ERG volume

ERG volume

Oct 18, 2025

GRG Sphere — Points • Surface • ERG Volume • Smooth

Abstract. A sphere of radius $R$ is generated as a quasi-uniform point set. With sampling step:

$$ h_{\max}(\kappa,\varepsilon)\;=\;2\sqrt{\frac{\varepsilon}{\kappa}}, \qquad \kappa=\frac{1}{R}, $$

the reconstructed surface is perfect at scale $\varepsilon$ (geometric error $\le \varepsilon$); area and ERG volume converge with $O(h^2)$ error.

1) Definitions

  • Resolution $\varepsilon$: admissible indistinguishability threshold.
  • Step $h$: distance between nearby surface samples.
  • Curvature $\kappa$: for a sphere, $\kappa = 1/R$.
  • Perfect at scale $\varepsilon$: Hausdorff distance (ideal ↔ reconstructed) $\le \varepsilon$.

Geometric reminder (sagitta of an arc of radius $R$):

$$ s \;\approx\; \frac{h^{2}}{8R} \;=\; \frac{\kappa h^{2}}{8} $$
$$ \Rightarrow\quad h_{\max} \;=\; 2\sqrt{\frac{\varepsilon}{\kappa}} \;=\; 2\sqrt{\varepsilon R}. $$

2) Points → Surface → ERG volume

2.1 Points (quasi-uniform sampling)

  • Spherical Fibonacci lattice: simple and isotropic.
  • Icosphere (subdivided icosahedron): regular connectivity.

Point count (triangles near-equilateral, area $(\sqrt{3}/2)\,h^2$):

$$ N \;\approx\; \frac{4\pi R^{2}}{(\sqrt{3}/2)\,h^{2}} \;=\; \frac{2\pi}{\sqrt{3}}\,\frac{R}{\varepsilon}. $$

It grows linearly with $R/\varepsilon$.

2.2 Surface (closed, oriented mesh)

Triangulate coherently on the spherical manifold. Face normal: $\mathbf{n}_f \propto (p_j-p_i)\times(p_k-p_i)$.

Discrete area:

$$ A \;=\; \sum_{\triangle ijk} \frac{1}{2}\,\big\|\, (p_j-p_i)\times(p_k-p_i)\, \big\|. $$

2.3 ERG volume (from closed surface)

With an oriented closed mesh (tetrahedra w.r.t. the origin):

$$ V_{\mathrm{ERG}} \;=\; \frac{1}{6}\sum_{\triangle ijk}\big((p_i\times p_j)\cdot p_k\big). $$

Convergence: $A$ and $V_{\mathrm{ERG}}$ approach the ideal values with $O(h^2)$ error.

3) Smooth$_{\varepsilon}$ (when and how)

Goal: stabilize normals and reflections without exceeding $\varepsilon$.

  1. Local Taubin/MLS filter (forward/backward pass to avoid shrinkage).
  2. Projection onto the sphere ($\lVert v\rVert \to R$).
  3. Check: Hausdorff (pre/post) $\le \varepsilon/2$.
  4. Smoothed normals from local MLS fit ($\ge C^{1}$ continuity).

When it’s not needed: if you only measure area/ERG volume and faceting is already below the $\varepsilon$-scale threshold.

4) Algorithm (concise)

input: R, ε
κ ← 1/R
h_max ← 2*sqrt(ε/κ)       # = 2*sqrt(εR)
N    ← round((2π/√3) * R/ε)

P ← fibonacci_sphere(N)    # or icosphere(h_max)
T ← triangulate_on_sphere(P)
orient(T)                  # outward normals

A ← sum_areas(T)
V_ERG ← tetra_volume(T)

if need_smooth_normals:
    P' ← taubin_mls(P, T)  # controlled smoothing
    P' ← project_to_sphere(P', R)
    assert hausdorff(P, P') ≤ ε/2
    recompute_normals(P', T)
output: (P, T, A, V_ERG, normals)

5) Checks & validation

  • Hausdorff distance (ideal ↔ mesh) $\le \varepsilon$.
  • Triangle quality (avoid extreme angles).
  • No self-intersections.
  • $O(h^{2})$ convergence for area and ERG volume.

6) Why it matters

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)

Sagitta geometry: s ≈ h^2/(8R) Arc of radius R, chord of length h, sagitta s; all lengths in the same units. R O s | ⟂ h
F3 — Sagitta relation used to bound $h_{\max}$.
Quasi-uniform points on a sphere Wireframe sphere with dots to suggest Fibonacci vs icosphere distributions.
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 grid icon Concentric arcs with dyadic ticks representing ERG refinement.

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 level b ∈ ℕ≥0 (recommended range: 0–18)
  • Optional: recent estimates V2b-1 and V2b to check the dyadic stopping rule
\[ \Delta u \;=\; 2^{-(b+2)}, \qquad K \;=\; \left\lfloor \frac{1}{\Delta u} \right\rfloor \;=\; 2^{\,b+2}. \]
\[ \text{Stop if}\quad \bigl|V_{2^{b}} - V_{2^{b-1}}\bigr| \le \varepsilon. \]

Calculator

Results

  • Δu = — (exact: —)
  • K = — (i.e., number of grid points)
  • N = — (turns-first level, N = 2b)
Preview of u-grid uk = k · Δu

First 8: —

Last 8: —

—

Practical checklist

  • Use dyadic refinement: increase b until the dyadic difference falls below ε.
  • Quarter-sampling is assumed: Δu = 2-(b+2).
  • Grid is deterministic; CSV can be versioned for reproducibility.
Methods

Methods

Sep 12, 2025

Methods and Their Scope

Current Binary Radial Kernels

The current formulation defines regular equal sectors, local chord–tangent lengths, triangular areas, and weighted kernels. Its proofs concern those definitions.

