ballistic-solver computes the launch angles to hit moving targets under gravity, quadratic air drag, and optional wind — for games, simulation, robotics, and targeting.

Unlike vacuum / closed-form solvers, it simulates the projectile and solves the intercept numerically, so it stays accurate on strongly curved trajectories. Native C/C++ core with a stable C ABI and a modern C++ API (bs::solve), plus Python (pip install ballistic-solver), C#/Unity, and Godot bindings.

godot_intercept.mp4

Real-time 3D showcase — the built-in tracker leading a noisy, diving target with range rings, tracer trails, and a live hit-rate HUD (examples/godot/).

Applications. The solver only sees relative kinematics, so the same code drops into many domains — most visibly games and simulation (turret lead-targeting, projectile interception; see the demo above), and equally robotics (interception / capture), tracking, and other aiming / targeting problems. An optional position-only tracker (bs.TargetTracker) adds lead-fire from a noisy track. See benchmarks/intercept_eval.py and examples/viz/ for cross-domain examples and accuracy comparisons.

---

Quick start

Python (PyPI)

pip install ballistic-solver

Requires Python >= 3.10.

import ballistic_solver as bs

result = bs.solve(
    relPos0=(120, 30, 5),
    relVel=(2, -1, 0),
    v0=90,
    kDrag=0.002,
)

print(result["theta"], result["phi"], result["miss"])
print(result["success"], result["status"], result["message"])

For tighter convergence without manually tuning every knob:

params = bs.params_preset("precise")
result = bs.solve((120, 30, 5), (2, -1, 0), 90, 0.002, params=params)

For repeated solves, Python also provides a thin convenience wrapper:

solver = bs.Solver.preset("precise")
result = solver.solve((120, 30, 5), (2, -1, 0), 90, 0.002)

---

Demo (Unity)

Highly curved trajectories under strong air drag, still converging to a hit against moving targets.

ballistic_demo_low.mp4

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Why this solver

Many launch-angle solvers either assume a vacuum (no drag) or linearize the dynamics. Each common approach trades away something:

Approach	Moving target	Drag	Wind	No per-shot tuning	Real-time speed	Robust
Vacuum / closed-form	○	✗	✗	◎	◎	○
Brute-force angle grid	◎	◎	◎	○	✗	◎
Generic optimizer (e.g. least-squares)	◎	◎	◎	△	△	○
ballistic-solver	◎	◎	◎	◎	◎	◎

◎ strong · ○ workable · △ limited · ✗ not supported

Vacuum solvers are the fastest but ignore drag and wind. Brute-force grids handle everything but are too slow for real-time use. Generic optimizers work on the same residual but need a good initial guess and per-problem tuning, and are more prone to local minima. ballistic-solver is purpose-built for the moving-target + drag + real-time corner: a closed-form vacuum warm start, an analytic Jacobian preconditioner, and a multistart fallback for the hard cases. See docs/numerical_method.md for the full method.

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Key properties

• Moving targets supported
• Constant-acceleration targets supported via the extended API
• Strong air resistance (quadratic drag) supported
• Low / High arc selection (since v0.2)
• Wind vector supported (since v0.3)
• Extended C ABI utilities (since v0.4)
• Quartic vacuum-lead initialization for moving targets (since v0.6)
• High-arc moving-target auxiliary multi-start fallback (since v0.6.1)
• Coordinate-residual Gauss–Newton core with vacuum-seeded Jacobian (default since v1.0; ~15–25% faster, auxiliary-residual method kept as solve_aux)
• Fast / balanced / precise solver presets
• Physical drag helper: kDrag = 0.5 * rho * Cd * area / mass
• Robust in strongly nonlinear regimes (no analytic assumptions)
• Best-effort result returned even without perfect convergence
• Explicit success / failure reporting (+ diagnostic message)
• Stable C ABI for multi-language use
• Modern C++ API (bs::solve / bs::solve_aux)
• Easy install via PyPI: pip install ballistic-solver

---

Python API

solve(...)

solve(relPos0, relVel, v0, kDrag, arcMode=None, params=None, relAcc=None) -> dict

• relPos0: target relative position at t=0 (x,y,z)
• relVel: target relative velocity (x,y,z)
• v0: muzzle speed (scalar)
• kDrag: quadratic drag coefficient
• arcMode: 0/1 or "low"/"high" (case-insensitive); None keeps the params value
• params: optional BallisticParams for advanced tuning (gravity, wind, integrator and solver knobs)
• relAcc: optional constant target relative acceleration (x,y,z); same as solve_accel

Returned dict keys include:

