cancel out. You can therefore display: “Estimated error ≈ \( \tfrac{4}{3}\,\Delta_N \)”.
  3. Internal acceleration (Richardson). Cancel the \(1/N^{2}\) term by combining two steps: $$ \pi_{\text{ex}} \;=\; \pi_{2N} \;+\; \frac{\pi_{2N}-\pi_N}{3}, \qquad \pi - \pi_{\text{ex}} \;=\; O(N^{-4}). $$ Then compare with \( \pi_{4N} \): $$ E_N := \big|\pi_{\text{ex}} - \pi_{4N}\big|, \qquad \text{verified digits} \;\approx\; \big\lfloor -\log_{10} E_N \big\rfloor. $$ If verified digits stop growing when you double N, the numerical checker is limiting you (inversion).

Takeaway

  • Intellectual simplicity (GRG): one knob N, monotone inside-approximation, predictable error.
  • Honest inversion: beyond a threshold, tools used to verify are the limit — not GRG.
  • Self-certification: via \( \Delta_N \), \( \pi_{\text{ex}} \), and \( E_N \), no external \( \pi \) needed.

In the calculator above, when the internal checks suggest inversion, the panel turns amber and an explanatory note appears. Prefer powers of two for N, or increase working precision, if you need to push further.

Context note (as of 10 Sep 2025)

GRG’s turn-based, chord-sum definition of \( \pi \) is presented here as a new approach that increases accuracy by raising a single internal parameter N. This page documents the method and its error behavior; it is not intended to compete with record-breaking \( \pi \) algorithms.

For extremely large digit counts, established methods (AGM/Salamin–Brent, Borwein, Ramanujan/Chudnovsky) remain the fastest due to decades of optimization. GRG is primarily valued for conceptual clarity, monotone bounds, and controlled \( N\!\to\!\infty \) convergence.

Concrete examples (GRG)

Here is how the value improves as you raise N (rounded):

  • N = 16 → πN ≈ 3.140331156954753 (error ≈ 1.261×10⁻³)
  • N = 128 → πN ≈ 3.141572940367091 (error ≈ 1.971×10⁻⁵)
  • N = 1024 → πN ≈ 3.141592345570118 (error ≈ 3.080×10⁻⁷)

These numbers come straight from the turn-based chord-sum formula (angles in turns \(u\in[0,1)\) with \(s(u):=\sin(2\pi u)\)):

$$ \pi_N \;=\; 4N\,s\!\Big(\tfrac{1}{8N}\Big), \qquad \pi - \pi_N \;\approx\; \frac{\pi^3}{96\,N^2}. $$

Tip: use powers of two for N (16, 32, 64, …) for cleaner Δ-tests and steadier convergence; if verified digits stop increasing when doubling N, raise working precision.

How other methods reach π (in plain words)

Long alternating sums (Gregory–Leibniz)

This is the famous series \( \pi/4 = 1 - 1/3 + 1/5 - 1/7 + \dots \). After \(n\) terms, the error in \( \pi \) is about \( \frac{4}{2n+1} \approx \frac{2}{n} \). It’s beautiful but very slow: ~10,000 terms give ≈4 correct decimals; ~1,000,000 terms ≈6 decimals. A mountain of fractions.

Nilakantha’s refinement

Nilakantha’s series \( \pi = 3 + \frac{4}{2\cdot3\cdot4} - \frac{4}{4\cdot5\cdot6} + \frac{4}{6\cdot7\cdot8} - \dots \) converges much faster: the error shrinks like \(1/n^2\). Ten terms already land close; one hundred or one thousand terms push several more digits — still, you juggle many fractions.

Machin-like formulas (arctangents)

Here \( \pi \) is a combination of arctangents, e.g. \( \pi = 16\arctan(1/5) - 4\arctan(1/239) \). The arctan series converges fast because powers \(1/5^{\,2k+1}\) and \(1/239^{\,2k+1}\) drop quickly. With just a handful of terms you already see many correct digits.

Ramanujan, Chudnovsky & co.

“Rocket” series: each term yields many digits (≈8 per term for several Ramanujan formulas, ≈14 per term for Chudnovsky). They power modern record computations — superb for huge digit counts, but conceptually heavier than turning a single \(N\).

