The binary radial formulation combines complementary internal and external sector triangles. The area kernel uses one internal contribution and two external contributions, unlike the circumference kernel’s two-to-one chord–tangent balance.
Geometric Elements
Resolve one Turn into N equal symmetric sectors. Let a be the internal triangle’s height, h half its chord, and e half the external tangent side. Define q = h/a.
$$ N=2^b,\quad b\ge2,\quad R>0,\qquad a^2+h^2=R^2,\quad e=R\frac ha. $$
Regular geometric bisection yields a scale bound. Together with the expansion, it proves second-order total error for the arithmetic kernel and fourth-order total error for the weighted kernel in exact arithmetic.
The earlier version introduced this quarter-step sampling parameter inside each module. The proof here concerns the complete sector triangles defined above. A different sample-based area construction requires its own explicit definition and proof.
Internal–external weights are reversed between the circumference and area kernels. This diagram is schematic; the triangle definitions determine the calculation.
Research Status
The cancellation is established for the regular triangular elements above. This does not establish original authorship of the weighting, numerical stability of an implementation, or a computational advantage. Those questions require separate literature and implementation comparisons.
Earlier revisions of this page described fourth-order behaviour as a candidate. The current analytical clarification identifies the geometric assumptions under which that behaviour follows.
A GRG Formulation: Circle and Area from a Binary Radial Framework
Generative Radial Geometry describes circular measures through a finite binary radial framework. Its working quantities are a radius, a full Turn, a binary resolution, and local chord–tangent constructions.
About the Current GRG Formulation
This page presents the documented binary radial formulation, with geometric definitions and an explicit verification of its measurement kernels. Earlier approaches, including Original Projection and previous Turns formulations, remain part of the research history.
The geometric construction, kernel accuracy, and computational efficiency are separate subjects. The analysis below establishes accuracy for regular symmetric sectors. It does not establish a performance advantage or prescribe an implementation.
The weighted area proposal is documented in A Symmetric GRG Kernel for Area. The geometric ratio notation and proofs below are an analytical clarification of the documented triangles and kernels.
Binary Turns and Resolution
A full rotation is one Turn. Resolve it into equal sectors. The nondegenerate chord–tangent formulas on this page require at least four sectors.
Let a be the distance from the centre to the internal chord, h half that chord, and e half the external tangent side. The tangent is perpendicular to the central radius. The two boundary rays intersect both the circle and this tangent.
$$ a^2+h^2=R^2,\qquad e=R\frac ha. $$
Figure 1 — Symmetric sector: internal chord, central radius, and external tangent side.
Define dimensionless ratios from these constructed lengths:
$$ c_b=\frac aR,\qquad s_b=\frac hR,\qquad q_b=\frac ha. $$
The weighting is a known local arc approximation. This formulation uses it as a kernel within the binary radial framework; the weighting itself is not claimed as an original GRG result.
Area Elements and Kernels
The internal triangle has base 2h and height a. The external triangle has base 2e and height R.
The following identity is used to verify regular equal-sector bisection. It is a geometric consequence, not an attribution of a particular numerical algorithm to GRG.
At four sectors the half-sector triangle has equal legs, so q₂ = 1. Bisecting its direction gives the following ratio, obtained by normalising the boundary direction and adding it to the central unit direction:
$$ q_{b+1}=\frac{q_b}{1+\sqrt{1+q_b^2}}. $$
The denominator is at least two. Induction therefore gives:
$$ 0\lt q_b\le 2^{2-b}=\frac4N. $$
This proves the scale bound needed for the global error estimates. It uses the regular bisection geometry; it does not require a precomputed full-Turn length or area.
Kernel Accuracy: Analytical Verification
Define the local integral below solely as an analytical reference for the exact sector measures. It is not an input to the measurement kernels.
$$ F(q)=\int_0^q\frac{dt}{1+t^2}. $$
Parameterising the circular boundary by its slope gives exact sector arc length 2RF(q) and exact sector area R²F(q). Thus the fixed full-Turn reference measures satisfy:
$$ C_T(R)=2NRF(q_b),\qquad A_T(R)=NR^2F(q_b). $$
$$ F(q)=q-\frac{q^3}{3}+\frac{q^5}{5}+O(q^7). $$
Substitution into the circumference kernels yields:
The area weighting cancels the cubic local residual. Combining these expansions with the geometric scale bound proves, for fixed radius and exact arithmetic:
Both derivatives are nonnegative for 0 ≤ q ≤ 1, which covers the sectors considered here. Consequently the bracket follows for every b ≥ 2, rather than only for tested resolutions.
