Physics Engine

This is the multi-stage process for identifying and resolving intersections between objects. It is typically divided into three phases:

• Broadphase: Uses spatial data structures like Bounding Volume Hierarchies (BVHs) or spatial hashing to quickly cull pairs of objects that are too far apart to possibly collide.
• Narrowphase: Performs precise geometric tests on the remaining potential pairs using algorithms like GJK (Gilbert–Johnson–Keerthi) or SAT (Separating Axis Theorem) to compute exact contact points, normals, and penetration depths.
• Contact Generation: Creates persistent contact manifolds from the narrowphase results, which are passed to the constraint solver.

The constraint solver is the subsystem responsible for resolving forces and impulses to satisfy physical rules and connections. It ensures simulated bodies behave correctly under constraints such as:

• Contact Constraints: Prevent interpenetration and model friction (e.g., a box resting on a floor).
• Joint Constraints: Connect bodies, defining their relative motion (e.g., hinges, sliders, or ball joints in a robotic arm).
• Motor Constraints: Apply specific forces or target positions/velocities. Popular iterative solvers include the Projected Gauss-Seidel (PGS) solver, which handles these problems as a Linear Complementarity Problem (LCP).

Time integration is the numerical method that advances the simulation state forward in time. It solves the differential equations of motion (Newton's second law, F=ma) to update object positions and velocities. Common integrators include:

• Explicit Euler: Simple but can be unstable with large time steps.
• Semi-Implicit Euler (Symplectic Euler): More stable and commonly used in real-time engines.
• Runge-Kutta Methods (RK4): Higher accuracy for demanding simulations but more computationally expensive. The choice of integrator directly affects simulation stability, accuracy, and performance.

This component manages the fundamental state and motion of non-deformable objects. For each rigid body, it tracks:

• Position & Orientation: The body's location and rotation in world space.
• Linear & Angular Velocity: How the body is moving and spinning.
• Mass & Inertia Tensor: The body's resistance to linear and rotational acceleration.
• Center of Mass (COM): The point where mass is concentrated for force application. The dynamics core applies forces and torques, then uses the time integrator to compute new velocities and positions, forming the backbone of most real-time simulations.

These are specialized data organizations critical for performance, primarily accelerating the broadphase of collision detection. They spatially partition the simulation world to minimize the number of expensive pairwise checks. Common structures include:

• Bounding Volume Hierarchy (BVH): A tree of nested bounding volumes (spheres, AABBs) that allows for hierarchical culling.
• Spatial Hash Grid / Uniform Grid: Divides space into fixed-size cells; objects are checked against others in the same or adjacent cells.
• Octree/Quadtree: Recursively subdivides space into eight (3D) or four (2D) children, adapting to non-uniform object density.

Beyond basic rigid bodies, advanced engines incorporate methods for simulating continuous media and deformable objects. These are often separate, modular systems:

• Soft Body Dynamics: Simulates deformable materials like cloth, rubber, or flesh using mass-spring systems or the Finite Element Method (FEM).
• Fluid & Particle Systems: Models liquids, gases, and granular materials. Smoothed-Particle Hydrodynamics (SPH) is a common mesh-free Lagrangian method for fluids.
• Position-Based Dynamics (PBD): A constraint-based approach popular for real-time simulation of cloth, ropes, and fluids due to its stability and controllability.

The core simulation of non-deformable objects. It calculates the translational and rotational motion of solid bodies based on applied forces, torques, and mass properties.

• Governed by Newton-Euler equations.
• Essential for simulating robots, vehicles, and most mechanical objects.
• Relies on accurate mass, center of mass, and inertia tensor calculations.

The computational process of identifying when simulated objects intersect. It is typically a two-stage pipeline:

• Broadphase: Uses spatial structures like a Bounding Volume Hierarchy (BVH) to quickly find potentially colliding pairs.
• Narrowphase: Precisely tests pairs using algorithms like Gilbert–Johnson–Keerthi (GJK) to compute contact points, normals, and penetration depth.
• Continuous Collision Detection (CCD) prevents fast-moving objects from tunneling through thin geometry.

An algorithmic component that resolves forces to satisfy physical limits between bodies. It enforces rules such as:

• Contact constraints (non-penetration).
• Joint constraints (e.g., hinges, sliders, ball joints).
• Friction models. Common solvers formulate this as a Linear Complementarity Problem (LCP) and use iterative methods like the Projected Gauss-Seidel (PGS) solver to find a feasible solution.

An O(n) algorithm for efficiently computing the forward dynamics of kinematic chains, such as robotic arms. Part of Featherstone's algorithm family.

• Calculates how a multi-joint system will move given applied joint torques.
• Significantly faster than naive O(n³) methods for complex robots.
• Critical for real-time simulation of articulated systems in robotics.

The numerical method that advances the simulation state forward in time. It solves the differential equations of motion from one time step to the next.

• Common methods: Semi-implicit Euler, Runge-Kutta 4 (RK4).
• Choice affects simulation stability, accuracy, and performance.
• A deterministic simulation requires a fixed, reproducible integration scheme.

Techniques for simulating deformable materials and fluids.

• Finite Element Method (FEM): High-accuracy simulation of stress and strain in deformable solids.
• Position-Based Dynamics (PBD): Fast, stable method for cloth, rubber, and character flesh.
• Smoothed-Particle Hydrodynamics (SPH) & Material Point Method (MPM): For simulating fluids, sand, snow, and other complex materials with large deformations.