AGM (Salamin–Brent) and Borwein algorithms

Iterative methods with quadratic (or higher) convergence: each step roughly multiplies the correct digits. A dozen iterations already exceeds everyday needs — fantastic speed, less “from-the-inside” geometry.

Monte Carlo & Buffon’s needle

Random points/needles relate probabilities to \( \pi \). Charming and educational, but accuracy grows like \(1/\sqrt{T}\) (here \(T\) = trials): even millions of trials yield only a few decimals.

Why GRG feels intellectually simple

GRG is not about naming \( \pi \); it is about holding the circle. You dial only one parameter, \(N\), and everything else follows. The value increases from below (no overshoot), and its error is easy to estimate. If you truly need \( \pi \), this is the most direct, inside-out way I know — increasing a single internal factor means the computation belongs to the circle, not to an external \( \pi \).

Honest side-by-side (objective)

  • What you increase — GRG: just N. Archimedean polygons: sides m (inside/outside). Gregory–Leibniz / Nilakantha: number of terms. Machin / Ramanujan / Chudnovsky: number of series terms. AGM / Borwein: iteration count.
  • Speed of accuracy — GRG: error ≈ \(C/N^2\) (predictable, monotone). Gregory–Leibniz: \(O(1/n)\) (very slow). Nilakantha: \(O(1/n^2)\). Machin: fast; Ramanujan / Chudnovsky: very fast (many digits per term). AGM / Borwein: digits multiply each step.
  • When to use — GRG: concept, clarity, control “from the inside”, and up to many digits without extra machinery. For records or industrial-scale digits, AGM / Borwein / Chudnovsky win on raw speed.

Editor’s note (external view)

This note reflects an external opinion I chose to publish for fairness. The GRG approach is conceptually elegant: it defines \( \pi \) from inside the circle and lets accuracy grow by increasing only N. That simplicity is a real strength for understanding and for controlled, monotone inside-approximations.

As of August 25, 2025, GRG and this GRG-based \( \pi \) formula are newly introduced here. Researchers and practitioners still rely on established algorithms (AGM/Salamin–Brent, Borwein, Ramanujan/Chudnovsky) partly because these tools have been analyzed and optimized for decades and deliver record-speed convergence for very large digit counts.

The two views are compatible: GRG offers a clear, internal definition and a practical “increase only N” pathway with transparent self-checks; record algorithms remain the fastest instruments when one needs extreme digits or industrial-scale computations.

When verification becomes the limit (“inversion”)

As you increase N, GRG’s error drops predictably—until a point where the numerical tools used to check your result (floating point, auxiliary series, etc.) become the real bottleneck. From there on, GRG may be more precise than your verifier: that is the “inversion”.

Angles are measured in turns \(u\in[0,1)\) with period-1 trigonometry \( s(u):=\sin(2\pi u) \), \( c(u):=\cos(2\pi u) \).

What really happens — three regimes

  • Model error dominates. The GRG approximation $$ \pi_N \;=\; 4N\,s\!\Big(\tfrac{1}{8N}\Big) $$ satisfies $$ \pi - \pi_N \;\approx\; \frac{\pi^3}{96\,N^2}. $$ Doubling N quarters the error.
  • Transition zone. The theoretical error keeps falling, but rounding and limited working precision begin to matter. If “shown decimals” or internal precision don’t increase, visible gains slow down.
  • Inversion. Numerical/verification error exceeds the GRG error. Philosophically: the method is now better than your ruler.

