Zero-Angle Solver

Zero-Angle Solver

Problem

Find the bore-vs-sight angle θ that makes the trajectory hit a desired impact height at range R:

f(θ) = y(R; θ) − optic_height − impact_height = 0

y(R;θ) is the height reached by integrating from x=0 at v = v0·(cos(θ+aj), sin(θ+aj), 0) with aj = aerodynamic_jump (source/solve_angle.hpp). R is the zero distance (or any range in the inverse path), impact_height is usually 0 or the caller's zero_impact_height. The solver must be robust to poor seeds and report unreachability rather than a spurious angle.

Method

SolveAngle (source/solve_angle.hpp) is a fixed-point iteration with a geometric one-step:

FastInverseAngle(θ, f, R) = atan(tan θ − f/R)

(source/solve_angle.hpp). The fixed point is θ_{n+1}=FastInverseAngle(θ_n, f(θ_n), R), which is the angle that would zero the residual if the trajectory were a straight line — then corrected by re-integrating.

Pseudocode:

θ = seed
f = fire_to_target(θ)          // integrate to R, NaN if vx≤0 / TOF≥max / vacuum
for iter in 0..9
    θ' = FastInverseAngle(θ, f, R)
    if θ'∉[−45°,45°] or NaN → return NaN
    if |θ'−θ| ≤ tol           → return θ' (as MOA)
    θ = θ';  f = fire_to_target(θ)
return NaN                      // no convergence

Constants (source/solve_angle.hpp):

• kMaxIterations = 10
• kDefaultAngleTolerance = 0.01 MOA (≈ 2.9 µrad)
• kMaxAngle = 45°, kMinAngle = −45° — lob targets flat-fire trajectories; above 45° a high (mortar) and a low direct-fire trajectory can both reach R. The solver restricts search to the flat-fire branch, treating >45° as out-of-bounds (returns NaN / kLobErrorZeroUnreachable).

fire_to_target integrates step-by-step via SolveStep until x ≥ R and returns the vertical miss; any step where vx ≤ 0, TOF ≥ max_time, or speed ≤ minimum_speed yields NaN and the iteration becomes non-finite, propagating to a NaN return (unreachable).

Seed

BuildZeroAngle (source/lob_builder.cpp) supplies the seed when the caller gave ZeroDistanceYds instead of ZeroAngleMOA:

θ_vac = 0.5·g·R / v0²           (vacuum parabola angle)
seed  = clamp(θ_vac, −45°, 45°)

The same seed construction is used in test/source/solve_angle_test.cpp. A poor seed (e.g. 0 or even −30°) still converges for the validated ranges (test/source/solve_angle_test.cpp), but the vacuum seed cuts typical iterations to 2–4.

Validation

• test/source/solve_angle_test.cpp checks convergence from 0°, ±30°, 44° seeds, round-trip with ±3 in impact heights, very short range, tight 0.001 MOA tolerance, and unreachable detection.
• test/source/solve_angle_test.cpp cross-checks the builder's zero angle against a direct SolveAngle call (BuilderZeroAngleTest), while test/source/lob_api_test.cpp verifies weak BC=0.1, v=600 fps at 2000 yd exhausts iterations and returns NaN via kLobErrorZeroUnreachable.

Reuse

The same solver backs inverse adjustments (Inverse Solver) and is the point of the Shared Angle Solver decision. SolveAngle is inline in a header so both builder and solve translation units share one definition.