Game Custom Physics Solvers

Custom Physics Solvers & Advanced Math

Build a physics simulation by hand — collision detection, integration, and constraint resolution from first principles — when a built-in engine is unavailable, too heavy, or doesn't fit (deterministic lockstep, specialized soft bodies, custom fluids, tiny dependency-free builds).

When to Use

Activate this skill when the task involves writing the solver yourself rather than configuring an engine's built-in physics. Specific triggers:

• Collision detection from scratch: SAT for convex polygons/polyhedra, GJK + EPA for general convex shapes, circle/AABB/capsule tests.
• Integration: choosing and implementing semi-implicit (symplectic) Euler, Verlet, or RK4 with a fixed timestep.
• Resolution: impulse-based response (restitution + friction) or constraint solvers (sequential impulses / Gauss–Seidel, Position-Based Dynamics).
• Soft bodies: Verlet cloth, rope, and pressure-based soft bodies built from particles + distance constraints.
• Broadphase: spatial hashing/uniform grids, quadtrees/octrees, sweep-and-prune to avoid O(n²) pair checks.
• Fluids: particle systems and SPH (smoothed-particle hydrodynamics) solvers.
• Deterministic lockstep: cross-platform reproducible simulations requiring fixed-point or carefully managed float strategies.

Do Not Use For

If a production engine's physics fits, use it — a hand-rolled solver is more code, more bugs, and slower than Jolt/PhysX unless you specifically need determinism, custom behavior, or a tiny dependency-free build.

Prerequisites

• Solid understanding of vector math (dot/cross products, projection, normalization). See math-and-vectors skill if needed.
• The target language has no suitable physics library, or the project explicitly requires a custom solver (determinism, size, specialization).
• A profiling setup (even a simple frame-time counter) — custom solvers need performance validation as body counts grow.

Procedure

1. Establish the Fixed Timestep Loop (Non-Negotiable)

A variable dt makes integration non-deterministic and explosive — the same collision resolves differently at 30 vs 144 fps, and large frames tunnel through walls. Always accumulate real time and step the simulation at a constant dt, then interpolate the render state.

accumulator += frameDeltaTime;
while (accumulator >= FIXED_DT) {     // e.g. FIXED_DT = 1/60
    step(FIXED_DT);                   // ALL physics advances by a constant dt
    accumulator -= FIXED_DT;
}
float alpha = accumulator / FIXED_DT; // interpolate rendering between states
render(lerp(prevState, currState, alpha));

This single rule prevents most "physics feels different on my machine" bugs. Set FIXED_DT once and never change it at runtime.

2. Choose and Implement the Integrator

Semi-implicit Euler — update VELOCITY first, then position with the NEW velocity:

v += (force / mass) * dt;
x += v * dt;                 // stable; plain (explicit) Euler updates x with OLD v and drifts/explodes

Verlet — velocity is implicit in (current - previous) position:

Vec2 temp = pos;
pos = pos + (pos - prevPos) + accel * dt * dt;   // (pos - prevPos) carries momentum
prevPos = temp;

HARD RULE: Never use explicit (forward) Euler for dynamics. It gains energy and blows up. Semi-implicit Euler or Verlet only.

3. Implement Broadphase (Before Narrowphase)

Never run narrowphase on all N² pairs. Choose based on your scene:

• Uniform grid / spatial hash: fast for evenly sized objects. Hash floor(x/cellSize), floor(y/cellSize) into a dictionary; check neighboring cells.
• Quadtree / Octree: varied sizes; subdivides adaptively.
• Sweep-and-prune: good for many bodies along one axis. Sort by min-AABB on one axis, sweep, test overlap on remaining axes.

4. Implement Narrowphase Collision Detection

SAT (Separating Axis Theorem) — for two convex shapes, if there exists an axis (each face normal of both shapes) on which their projections don't overlap, they're separated. The axis of minimum overlap gives the collision normal and penetration depth.

// SAT core idea (2D convex): project both shapes onto each candidate axis.
for (Vec2 axis : allEdgeNormals(a, b)) {
    auto [minA, maxA] = project(a, axis);
    auto [minB, maxB] = project(b, axis);
    float overlap = std::min(maxA, maxB) - std::max(minA, minB);
    if (overlap <= 0) return NO_COLLISION;        // a separating axis exists
    if (overlap < minOverlap) { minOverlap = overlap; collisionNormal = axis; }
}
// minOverlap + collisionNormal define the MTV (minimum translation vector).