Self-checking from the inside (no external π)

  1. Δ–ratio test (asymptotics). Define $$ \Delta_N := \pi_{2N}-\pi_N, \qquad R_N := \frac{\pi_{4N}-\pi_{2N}}{\pi_{2N}-\pi_N}. $$ For clean numerics, \( R_N \to \tfrac{1}{4} \). If you see ≈ 0.25, you are still in the \(1/N^2\) regime (all good). If it wanders far from 0.25 (oscillates, collapses to 0, or behaves oddly), numerics have taken over ⇒ inversion.
  2. Internal error estimate (no π). From the series expansion, $$ \pi - \pi_N \;\approx\; \frac{4}{3}\,\Delta_N. $$ The constant \( \tfrac{4}{3} \) comes from analysis; the factors \( \pi^3 \) cancel out. You can therefore display: “Estimated error ≈ \( \tfrac{4}{3}\,\Delta_N \)”.
  3. Internal acceleration (Richardson). Cancel the \(1/N^2\) term by combining two steps: $$ \pi_{\text{ex}} \;=\; \pi_{2N} + \frac{\pi_{2N}-\pi_N}{3}, \qquad \pi - \pi_{\text{ex}} = O(N^{-4}). $$ Then compare with \( \pi_{4N} \ ): $$ E_N := \big|\pi_{\text{ex}} - \pi_{4N}\big|, \qquad \text{verified digits} \;\approx\; \big\lfloor -\log_{10} E_N \big\rfloor. $$ If verified digits stop growing when you double N, the numerical checker is limiting you (inversion).

Takeaway

  • Intellectual simplicity (GRG): you turn a single knob, N, get a monotone inside-approximation, and a predictable error.
  • Honest inversion: beyond a certain threshold, it’s not GRG that runs out of accuracy—it’s the tools used to verify it.
  • Self-certification: using \( \Delta_N \), \( \pi_{\text{ex}} \), and \( E_N \), the method checks itself without resorting to an external π.

In the calculator above, when the internal checks suggest inversion, the panel turns amber and an explanatory note appears. Prefer powers of two for N, or increase working precision, if you need to push further.

ERG – Trapezoidal

ERG – Trapezoidal

Aug 24, 2025

Document Status and Current Reference

This page preserves an earlier formulation or research method. Its notation, formulas, and embedded demonstrations belong to that version and should not be treated as the specification of the current binary radial kernels.

Read the current formulation and its analytical verification. Historical performance statements and implementation behaviour require separate validation. Earlier references to circular constants remain in this historical document; they are not inputs to the current geometric kernel definitions.

Δx Trapezoidal (Cartesian)

Area from the inside on the quarter of a circle. Sample uniformly on the x-axis with $\Delta x=r/N$, build trapezoids strictly under the arc $y(x)=\sqrt{r^2-x^2}$, then multiply by 4 for the full circle. No angles and no external constants appear in the primary operators.

Setup (quarter)

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

Quarter area (→ ×4 for the circle)

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

Because $y(x)$ is concave on $[0,r]$, these trapezoids give a monotone under-approximation. The error typically decays like $O(N^{-2})$. A quick check is step-doubling: compare $A_{\text{circle}}(N)$ vs $A_{\text{circle}}(2N)$.

Quarter with Δx trapezoids (Cartesian x–y) Strict Cartesian axes (equal units). r=100, N=5 ⇒ Δx=20. Trapezoids are inside the quarter arc. x y
Fig. 1 — Δx trapezoids strictly inside the quarter arc (example: $r=100$, $N=5$).

Single strip geometry

Between $x_k$ and $x_{k+1}$ the trapezoid has bases $y_k$ and $y_{k+1}$ and width $\Delta x$; the arc segment lies strictly above it.

One Δx trapezoid under the arc Strip between x=60 and x=80 for r=100. Heights are y_k=80 and y_{k+1}=60. Dashed arc segment shown above the trapezoid. x y
Fig. 2 — One Δx strip (here $k=3$): the arc is strictly above the trapezoid.

Pseudocode

Input: r, N
Δx = r / N
Aq = 0
for k = 0..N-1:
    yk   = sqrt(r*r - (k*Δx)*(k*Δx))
    yk1  = sqrt(r*r - ((k+1)*Δx)*((k+1)*Δx))
    Aq  += 0.5 * Δx * (yk + yk1)
A_circle = 4 * Aq

Notes

  • Monotone: $A_{\text{circle}}(N)$ increases with $N$ and approaches the true area from below.
  • Accuracy: smooth $O(N^{-2})$ decay; step-doubling is a simple diagnostic.
  • Implementation: works cleanly with fixed-point or vectorized code (just sums and square roots).