• success (bool)
• theta, phi (radians)
• miss (closest-approach distance)
• tStar (time of closest approach)
• relMissAtStar (3-vector miss at tStar)
• status (SolveStatus integer)
• message (short diagnostic string)
• plus convergence diagnostics (iterations, acceptedSteps, lastLambda, lastAlpha)

solve uses the coordinate-residual core (Gauss–Newton on the 3D closest-approach miss, vacuum-seeded Jacobian, multistart fallback). It is the same signature as before and a drop-in replacement, typically ~15–25% faster at equal robustness.

solve_aux(...)

solve_aux(relPos0, relVel, v0, kDrag, arcMode=None, params=None, relAcc=None) -> dict

The previous default: the auxiliary-residual method (re-aim through the vacuum inverse, solved by damped least squares with a multistart fallback). Same return dict as solve. Kept for compatibility and for reproducing the auxiliary-residual results.

solve_predicted(...)

solve_predicted(predictor, v0, kDrag, arcMode=None, params=None) -> dict

Lead-fire against a position-only trajectory predictor instead of explicit relVel/relAcc. predictor(t) returns the predicted relative position (x, y, z) at time t >= 0 (e.g. from a Kalman/IMM track). Internally a local constant-acceleration model is fitted at the intercept time and iterated to a fixed point — exact for constant-velocity / constant-acceleration predictors and second-order accurate at the intercept for smoothly curving tracks. Returns the same dict as solve.

TargetTracker

tracker = bs.TargetTracker(processNoise=1.0, measNoise=0.25)
for t, pos in measurements:        # (timestamp, relative position)
    tracker.update(t, pos)
lead = tracker.predict(1.5)        # predicted relative position 1.5 s ahead
result = tracker.solve(v0=95.0, kDrag=0.0015)   # lead-fire via the predictor seam

A position-only constant-acceleration Kalman tracker (per-axis, white-noise-jerk model). It denoises the track and recovers velocity/acceleration without finite-difference noise, then feeds the predictor seam — so callers never supply relVel/relAcc. See benchmarks/predictor_eval.py for a lead-prediction accuracy comparison against a constant-velocity baseline.

The real-time 3D showcase at the top of this README is built on this tracker — the full project is in examples/godot/ (default scene demo/intercept_demo.tscn).

Utilities

params = bs.params_preset("fast")  # or "balanced", "precise"
kDrag = bs.k_drag_from_physical(
    airDensity=1.225,
    dragCoefficient=0.30,
    area=0.00426,
    mass=0.145,
)

For constant-acceleration targets:

result = bs.solve_accel(
    relPos0=(120, 30, 5),
    relVel=(2, -1, 0),
    relAcc=(0, 0.2, 0),
    v0=90,
    kDrag=0.002,
    params=bs.params_preset("precise"),
)

---

Solver validity note

The solver internally integrates projectile motion using:

• 4th-order Runge–Kutta (RK4)
• Fixed timestep dt
• Quadratic drag: a = (0, 0, -g) - kDrag * |v - wind| * (v - wind)
• Wind as air velocity

To match in-game ballistics, your runtime simulation must use the same physical model and integrator configuration.

If your game uses a different integrator (e.g., Euler) or a different timestep, the computed launch angles may not hit even if the solver reports success.

---

C ABI (stable interface)

Primary intercept API:

void ballistic_inputs_init(BallisticInputs* in);
int32_t ballistic_solve(const BallisticInputs* in, BallisticOutputs* out);

Return value policy:

• 0: API call completed and out was filled.
• <0: API-level failure, such as null pointers or an internal exception.
• Numerical solve success is reported separately by out->success and out->status.
• ABI v3 adds convergence diagnostics to BallisticOutputs: iterations, acceptedSteps, lastLambda, and lastAlpha.
• ABI v4 adds a preset field to BallisticInputs (0=Fast, 1=Balanced, 2=Precise). ballistic_inputs_apply_preset records it so that the full preset tuning (line-search, lambda tries, finite-difference step, golden-section) is carried into ballistic_solve, not only dt/tMax/tolMiss/maxIter.

