ShootPattern — Engineering Summary (concise)Purpose:ShootPattern computes and simulates shotgun pellet patterns at user-selected distances, providing deterministic hit estimates and Monte-Carlo visualizations for engineering validation and field calibration.Core assumptions:• Angular spread ≈ constant → pattern diameter scales linearly with distance.  • Base stochastic model: uniform-in-area pellet distribution inside the circular pattern. (Advanced option: center-biased kernels.)Inputs:N = number of pelletsD_ref = reference pattern diameter (cm) measured at R_ref (m)R = target distance (m)T = target diameter (cm)Primary formulas:1. Pattern diameter at range (linear scaling)   D_p = (D_ref / R_ref) · R2. Areas   A_group = π · (D_p / 2)^2   A_target = π · (T / 2)^23. Deterministic expected hits (clipped)   fraction = clamp( A_target / A_group, 0, 1 )   E_hits = N · fractionSimulation (uniform-in-area, Monte-Carlo):• Sample for i = 1..N:   u ∼ Uniform(0,1)   r = sqrt(u) · (D_p/2)   θ ∼ Uniform(0, 2π)   x = r cosθ,  y = r sinθ• Count simulated hits inside target circle to validate E_hits and produce visuals.Optional advanced kernel (center-biased / Weibull):• Radial intensity λ(r) = λ0 · exp( − (r/σ)^k ) with normalization  λ0 = N / (2π σ^2 Γ(2/k))• Use for heatmaps, probability contours and POI offsets.Notes:• Calibrate using measured (D_meas, R_meas) to replace factory reference for higher accuracy.  • Model ignores pellet loss, wind, deformation and choke dynamics unless explicitly modeled.  • For probabilistic statements (P(≥k hits)) use repeated Monte-Carlo draws or binomial approximations based on per-pellet hit probability p = fraction.Implementation primitives (API):- pattern_diameter_at_distance(D_ref, R_ref, R) → D_p- expected_pellets_in_target(N, D_p, T) → E_hits- simulate_pellet_positions(N, D_p, seed=None) → [(x,y), ...]- advanced_kernel_heatmap(N, σ, k, POI) → density mapSafety: analytical tool only — validate with live testing.Credits: SiNiCore — Simba Nenov & Nikola Nenov