Scientific correction — 13 October 2026
Exact cell counting does not establish exact continuous geometry. Ring additivity requires nested domains with disjoint added rings; closure and symmetry require an explicit generation rule. Area convergence requires geometric consistency. Perimeter convergence additionally requires a specified metric and weights: unweighted horizontal/vertical steps approximate a Manhattan perimeter rather than a Euclidean perimeter.
Current reference: Circle and Area from Binary Radial Modules.
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; counting identities alone do not control geometric approximation or numerical rounding.
- 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: Convergence requires a defined generation rule, metric, and consistency proof.
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:$$\mu_s\!\left(D(r{+}1; s)\right) - \mu_s\!\left(D(r; s)\right) \;=\; \lvert A_{r+1} \rvert,$$where \(A_{r+1}\) is the ring generated at radius \(r{+}1\). This identity holds when the domains are nested and the new ring is exactly their disjoint set difference. - Arc closure.
Closure is a condition to verify for 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. - Proposed convergence to the continuous limit — conditional on consistency and metric assumptions.
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., \( \pi r^{2} \), \( 2\pi r \)).