Continuous

Sep 12, 2025

Scientific correction — 13 October 2026

Local Lipschitz regularity away from the seed gives local existence and uniqueness. Positive radial bias alone does not guarantee finite-time boundary arrival: in the unit disk, radial speed 1-r approaches the boundary asymptotically. Seed perturbations change the field as well as the initial data and require a bound on that field dependence. Convergence and monotonicity of discrete measures require a separate consistency proof.

Current reference: Circle and Area from Binary Radial Modules.

Continuous Model: Radial Fields, Well-posedness & Stability

Category: Generative Radial Geometry (GRG)

We describe a continuous model that interprets GRG’s inside-out construction as integral curves of a center-seeded field on a domain Ω ⊂ ℝ². Let the seed be s ∈ Ω. For any point x ≠ s define the unit radial direction e_r(x) = (x−s)/‖x−s‖ and a tangential direction e_t(x) orthogonal to e_r. The guidance field blends radial and tangential components.

Radial Guidance Field

The model uses a field F: Ω\\{s} → ℝ² with radial bias

$$ \mathbf{F}(x) \;=\; a(x)\, \mathbf{e}_r(x) \;+\; b(x)\, \mathbf{e}_t(x), \qquad a(x) > 0 $$

Integral curves γ(τ) solve

$$ \dot{\gamma}(\tau) \;=\; \mathbf{F}\big(\gamma(\tau)\big), \qquad \gamma(0)=x_0,\quad x_0\in\Omega\setminus\{s\} $$

Each turn-angle u ∈ [0,1) selects an initial direction, producing local trajectories; boundary arrival requires additional hypotheses. The discrete GRG construction samples these trajectories at uniform steps in turns.

Well-posedness

Existence & Uniqueness. If F is locally Lipschitz on Ω\\{s} then trajectories starting away from the seed have local existence and uniqueness while they remain in that domain. Global continuation and boundary arrival require additional hypotheses.
Termination. Trajectories stop when they hit ∂Ω. Dirichlet-type constraints prescribe a target boundary, while Neumann-type constraints control flux through ∂Ω.

Stability to Perturbations

If F has Lipschitz constant L, perturbing the seed by ε yields trajectories with deviation bounded by

$$ \|\gamma_\varepsilon(\tau) - \gamma(\tau)\| \;\le\; C\,\varepsilon\, e^{L\tau} $$

In practice we keep L moderate via smoothing of a(x), b(x) and by capping curvature of trajectories.

Regularization & Safety

Step lower bound. Minimum angular step Δu_{\min} to avoid local collapse.
Curvature damping. Add −λ\,κ(\gamma)\,\mathbf{e}_t to reduce spiraling near thin features.
Line-search. Backtrack the last advance if distance to boundary decreases below a threshold.

Multi-Seed and Holes

For non-simply connected domains or shapes with holes we blend fields from auxiliary seeds s_i with weights w_i(x):

$$ \mathbf{F}(x) \;=\; \sum_{i=1}^{m} w_i(x)\,\Big(a_i(x)\,\mathbf{e}_r^{(i)}(x) + b_i(x)\,\mathbf{e}_t^{(i)}(x)\Big), \qquad \sum_i w_i(x)=1,\ \ w_i(x)\ge 0 $$

Hole guards choose weights that vanish across cut-loci to prevent ambiguous paths.

Discrete Connection

Sampling each trajectory at uniform steps in turns yields the GRG primitives used for chord sums and radial strips. The continuous model proposes a connection; it does not by itself prove convergence of discrete measures. With turns-first sampling and dyadic refinement, the discrete GRG primitives converge monotonically as b increases.

Center-seeded field with trajectories to the boundary, auxiliary seed for a hole, and example paths.

Practical Defaults

Field coefficients. a(x)=1 near the seed, tapering to maintain stable speed; b(x) small and smooth to avoid spirals.
Adaptive control. Increase samples where the curvature proxy exceeds a threshold; cap Δu.
Refinement (dyadic). Use the dyadic level b ∈ ℕ_≥0 with N = 2^b samples and step Δu = 2^{-b}; increasing b yields monotone refinement.
Reconstruction. Quarter symmetry reconstructs the full figure; multi-seed blending covers non-simply connected domains.