Exactness at the Chosen Scale (GRG)

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 (L_s): a lattice at scale s (e.g., a square grid) with unit cell measure 1.
• Radial generation rule (G_s): 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 μ_s(D) = number of cells in D; Perimeter P_s(D) as counts of admissible boundary steps; Angle count Θ_s as discrete angular increments used to build and close arcs.

Core properties

1. Ring additivity (no drift).
   For each integer step in radius:
   $$\mu_s\!\left(D(r{+}1; s)\right) - \mu_s\!\left(D(r; s)\right) \;=\; \lvert A_{r+1} \rvert,$$
   where \(A_{r+1}\) is the ring generated at radius \(r{+}1\). This identity holds by construction for all r.
2. Arc closure.
   Discrete arcs generated with \(G_s\) close exactly at their intended angular multiples. Looping paths return to the origin without residual.
3. Symmetry and monotonicity.
   The rule preserves lattice symmetries and grows rings without self-intersections, ensuring stable counts for all radii.
4. Convergence to the continuous limit.
   As \(s \to 0\):
   $$\mu_s(D)\,s^{2} \;\longrightarrow\; \text{continuous area}$$
   $$P_s(D)\cdot s \;\longrightarrow\; \text{continuous perimeter}$$
   $$\Theta_s \;\longrightarrow\; \text{continuous angle}$$
   π appears here as the analytic reference (e.g., \( \pi r^{2} \), \( 2\pi r \)).