Physics Engine

Physics Engine Internals — Deep Guide

A physics engine simulates rigid bodies through three pipelines every tick: collision detection (find contacts), collision resolution (apply impulses/constraints), and integration (advance positions/velocities). This guide gives a lead engineer's view of how real engines (PhysX, Jolt, Bullet, Box2D) work underneath.

1. The Physics Step Pipeline

1. For each body: apply forces -> linear/angular velocity (accelerate)
2. Broad-phase:  generate candidate pairs of AABB-overlapping bodies
3. Narrow-phase: exact contact detection -> contact manifold (points, normals, penetration)
4. Constraint solver: iterate contacts + joints to correct penetration & velocities
5. Integration:   x += v * dt; Rot += omega * dt (per body)
6. Sleep/wake:    mark bodies nearly at rest for deactivation

1.1 Determinism vs Performance

• Fixed timestep required (see skills/game/game-engine/patterns/SKILL.md → game-loop section).
• Deterministic engines (rollback netcode, replays) must avoid unordered map iteration, use fixed float ops, and sort contact manifolds deterministically.

2. Broad-Phase Collision Detection

Goal: reject distant pairs cheaply; output pairs of possibly-intersecting AABBs.

2.1 Sweep-and-Prune (SAP)

• Sort AABB min/max along one/main axis; maintain lists; in the sorted order, intervals that overlap can be tracked incrementally.
• O(N log N) sort + O(N + P) query.
• Great when most overlap is on one axis and lists are already nearly sorted frame-to-frame.

2.2 Dynamic Bounding Volume Hierarchy (DBVH)

• Tree of AABBs; leaf = one body; internal node = union.
• Insert/remove/update with surface-area-heuristic (SAH) rebalancing.
• O(log N) queries; supports fast updates per moving-body (refit, not rebuild).
• Used by PhysX, Jolt, most modern engines.

class DbvhNode {
    AABB box;
    DbvhNode* parent;
    DbvhNode* children[2];   // internal
    int32_t bodyIndex;       // leaf: body id
    int32_t height;
};
// update: if (invalidated) reinsert; else refit bbox walking up the tree

2.3 Uniform Grid / Hash Grid

• Cell-keyed hash of body IDs in each cell; only adjacent cells checked.
• Best for uniform-distribution particles (SPH, crowds). Non-uniform scenes degenerate — use hierarchical grid.

3. Narrow-Phase Collision

Given a pair, compute contact geometry: point(s), normal, penetration depth.

3.1 The Collider Zoo

3.2 SAT (Separating Axis Theorem)

For convex vs convex: if a separating axis exists, no collision. Project both shapes onto each candidate axis (edge + face normals); if separated → done; else find minimum overlap axis → contact plane.

3.3 GJK + EPA

• GJK: fast convex-vs-convex boolean test using Minkowski difference and evolving simplex; O(log N) iterations.
• EPA: extends GJK to extract penetration depth + normal (expanding polytope).
• Best for arbitrary convex hulls; less numerically fragile than SAT for many axes.

3.4 Contact Manifold

Result of narrow-phase: a contact manifold:

struct ContactPoint { Vec3 localA, localB, normal, position; float penetration; };
struct Manifold {
    ContactPoint points[4];   // up to 4 (box-box), reduced for others
    Vec3 normal;              // shared frame
    Body* a, * b;
};

Simplify: reduce to 1-1 points for spheres, up to 4 for box/box. Give the solver a small set of contact points, NOT the full polygon intersection.

4. Collision Resolution — the Solver

4.1 Impulse-Based vs Penalty

4.2 The Sequential Impulse Solver (per contact)

For each contact: relative velocity at contact point

v_rel = (v_b + w_b x r_b) - (v_a + w_a x r_a)
vn = dot(v_rel, n)                // approach velocity along normal

// bias: penetration restitution (bounce)
e = restitution(a,b)
j = -(1 + e) * vn / (invMass_a + invMass_b + r cross terms)
v_a -= j * invMass_a * n ; v_b += j * invMass_b * n
// + angular impulse terms

Iterate all contacts several times (4-20 iterations) to converge a global solution — each contact affects others.