Earlier Methods

Algorithm Specifications

A runnable method must identify initial values, refinement or direct evaluation, arithmetic format, outputs, and stopping conditions. A geometry identity alone is not a runtime benchmark.

Extensions

Adaptive sampling, general curves, multiple seeds, and three-dimensional constructions require their own definitions and validation. No error order or performance advantage is transferred automatically from the regular-circle proof.

Foundations

Foundations

Sep 12, 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.

Mathematical Foundations: Turns Trigonometry & Radial Discretization

Category: Generative Radial Geometry (GRG)

GRG measures angles in turns with period‑1 primitives c(u), s(u) (no π assumed in discrete steps). Constructions proceed on quarters/sectors, summing center‑seeded primitives (chords and radial strips). Below are the core identities used across methods.

Turns Trigonometry (period‑1)

Let a full rotation be one turn. The functions c(u), s(u) are period‑1 and satisfy the usual symmetries and addition rules:

$$ c(u+1)=c(u),\quad s(u+1)=s(u),\quad c(-u)=c(u),\quad s(-u)=-s(u) $$
$$ c(u+v)=c(u)c(v)-s(u)s(v),\qquad s(u+v)=s(u)c(v)+c(u)s(v) $$

Quarter symmetry and parity yield the relations needed for GRG’s inside‑out sums, without introducing π in the discrete formulas.

Chord, Sagitta, and Radial Strip Area

For a unit circle with step Δu between consecutive samples u_L and u_R=u_L+Δu:

$$ \operatorname{chord}(\Delta u)=2\,s\!\left(\tfrac{\Delta u}{2}\right) $$
$$ \operatorname{sagitta}(\Delta u)=1-c\!\left(\tfrac{\Delta u}{2}\right) $$

The area of the center‑seeded strip (triangle O–p(u_L)–p(u_R)) equals half the cross‑product magnitude, i.e. a period‑1 sine of the step:

$$ A_{\text{strip}}(\Delta u)=\tfrac{1}{2}\,\Big(c(u_L)\,s(u_R)-s(u_L)\,c(u_R)\Big) =\tfrac{1}{2}\,s(\Delta u) $$

Discrete Perimeter and Area on a Quarter

With uniform steps Δu=\tfrac{1}{4N} on a quarter (N samples):

$$ P_{\tfrac14}(N;\text{rule})=\sum_{k=1}^{N}\operatorname{chord}(\Delta u),\qquad P(N)=4\,P_{\tfrac14}(N;\text{rule}) $$
$$ A_{\tfrac14}(N)=\sum_{k=1}^{N}\tfrac{1}{2}\,s(\Delta u)=\tfrac{N}{2}\,s\!\left(\tfrac{1}{4N}\right),\qquad A(N)=4\,A_{\tfrac14}(N) $$

Rule choice (midpoint / right‑endpoint / averaged) governs bias for arc length; strip areas use the exact triangle formula per step, remaining π‑free at finite N.

π as an Emergent Limit (turns)

From the full polygon with Δu=1/N:

$$ \pi=\lim_{N\to\infty} N\,s\!\left(\tfrac{1}{2N}\right) $$

Background & Positioning

Category: Generative Radial Geometry (GRG)

Generative Radial Geometry (GRG) builds shapes from the inside via discrete radial scans. Measurements arise by summing chords and radial trapezoids without assuming π in advance. Angles are measured in turns (period‑1), using primitives c(u), s(u) with u \in [0,1). We present GRG as a digital‑native (2025) center‑seeded approach—π‑free and complementary to classical, formula‑first geometry.

Motivation

Classical geometry is top‑down, continuous, formula‑centric. GRG is inside‑out: pick a resolution N, rotate a central ray with step Δu, and accumulate discrete contributions that converge as N grows.

Key Idea (Quarter Aggregation)

Build a quarter from center‑seeded primitives, then use symmetry.

$$\operatorname{chord}(\Delta u)=2\, s\!\left(\tfrac{\Delta u}{2}\right)$$

Quarter arc length follows from summing chords (upper/lower/midpoint/average rules). Area uses radial trapezoids on the quarter, scaled ×4. The one‑step sagitta is

$$\operatorname{sagitta}(\Delta u)=1-c\!\left(\tfrac{\Delta u}{2}\right)$$

π as an Emergent Limit (in turns)

Let the full polygon have N chords with Δu = 1/N. The chord between consecutive samples is 2·s(Δu/2), hence the polygon perimeter is C_N = 2N·s(1/(2N)). Define π (in turns) as half the circumference of the unit circle. Then:

$$\pi \,=\, \lim_{N\to\infty} N\, s\!\left(\tfrac{1}{2N}\right)$$

Dyadic variant: letting N = 2^B, we also have π = lim_{B→∞} 2^{B}·s(2^{-(B+1)}).

What GRG Adds

  • Deterministic control: explicitly choose N, the step Δu, and the rule (midpoint / right‑endpoint / average / radial‑trapezoidal Δu).
  • Turns‑based formalism: period‑1 trigonometry with c(u), s(u); quarter aggregation unifies perimeter and area.
  • Inside‑out convergence: measurements as discrete sums of center‑seeded primitives.

Page Takeaways

  • GRG is a center‑seeded, π‑free discretization framework (2025).
  • Angles are measured in turns with period‑1 primitives c(u), s(u).
  • Quarter aggregation gives simple, reproducible formulas for perimeter and area.