SAT works great for boxes/convex polygons; axis count explodes for high-vertex 3D shapes.

GJK — tells you if two convex shapes intersect (and the closest distance if not) using support functions and a simplex. Scales to any convex shape. Pair with EPA to recover penetration depth + normal when they do intersect.

Compute and store the contact manifold — normal, penetration, contact point(s). Resolution needs all three.

5. Implement Resolution

Impulse-based (rigid bodies): along the contact normal, apply an impulse j from relative velocity, restitution e, and inverse masses; then apply a friction impulse along the tangent clamped by the Coulomb cone (|jt| ≤ μ·jn).

// Impulse magnitude along the contact normal.
float invMassSum = a.invMass + b.invMass;
Vec2 rv = b.vel - a.vel;
float velAlongNormal = dot(rv, normal);
if (velAlongNormal > 0) return;                  // already separating — do nothing
float j = -(1 + e) * velAlongNormal / invMassSum;
Vec2 impulse = j * normal;
a.vel -= a.invMass * impulse;
b.vel += b.invMass * impulse;                    // static bodies use invMass = 0

Positional correction: velocity impulses alone leave objects slightly sunk. Add Baumgarte/slop correction to push out residual penetration without injecting energy. Use a small slop (e.g. 0.01–0.05 units) and a correction percent like 0.2–0.8.

Constraint solvers (joints, stacks, soft bodies): solve constraints iteratively:

• Sequential impulses / Gauss–Seidel: run multiple iterations per step; more iterations = stiffer, more stable stacks.
• Position-Based Dynamics (PBD): projects positions to satisfy constraints directly; backbone of cloth/rope.

6. Soft Bodies & Cloth (Verlet + Constraints)

1. Represent the body as particles (mass points) integrated with Verlet.
2. Connect them with distance constraints (structural, shear, bend springs for cloth).
3. Each step, relax constraints several iterations: move particle pairs back toward their rest length. More iterations → stiffer cloth. This is PBD in miniature and is far more stable than stiff spring forces.

HARD RULE: Do not use stiff springs to simulate rigid connections. High spring constants force a tiny timestep and introduce instability. Prefer PBD/constraint projection for stiff behavior.

7. Fluids (SPH)

• Treat fluid as particles carrying mass; each frame compute density and pressure from neighbors within a smoothing radius h, then pressure + viscosity forces drive motion.
• A neighbor search over a spatial grid is mandatory — naive all-pairs SPH is O(n²) and unusable past a few hundred particles. Hash particles into a grid keyed by the smoothing radius.
• SPH is touchy: timestep, smoothing radius, and stiffness are coupled. Start from known-good parameters and change one at a time.

Pitfalls

1. Variable timestep: the root cause of non-determinism and explosions. Always fixed-step + accumulator + render interpolation. No exceptions.
2. Explicit (forward) Euler for dynamics: gains energy and blows up. Use semi-implicit Euler or Verlet.
3. Skipping broadphase: O(n²) narrowphase dies past a few hundred bodies. Add a grid/quadtree/SAP first.
4. Velocity-only resolution: objects sink and jitter. Add positional correction with slop.
5. Tunneling: fast/thin objects pass through thin walls in one step. Use sub-stepping or continuous collision (conservative advancement / swept tests).
6. Too few solver iterations: stacks and joints feel mushy or jitter. Increase Gauss–Seidel iterations (trade CPU for stability).
7. Floating-point determinism assumptions: if you need cross-platform lockstep, plain floats may diverge — consider fixed-point or a single reference platform. Do not assume float arithmetic is identical across CPUs, compilers, or build settings.
8. Stiff springs instead of constraints: high spring constants force a tiny timestep. Prefer PBD/constraint projection for stiff behavior.
9. All-pairs SPH neighbor search: unusable at scale. Hash particles into a grid keyed by the smoothing radius.
10. Forgetting friction tangent impulse: applying only the normal impulse makes objects slide forever on surfaces. Compute the tangent impulse and clamp by the Coulomb cone.
11. Not handling invMass = 0 for static bodies: division by zero or incorrect impulse distribution. Static/kinematic bodies must have invMass = 0 and invInertia = 0.

