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
\[ \Delta u \;=\; 2^{-(b+2)}, \qquad
K \;=\; \left\lfloor \frac{1}{\Delta u} \right\rfloor \;=\; 2^{\,b+2}. \]
The sphere is an ideal testbed: constant curvature and closed-form $A,V$.
The chain points → surface → ERG volume → smooth at scale $\varepsilon$ sets the general GRG/ERG procedure for solids.
7) Figures (inline SVG)
F3 — Sagitta relation used to bound $h_{\max}$.F1 — Point distribution (Fibonacci vs icosphere idea) at equal $h$.
8) FAQ
Is “perfect” absolute? No: it is perfect at scale $\varepsilon$. Details $<\,\varepsilon$ are not guaranteed.
Do we always need smoothing? No: needed for normals/optics stability; otherwise optional.
Fibonacci or icosphere? Both work: Fibonacci is simple and isotropic; icosphere offers regular connectivity.
How do I choose $\varepsilon$? Based on use: measurement accuracy, rendering, fabrication, etc.
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
\[ \Delta u \;=\; 2^{-(b+2)}, \qquad
K \;=\; \left\lfloor \frac{1}{\Delta u} \right\rfloor \;=\; 2^{\,b+2}. \]
Method block (turns-based). Angles are measured in turns (fractions of a full rotation, u∈[0,1)). We write c(u), s(u) for period-1 cosine/sine in turns (so c(0)=1, s(0)=0; c(1/4)=0, s(1/4)=1). No sexagesimal units and no external constants appear in the primary formulas.
Use this block when you want a pure angular build; for Cartesian builds see Δx Trapezoidal and Original Projection pages.
Angle unit & precision (turns, binary fixed-point)
We measure angles in turns: a full rotation is 1 turn. Hence
quarter-turn = 1/4, half-turn = 1/2, etc. Trigonometric functions are period-1 in turns:
c(u), s(u) with c(0)=1, s(0)=0, c(1/4)=0, s(1/4)=1.
Equal-angle partition
Choose an integer N (number of sectors) and set the angular step
Δu = 1/N. The sampling angles are uk = k/N, k=0,…,N−1.
Binary fixed-point turns (B bits)
For hardware/software precision, store angles as u = k / 2^B turns with
k ∈ {0,…,2^B−1}. The angular resolution is ΔuB = 1 / 2^B.
If you want exactly N equal sectors, pick N that divides 2^B
(e.g., N = 2^B), and use indices k = m·j with m = 2^B / N, j = 0,…,N−1.
Why more bits ⇒ more accuracy
Circumference (chords):CN = 2 N r · s(1/(2N)) is the perimeter of the inscribed regular N-gon and increases to the circle’s circumference. With N=2^B, the step halves when B→B+1 and the error typically drops ~×4 (quadratic in the step).
Area (sectors):AN = ½ N r² · s(1/N) is the area of N equal sectors and increases to the circle’s area; same step-squared behaviour.
Example with B = 3 bits ⇒ 23 = 8 ticks. Increasing B doubles the ticks and the precision.
Pseudocode (B-bit angles)
Input: r, B
N = 1 << B // N = 2^B sectors
C = 2 * N * r * s( 1.0 / (2 * N) )
A = 0.5 * N * r^2 * s( 1.0 / N )
Implementation tip: if you keep an angle register k modulo 2^B, additions are wrap-around on B bits. For LUTs of size 2^M with M ≤ B, you can index by k >> (B−M).
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.
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
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.
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.
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.
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:
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.
Define the chiral center C and the four substituents (A, B, C, D).
Choose distance bounds for C–X: e.g., 1.45–1.55 Å.
Partition the interval into k concentric shells (e.g., 5).
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.
Score with a light force field (e.g., UFF) and retain the lowest-energy 10 conformers.
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.
Place a metal node M (e.g., Zn2+) at the cell center; four linkers L at the corners of a radial tetrahedron.
Set coordination bounds (e.g., M–O = 1.9–2.2 Å).
For each L site, sample positions on a spherical shell about M, keeping \(\angle\text{L–M–L}\approx 109.5^\circ\).
Tile the cell periodically with vectors \((a,0,0)\), \((0,a,0)\), \((0,0,a)\).
Run a short geometry minimization (e.g., Powell) to relieve strain.
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.
Locate the solute center of mass and choose \(R_1\) (solute + first shell) and \(R_2\) (outer boundary).
Split \([R_1,R_2]\) into \(N\) shells (typically 3–5).
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 \(d_{\min}(\mathrm{O}, \text{solute}) \ge 2.5\ \text{Å}\).
Assign water orientations (H–O–H) with small random perturbations.
Export to PDB and run a short heating/minimization to relax the sphere.
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.
Center a pocket reference and define shells \([r_k,r_{k+1}]\) plus ring samples at turns \(u_j=j/N\).
Precompute coarse electrostatic potential on these cells; score ligand fragments by shell and ring.
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.
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”):
\(R_{n}^{-}=\min\{R:\,N^{\uparrow}(R)\ge n\}\) is a lower bound of the true \(R_n\).
\(R_{n}^{+}=\min\{R:\,N^{\downarrow}(R)\ge n\}\) is an upper bound of the true \(R_n\).
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.
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.
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.
