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 $R$):

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 $(\sqrt{3}/2)\,h^2$):

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:

2.3 ERG volume (from closed surface)

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

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)

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.