Verification

Run through this checklist after implementing the solver:

• Simulation runs on a fixed timestep with an accumulator; rendering interpolates between states using alpha.
• Dynamics use semi-implicit Euler / Verlet / RK4 (never explicit Euler).
• A broadphase culls pairs before narrowphase (grid / quadtree / sweep-and-prune).
• Narrowphase produces normal + penetration + contact point (SAT or GJK/EPA).
• Resolution applies normal + friction impulses and positional correction (Baumgarte/slop).
• Constraint/PBD solver runs enough iterations for stable stacks/cloth (test with a stack of 10+ boxes).
• Fast objects don't tunnel (sub-stepping or swept tests at the chosen dt — test with a high-velocity projectile vs. thin wall).
• SPH/soft-body neighbor search uses spatial partitioning, not all-pairs.
• If lockstep determinism is required, the float/fixed-point strategy is explicit and documented.
• Energy is conserved over long runs: place a bouncing ball with restitution 1.0 in a closed box and verify it doesn't gain or lose significant energy over 60 seconds.

Quick Smoke Test (PowerShell)

If you have a build output (e.g. a C++ executable or a script), run a timed test:

# Run the simulation binary for 60 seconds of sim time, capture output
$proc = Start-Process -FilePath ".\build\physics_sim.exe" -ArgumentList "--duration 60 --bodies 100" -NoNewWindow -Wait -PassThru
Write-Host "Exit code: $($proc.ExitCode)"
# Exit code 0 with stable output = pass; non-zero or NaN in output = fail

For a Godot/Unity script-based solver, run the scene and check the output log for NaN/Infinity:

Get-Content .\godot_project\logs\physics.log | Select-String -Pattern "NaN|Infinity|ERROR"
# No matches = clean

Examples

Minimal Fixed-Timestep Game Loop (C++)

const float FIXED_DT = 1.0f / 60.0f;
float accumulator = 0.0f;

void onFrame(float frameDeltaTime) {
    accumulator += frameDeltaTime;
    // Clamp to prevent spiral-of-death after stalls
    if (accumulator > FIXED_DT * 5) accumulator = FIXED_DT * 5;

    while (accumulator >= FIXED_DT) {
        previousState = currentState;
        step(FIXED_DT);
        accumulator -= FIXED_DT;
    }
    float alpha = accumulator / FIXED_DT;
    render(lerp(previousState, currentState, alpha));
}

Distance Constraint Relaxation (PBD, Pseudocode)

void satisfyConstraint(Particle& a, Particle& b, float restLength) {
    Vec2 delta = b.pos - a.pos;
    float dist = length(delta);
    if (dist == 0) return;
    Vec2 dir = delta / dist;
    float correction = (dist - restLength) / (a.invMass + b.invMass);
    a.pos += dir * correction * a.invMass * 0.5f;
    b.pos -= dir * correction * b.invMass * 0.5f;
}
// Call this N times per step (N = iteration count, e.g. 5–20)

Related Skills

• physics-system — Godot's built-in bodies/Jolt. Use it instead unless you specifically need a custom solver.
• unreal-engine / unity-engine — Production engines whose physics (Chaos/PhysX) you'd normally use over a hand-rolled solver.
• math-and-vectors — Supporting linear algebra these solvers build on.
• shader skills — GPU compute paths for particle/fluid solvers if CPU throughput is insufficient.

References

Load these reference files when you need deeper detail on a specific algorithm:

• Separating Axis Theorem, GJK, and EPA — collision-detection literature. Search for "SAT collision detection" and "GJK EPA penetration depth" for canonical implementations.
• Position-Based Dynamics (Müller et al.) — the PBD paper for constraint projection methods.
• Sequential-impulse constraint solving (Catto, Box2D) — for rigid-body constraint solvers.
• Verlet integration for cloth/rope — classic particle-based soft body methods.
• SPH (Müller, Charypar, Gross) — "Particle-Based Fluid Simulation for Interactive Applications" for real-time fluids.
• "Fix Your Timestep!" (Gaffer on Games) — the definitive reference for the fixed-step/interpolation loop.