ERG volume

ERG volume

Oct 18, 2025

GRG Sphere β€” Points β€’ Surface β€’ ERG Volume β€’ 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 β†’ Surface β†’ 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, Ξ΅
ΞΊ ← 1/R
h_max ← 2*sqrt(Ξ΅/ΞΊ)       # = 2*sqrt(Ξ΅R)
N    ← round((2Ο€/√3) * R/Ξ΅)

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') ≀ Ξ΅/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 β†’ surface β†’ ERG volume β†’ smooth at scale $\varepsilon$ sets the general GRG/ERG procedure for solids.

7) Figures (inline SVG)

Sagitta geometry: s β‰ˆ h^2/(8R) Arc of radius R, chord of length h, sagitta s; all lengths in the same units. R O s | βŸ‚ h
F3 β€” Sagitta relation used to bound $h_{\max}$.
Quasi-uniform points on a sphere Wireframe sphere with dots to suggest Fibonacci vs icosphere distributions.
F1 β€” Point distribution (Fibonacci vs icosphere idea) at equal $h$.

8) FAQ

Is β€œperfect” 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 Ξ”u and the grid size K from the dyadic level b, previews the uk values, and exports the grid to CSV.

Inputs

  • Dyadic level b ∈ β„•β‰₯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

  • Ξ”u = β€” (exact: β€”)
  • K = β€” (i.e., number of grid points)
  • N = β€” (turns-first level, N = 2b)
Preview of u-grid uk = k Β· Ξ”u

First 8: β€”

Last 8: β€”

β€”

Practical checklist

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