Sample $S_1$ and $S_2$ with $\Delta r$, $\Delta \theta$ such that local geodesic spacing never exceeds $h(u,v)$.
Density increases where $\kappa_{\max}$ is higher (small radii or high-curvature regions).
2.2 Surfaces and closure
Build two coherent meshes $S_1, S_2$ and close with a thin lateral ring (cylindrical rim) to obtain a closed, oriented solid.
2.3 ERG volume
Tetrahedra method (as for the sphere) on the closed mesh:
which matches the tetrahedra estimate with $O(h^{2})$ error when the sampling bound is enforced.
3) Smooth$_{\varepsilon}$ for optics (C² continuity)
Goal: remove micro-faceting and guarantee surface and derivative continuity ($C^{2}$) for stable normals and curvature.
Local MLS/RBF fit (degree $\ge 2$) on a radius $\rho\in[h,2h]$ for each face.
Resample vertices on the smoothed patches; deviation $\le \varepsilon/2$.
Normal filtering along geodesic neighborhoods, bandwidth $\propto \varepsilon$.
Optical check: angular variation of normals consistent with expected faceting and no edges (no $C^{1}$ discontinuities).
Topology check: no self-intersections after smoothing.
4) Algorithm (concise)
input: surfaces S1, S2 (or z1=f1(r), z2=f2(r)), ε
for each face S∈{S1, S2}:
compute κ_max(u,v) # or, for revolution, κ_mer(r), κ_circ(r)
set h(u,v) ← 2*sqrt(ε/κ_max(u,v))
generate adaptive grid (Δr, Δθ) satisfying h
T ← triangulate(S1) ∪ triangulate(S2) ∪ cylindrical_rim
orient(T)
A ← sum_areas(T)
V_ERG ← tetra_volume(T)
if optical requirements:
T' ← smooth_MLS_RBF_C2(T) with deviation ≤ ε/2
check_normals(T') # continuity & angular variation
check_self_intersections(T')
output: (closed mesh T', A, V_ERG, normals)
5) Checks & validation
Hausdorff (ideal ↔ mesh) $\le \varepsilon$.
$C^{2}$ continuity after smoothing (tests on numerical derivatives of patches).
Triangle quality and no self-intersections.
(Optional) qualitative optical test: no “steps”.
6) Why it matters
Lenses require $C^{2}$ continuity: even tiny edges degrade refraction. GRG adaptive sampling densifies where needed; Smooth$_\varepsilon$ controls deviation. The result is a lens “perfect at the chosen resolution”, with validated ERG volume.
7) Figures (inline SVG)
L2 — Meridional section (profiles $z_1$, $z_2$) and rim closure.L1 — Density map driven by $\kappa_{\max}$: smaller $h$ where curvature is higher.
8) FAQ
How do I choose $\varepsilon$ for optics?
Start from the required form tolerance (e.g., $\lambda/10$ or microns) and translate it into a geometric deviation limit.
Does smoothing alter dioptric power?
No, if capped at $\varepsilon/2$ and validated with deviation maps and derivative checks.
What about the rim?
Close with a coherent ring: no sharp or inverted edges; enforce correct orientation.
ERG Quick Calculator
This compact tool computes the quarter-sampling step Δu and the grid size K from the dyadic level b, previews the uk values, and exports the grid to CSV.
Inputs
Dyadic levelb ∈ ℕ≥0 (recommended range: 0–18)
Optional: recent estimates V2b-1 and V2b to check the dyadic stopping rule
\[ \Delta u \;=\; 2^{-(b+2)}, \qquad
K \;=\; \left\lfloor \frac{1}{\Delta u} \right\rfloor \;=\; 2^{\,b+2}. \]
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.
GRG in One Page: π-Free Corridors, Benchmarks, and a Lean Mini-App
This single page addresses reviewer questions with concrete numbers, one minimal code box, and a practical validation plan. All constructions are turns-first and dyadic: b ∈ ℕ≥0, N = 2^b, step Δu = 2^{-(b+2)} = 1/(4N). We use only sums and square roots.
1) Worked example (r = 100) with certified circumference/area corridors
Let c_N be the unit-radius chord spanning an arc of 1/N turns. Starting from the square, c_4 = √2, we refine dyadically by a π-free half-angle step:
Certified corridors for circumference and area using only sums and square roots (r = 100).
Chord (inscribed), tangent (circumscribed), and apothem define GRG corridors.
2) One minimal code box (Python) that reproduces the table
Drop-in, π-free implementation. Only square roots and sums via half-angle edge doubling.
# GRG: π-free corridors for circumference and area (dyadic N=2^b)
# r: radius; N: number of edges (dyadic)
import math
def c_dyadic(N):
# start from square (N=4): c_4 = sqrt(2)
if N & (N-1):
raise ValueError("N must be a power of 2")
if N < 4:
raise ValueError("N must be >= 4")
c = math.sqrt(2.0)
n = 4
while n < N:
c = math.sqrt(2.0 - 2.0*math.sqrt(1.0 - 0.25*c*c))
n *= 2
return c # unit radius
def corridors(r=100.0, b_list=(3,4,5,6)):
out = []
for b in b_list:
N = 2**b
c = c_dyadic(N)
t = c / math.sqrt(1.0 - (0.5*c)**2) # computed π-free
C_minus = N * r * c
C_plus = N * r * t
apothem = r * math.sqrt(1.0 - (0.5*c)**2)
A_minus = 0.5 * C_minus * apothem
A_plus = 0.5 * C_plus * r
pi_lo, pi_hi = C_minus/(2*r), C_plus/(2*r)
out.append(dict(N=N, C_minus=C_minus, C_plus=C_plus,
A_minus=A_minus, A_plus=A_plus,
pi_low=pi_lo, pi_high=pi_hi,
width=pi_hi-pi_lo, mid=0.5*(pi_hi+pi_lo)))
return out
if __name__ == "__main__":
for row in corridors():
print(row)
Why this matters: Reviewers asked for “real code.” The snippet above is reference-grade and numerically stable for dyadic N in double precision.