The related area article specifies quarter-step sampling within a module. The verification above concerns the complete sector triangles. An implementation using internal samples must specify how those samples evaluate the same elements; the sampling parameter alone does not define an additional area kernel.
Figure 2 — Binary sector refinement at N = 4, 8, and 16.
Computational Efficiency and Research Scope
Binary resolution is compatible with binary angular indices and power-of-two scaling. A performance advantage for the complete calculation must nevertheless be measured.
A reproducible comparison must use equal accuracy targets, include generation or retrieval of the local measures, account for precomputation and memory, and specify arithmetic precision, rounding, hardware, and execution time. Stability in fixed-point arithmetic requires a separate error analysis.
The results established here concern the documented regular chord–tangent geometry and its kernels. Historical originality and implementation efficiency are separate investigations.
Constructive Figures in GRG: Dyadic Quarter, Mirrored Circle, and Sector Area
This page collects three minimal, constructive drawings used throughout GRG: a dyadic quarter fan, a fully mirrored circle built from four quarters, and a sector-based area accumulator. All figures are π-free inside the loop; conversions happen only at setup or I/O.
1) Dyadic quarter fan (turns-first sweep)
Angle advances in turns. Refinement uses powers of two so that a quarter turn has N = 2^b rays with a step \Delta u = 2^{-b-2}. This aligns with binary hardware.
$$
N = 2^{\,b}, \qquad \Delta u = 2^{-\,b-2} \quad \text{(quarter sampling)}
$$
Quarter sweep with N = 2^b rays. Mirror to complete the circle.
2) Full circle via mirrored quarters (total symmetry)
Build one quarter, then mirror/rotate it three times to assemble the full circle with exact bilateral and rotational symmetry.
Four identical quarters enforce perfect symmetry and cache-friendly construction.
3) Sector-based area accumulation (π-free loop; classical limit)
Area is the sum of radial sectors. The loop aggregates in turn units; the classical value appears in the limit.
$$
\Delta A = \tfrac12 r^2\,\Delta\theta \qquad \text{with}\quad \Delta\theta = 2\pi\,\Delta u
$$
$$
A \;=\; \sum \Delta A \;\xrightarrow[\Delta u \to 0]{}\; \pi r^2
$$
Sum of ½ r² Δθ over dyadic steps converges to the classical area.
Notes on determinism and implementation
π outside the loop: the runtime loop uses recurrences; π appears only in calibration or reporting.
Bit-shift counts:N = 1 << b replaces divisions with a native shift.
Mirrored quarters: memory and symmetry benefits; identical update sequences aid reproducibility.
Summary.
Generative Radial Geometry (GRG) builds shapes ring-by-ring on a discrete support (pixels, steps, layers, gear teeth). In this setting, cell counts can be exact for the specified discrete object. Continuous area, Euclidean perimeter and angular measures require separate definitions and error analysis. Geometric approximation and numerical rounding must be distinguished from exact counting.
Why this matters
No drift: Each added ring contributes exactly its intended count; errors do not accumulate as shapes grow.
Implementation-first: The geometry matches how hardware and software actually operate on lattices, layers, and stepper motions.
Predictable outcomes: Offsets, masks, and toolpaths behave consistently across radii and angles.
Continuity bridge: As the scale refines, discrete measures converge to their continuous counterparts.
Setting and basic definitions
Chosen scale (s): the unit of discreteness (pixel size, machining step, layer height).
Discrete domain (Ls): a lattice at scale s (e.g., a square grid) with unit cell measure 1.
Radial generation rule (Gs): a constructive procedure that grows shapes by rings while enforcing symmetry (e.g., 4/8-fold on a square grid), monotone radius (rings do not cross), and arc closure (discrete arcs close exactly at intended angles).
Radial object (D(r; s)): the union of rings generated from the center up to discrete radius r.
Operational measures: Area μs(D) = number of cells in D; Perimeter Ps(D) as counts of admissible boundary steps; Angle count Θs as discrete angular increments used to build and close arcs.
Core properties
Ring additivity (no drift).
For each integer step in radius:
where \(A_{r+1}\) is the ring generated at radius \(r{+}1\). This identity holds by construction for all r.
Arc closure.
Discrete arcs generated with \(G_s\) close exactly at their intended angular multiples. Looping paths return to the origin without residual.
Symmetry and monotonicity.
The rule preserves lattice symmetries and grows rings without self-intersections, ensuring stable counts for all radii.
Convergence to the continuous limit.
As \(s \to 0\):
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
\[ \Delta u \;=\; 2^{-(b+2)}, \qquad
K \;=\; \left\lfloor \frac{1}{\Delta u} \right\rfloor \;=\; 2^{\,b+2}. \]
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