Callers should check both:

int32_t rc = ballistic_solve(&in, &out);
if (rc != 0) {
    /* API call failed */
}
if (!out.success) {
    /* Solver ran but did not satisfy the requested tolerance. */
}

Since v0.4.0, additional utility functions are available:

void ballistic_rk4_step(...);

int32_t ballistic_simulate_trajectory(...);
int32_t ballistic_simulate_trajectory_from_angles(...);

int32_t ballistic_find_closest_approach(...);

int32_t ballistic_vacuum_arc_angles_to_point(...);
void ballistic_initial_guess_vacuum_lead(...);

Additional extended APIs include:

void ballistic_accel_inputs_init(BallisticAccelInputs* in);
int32_t ballistic_solve_accel(const BallisticAccelInputs* in, BallisticOutputs* out);
int32_t ballistic_inputs_apply_preset(BallisticInputs* in, int32_t preset);
int32_t ballistic_k_drag_from_physical(...);
int32_t ballistic_make_relative_motion(...);

See ballistic_solver_c_api.h for full signatures and parameter definitions.

This enables usage from:

• C / C++
• Python (ctypes via the C ABI)
• C# / .NET / Unity (P/Invoke)
• Godot 4 (GDExtension addon-style example)
• Others via FFI

Prebuilt native binaries are provided via GitHub Releases.

---

Arc mode (since v0.2)

C ABI convention:

• arcMode = 0 → Low
• arcMode = 1 → High

High arc example:

ballistic_demo_high.mp4

---

Wind (since v0.3)

C ABI convention:

• wind[3] = air velocity vector (same frame as relPos0/relVel)
• Drag uses relative airspeed: v_rel = v_projectile - wind

Wind demo:

ballistic_demo_wind.mp4

---

Using prebuilt binaries (C ABI)

Download the archive for your platform from Releases.

Each release contains:

• Shared library

  • Windows: ballistic_solver.dll
  • Linux: libballistic_solver.so
  • macOS: libballistic_solver.dylib

• C ABI header: ballistic_solver_c_api.h

---

Use as a CMake dependency (C/C++)

Pull the library straight into your CMake build with FetchContent:

include(FetchContent)
FetchContent_Declare(
    ballistic_solver
    GIT_REPOSITORY https://github.com/ujinf74/ballistic-solver.git
    GIT_TAG v1.0.0
)
set(BUILD_TESTING OFF)  # don't build the library's own tests in your tree
FetchContent_MakeAvailable(ballistic_solver)

target_link_libraries(your_app PRIVATE ballistic_solver)

C++

#include <ballistic_solver.hpp>

bs::Problem problem;
problem.rel_pos0 = {120, 30, 5};
problem.rel_vel  = {2, -1, 0};
problem.v0       = 90;
problem.k_drag   = 0.002;

bs::Options options;
options.preset = bs::Preset::Precise;

bs::Intercept r = bs::solve(problem, options);
if (r)                       // explicit operator bool() == success
    aim(r.theta, r.phi);     // elevation, azimuth in radians

With C++20 designated initializers the same call is more compact:

auto r = bs::solve({.rel_pos0 = {120, 30, 5},
                    .rel_vel  = {2, -1, 0},
                    .v0       = 90,
                    .k_drag   = 0.002},
                   {.preset = bs::Preset::Precise});

The bs:: API is source-level (build it with the same toolchain as the library; none of the internal headers leak into your code). For a stable ABI or FFI from other languages, use the C API instead.

C (stable ABI)

#include "ballistic_solver_c_api.h"

BallisticInputs in; BallisticOutputs out = {0};
ballistic_inputs_init(&in);
in.relPos0[0] = 120; in.relPos0[1] = 30; in.relPos0[2] = 5;
in.relVel[0]  = 2;   in.relVel[1]  = -1; in.relVel[2]  = 0;
in.v0 = 90; in.kDrag = 0.002;

if (ballistic_solve(&in, &out) == 0 && out.success)
    printf("theta=%.4f phi=%.4f miss=%.4f\n", out.theta, out.phi, out.miss);

---

C# / Unity usage

A C# P/Invoke example is available in:

examples/dotnet/

On Windows, place ballistic_solver.dll next to the executable (or ensure it is discoverable via PATH), then call ballistic_solve via DllImport.

This works directly inside Unity.

---

Godot

The GDExtension addon in examples/godot/ exposes the solver to GDScript:

var solver := BallisticSolver.new()
var r := solver.solve(rel_pos0, rel_vel, v0, k_drag, 0)  # arc_mode: 0 low, 1 high
if r.success:
    aim_at(r.theta, r.phi)  # elevation, azimuth in radians

A position-only BallisticTracker (Kalman) with the same solve(v0, k_drag, arc_mode) seam is also provided — it powers the real-time demo at the top of this README.