4.3 Friction

Apply two orthogonal tangent impulses:

t = v_rel - dot(v_rel,n)*n   (tangent direction)
jt = clamp(-v_rel_t * 1/(totalInvMass...), -mu*jn, mu*jn)

Use mu = Coulomb coefficient (static/dynamic blend).

4.4 Solver Order & Warm Starting

• Order contacts (e.g., by penetration) matters for determinism.
• Warm starting: store previous-frame impulses and apply them as initial guesses → stable stacks, big speedup.
• Contact persistence: keep manifolds between frames (don't rebuild) so stacks stay stable.

5. Integration

5.1 Choosing the integrator

Semi-implicit (symplectic):

v += a * dt
x += v * dt        // velocity updated BEFORE position

5.2 Rigid Body State

struct RigidBody {
    Vec3 position;
    Quat rotation;
    Vec3 linearVelocity, angularVelocity;
    float invMass, invI[3];      // inverse inertia tensor (diagonal local space)
    Vec3 force, torque;          // accumulated this tick
    bool sleeping;
};

Integrate rotation: apply angular velocity about local axes via quaternion derivative and normalize.

5.3 CCD (Continuous Collision Detection)

High-speed small objects tunnel through thin walls at large dt. Solutions:

• Swept line/ball vs world (PhysX CCD): expand collider along velocity, find first hit.
• Substeps: split dt into smaller steps for fast movers only.
• Cast-double-check: sweep a test then narrow-phase at midpoint.

Rule: CCD only for fast bodies vs static world (never CCD-vs-dynamic — expensive/nonsensical).

6. Joints & Constraints

• Distance joint: keeps 2 anchor points at fixed distance.
• Revolute joint (hinge): rotate about one axis.
• Prismatic joint: slide along one axis.
• Ball/socket: 3-MOI constraint point.
• Pulley / gear: derived from distance/velocity ratios.

Constraint form: C = 0 (position) → Cdot = J * v = 0 (velocity). Solve with the same sequential impulse pattern using Jacobian J.

// generalized velocity constraint solve
lambda = -(J v + bias) / (J M^-1 J^T)
v = v + M^-1 J^T lambda

7. Sleeping & Activation

• Bodies accumulate kinetic energy; if below threshold for N ticks, mark sleeping.
• Wake via: any new contact/impulse, joint activity, or external query of sizable magnitude.
• Deterministic order: wake in fixed order, not map order.

8. Character Controllers

PhysX/Jolt expose character controllers that are NOT rigid-body player physics:

• Capsule body, ray/sweep down to floor.
• Move with move_and_slide-style: attempt move; slide along obstacles (collide-and-slide).
• Step over small obstacles (max step height), avoid vertical tunneling on slopes.

Implement yourself (matching forced checkpoints), or use engine built-in (Godot CharacterBody3D, Unreal CharacterMovementComponent, Unity CharacterController).

9. Scene Queries (Raycasts / Overlaps)

// raycast against static + dynamic
Hit hit;
physics.rayCast(origin, dir, maxDist, out hit);
// overlap test
physics.overlapSphere(pos, radius, filter, out bodies[]);

Rules:

• Batch queries; per-frame thousands of raycasts in a loop → use QueryFilterCallback and broad-phase fast paths.
• Respect layer masks so objects don't self-collide.
• Cache query results when they don't need per-frame freshness.

10. Determinism & Networking

• Game server physics: authoritative; clients never simulate physics that matters for gameplay.
• Replays: log the input stream + fixed-step physics; deterministic engines can reproduce.
• Float determinism: same binary across all clients; avoid sqrt in solver hot paths where precision differs (or order-preserve).
• Lockstep (RTS): same inputs ticked identically; requires tick-aligned inputs with delay.

11. Tuning Cheat-Sheet

12. Engine Comparison (when to choose what)

13. References

• skills/game/game-engine/patterns/SKILL.md — engine patterns (game loop, fixed timestep)
• skills/game/game-engine/patterns/references/spatial-partitioning.md — broad-phase data structures
• skills/game/game-engine/ecs-pattern/SKILL.md — ECS storage relevant to physics hotspots
• skills/game/game-engine/ecs-pattern/references/ecs-netcode.md — deterministic sim across the wire
• skills/game/multiplayer-netcode/SKILL.md — server-authoritative simulation