3) Mini-calculator (in-page): corridor and area, π-free
Try different radii and resolutions. The calculator only uses the recursion above (no trigonometric calls).
Geometry: regular polygon perimeter/area via trig (reference), rasterization by angular stepping, standard kernel approximations.
Optics: Zernike-based phase optimization (asphere/phase plate) and uniform samplers; compare against GRG dyadic turns.
Chemistry: radial distribution (isotropic) vs. anisotropic stress tests; compare GRG dyadic shells to uniform/log binning.
4.2 Datasets and tasks
Circles/Spheres: circumference/area/volume corridors at fixed sample budgets.
Optical toy problem: paraxial lenslet: maximize on-axis MTF at a target spatial frequency under a fixed simulator budget.
Chemical toy problem: radial distribution functions with/without anisotropy; report L_2 error vs. a high-precision reference.
4.3 Metrics and stats
Error vs budget: corridor width or absolute error vs. number of samples.
Time vs accuracy: seconds to reach a target error.
Robustness: worst-case error across seeds; bootstrap CI.
5) Scalability and complexity
Cost: computing c_N is O(log N) half-angle steps; producing C_N^{±}, A_N^{±} is O(1).
Beyond circles/spheres: For radially defined profiles, GRG’s dyadic turns yield nested certified bounds. For anisotropy, use a wedge-stratified scheme: split [0,1) turns into dyadic wedges and apply GRG within each wedge; corridors add up across wedges.
6) GRG Mini-App: current status and UI notes
Scope: live computation of C_N^{±}, A_N^{±}, π corridor, and dyadic overlays (“inscribed polygon preview”, “row layout”).
Roadmap: public demo + short tutorial video; CSV export of computed corridors; copy-paste of the Python above.
Accessibility: keyboard-operable sliders for b, ARIA live regions on numeric outputs (as in the mini-calculator).
7) What to expect next
Open-source drop: the single Python file (MIT/BSD-3) + tiny unit tests.
Benchmarks: publish results vs. Zernike and uniform samplers on the toy problems; scripts and seeds included.
This toy problem shows how GRG guides sampling while any preferred field solver evaluates performance. We optimize a rotationally symmetric height profile under a fixed evaluation budget.
Constraints: height range (manufacturability), smoothness (bounded second difference), and a hard cap on simulator calls.
8.2 GRG sampling policy
Choose dyadic level b; sample radii on N=2^b turns (rings). For anisotropic stress tests, split the pupil into dyadic wedges and apply the same policy per wedge.
At each refinement, reuse prior evaluations (nested design), cutting waste in flat regions.
8.3 Budget-fair comparison
Compare against equal-angle and equal-arc samplers at the same number of simulator calls.
Report \Delta \mathrm{MTF} = \mathrm{MTF}_{\text{GRG}} - \mathrm{MTF}_{\text{baseline}} and time-to-target MTF.
8.4 Reproducibility hook
The π-free recursion and nested sampling policy above serve as the public spec. A scalar paraxial PSF/OTF evaluator—or your preferred solver—can be dropped in without altering the sampling design.
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.
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:
$$
M
\begin{pmatrix}
a_{4}\\[2pt]
a_{6}\\[2pt]
a_{8}
\end{pmatrix}
\;=\;
b.
$$
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:
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:
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.
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.
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.
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.
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
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):
\(R_{\tau}^{-}=\min\{R:\,E^{\uparrow}(R)\ge \tau\}\) is a lower bound on the true \(R_{\tau}\).
\(R_{\tau}^{+}=\min\{R:\,E^{\downarrow}(R)\ge \tau\}\) is an upper bound on the true \(R_{\tau}\).
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.
To provide the best experiences, we use technologies like cookies to store and/or access device information. Consenting to these technologies will allow us to process data such as browsing behavior or unique IDs on this site. Not consenting or withdrawing consent, may adversely affect certain features and functions.
Functional
Always active
The technical storage or access is strictly necessary for the legitimate purpose of enabling the use of a specific service explicitly requested by the subscriber or user, or for the sole purpose of carrying out the transmission of a communication over an electronic communications network.
Preferences
The technical storage or access is necessary for the legitimate purpose of storing preferences that are not requested by the subscriber or user.
Statistics
The technical storage or access that is used exclusively for statistical purposes.The technical storage or access that is used exclusively for anonymous statistical purposes. Without a subpoena, voluntary compliance on the part of your Internet Service Provider, or additional records from a third party, information stored or retrieved for this purpose alone cannot usually be used to identify you.
Marketing
The technical storage or access is required to create user profiles to send advertising, or to track the user on a website or across several websites for similar marketing purposes.