---

How it works (high level)

1. Build a vacuum-lead initial guess from the moving-target quartic.
2. Seed the inverse Jacobian analytically from the vacuum-arc map (no trajectory integration), as a cheap preconditioner.
3. Simulate projectile motion using RK4 integration with drag (+ wind) and track the closest approach between projectile and target.
4. Use the 3D closest-approach miss vector itself as the residual (it is well-conditioned), and solve the launch angles by Gauss-Newton on it.
5. Refine the Jacobian with Broyden-style rank-1 updates; take full steps (no Levenberg–Marquardt damping, no line search).
6. If the warm start does not reach tolerance, retry from a small arc-appropriate theta-grid multistart.
7. Return the best solution found.

solve uses this coordinate-residual core. The earlier auxiliary-residual method (re-aim through the vacuum inverse, with damped least squares) remains available as solve_aux for compatibility and reproducibility.

Failure cases are explicitly detected and reported. See docs/numerical_method.md for a step-by-step description and docs/limitations.md for the modelling assumptions.

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When solving can fail

The solver returns a best-effort result even when it cannot satisfy tolMiss. Common causes include:

• invalid inputs (v0 <= 0, non-positive g, dt, tMax, or maxIter)
• geometrically unreachable vacuum targets
• targets that require an intercept beyond tMax
• strong drag or high-arc cases where the selected arc branch cannot be maintained
• iteration limits that are too tight for the requested tolerance

Use success, status, message, and miss together when deciding whether to accept a solution.

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Limitations

ballistic-solver is a point-mass solver with a deliberately scoped physics model:

• point mass — no spin drift / Magnus effect
• quadratic drag with a single coefficient — no Mach-dependent drag table
• constant gravity and uniform, constant wind
• solves launch angles only (muzzle speed and arc branch are inputs)
• no terrain / obstacle collision
• best-effort (not guaranteed) on unreachable or near-limit targets
• a numerical library for games, simulation, robotics, and research — not a certified fire-control system

See docs/limitations.md for the full list and the reasoning behind each.

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Status codes (SolveStatus)

BallisticOutputs.status / Python result["status"] corresponds to:

• 0 = Ok
• 1 = InvalidInput
• 2 = InitialResidualFailed
• 3 = JacobianFailed
• 4 = LMStepSingular
• 5 = ResidualFailedDuringSearch
• 6 = LineSearchRejected
• 7 = LambdaTriesExhausted
• 8 = MaxIterReached

The default coordinate-residual solve emits Ok, InvalidInput, JacobianFailed, or MaxIterReached. The Levenberg–Marquardt codes (4–7) are specific to the auxiliary-residual solve_aux path.

message contains a short diagnostic string.

---

Build from source

cmake -S . -B build
cmake --build build -j
ctest --test-dir build

The shared library target is ballistic_solver.

---

Regression and benchmark

Distributed regression and benchmark scripts are available:

python tests/random_regression.py
python benchmarks/linear_cases.py

The CI workflow runs CTest smoke coverage plus the Python regression script. The regression script includes analytic vacuum checks, constructed moving-target vacuum cases, unreachable-target checks, randomized linear cases, and constant-acceleration API smoke coverage.

Benchmark numbers depend on CPU, OS, compiler, build type, Python version, and whether native or Python entrypoints are measured. The local reference below was measured on Windows NT 10.0.26200.0, AMD Ryzen 9 6900HS, Python 3.12.8, MSVC 14.44.35207, x64 Release build, through the Python native extension.

Reference preset result from the same local Release build:

fast:     500/500, median 0.031 ms, p95 0.131 ms, p95 miss 2.810e-02 m
balanced: 500/500, median 0.057 ms, p95 0.261 ms, p95 miss 7.053e-03 m
precise:  500/500, median 0.087 ms, p95 0.356 ms, p95 miss 5.596e-06 m

Current default solver (coordinate-residual core) results from 10,000 generated cases with seed 20260503. The last row is the high-arc moving-target stress case:

Case set	Success	Median runtime	P95 runtime	P95 miss
Low arc, moving target	10000/10000 (100.00%)	0.089 ms	0.266 ms	6.450e-03 m
High arc, stationary target	10000/10000 (100.00%)	0.408 ms	0.846 ms	8.018e-03 m
High arc, moving target	10000/10000 (100.00%)	0.607 ms	1.264 ms	8.341e-03 m

For a difficulty-bucketed breakdown (low / high arc × drag strength, with P99 and the hard-corner tail) and the full methodology, see docs/benchmarks.md.

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ABI notes

• Plain C layout across the ABI boundary
• Fixed-size arrays only
• No dynamic allocation across the boundary

---

Official repository

The only official repository is https://github.com/ujinf74/ballistic-solver. Install from PyPI (pip install ballistic-solver) or this repository's Releases page — not from third-party copies, mirrors, or reuploads.

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License

MIT License