Unit
- Newton's First Law: Objects maintain their state of motion unless acted upon by a force. In simulation, this means velocity remains constant when no net force is applied, requiring careful resetting of force accumulators each frame.
- Newton's Second Law: Force equals mass times acceleration (F=ma). Simulators compute acceleration from net forces and mass, then integrate to update velocity and position each timestep using numerical methods.
- Newton's Third Law: For every action there is an equal and opposite reaction. When two bodies collide, equal and opposite impulses must be applied to both to conserve momentum accurately.
- Euler Integration: The simplest method updates position as position += velocity * dt. While fast, it is unstable for large timesteps and can cause energy gain in undamped systems.
- Force Accumulation: Each frame, forces such as gravity, friction, and user input are summed into a net force vector. Acceleration is then derived as net force divided by mass before integration.
- Mass as Parameter: Heavier objects require more force to accelerate and impart larger impulses during collisions. Tuning mass directly affects the perceived weight and realism of objects.
- Damping Effects: Linear damping reduces velocity proportionally to simulate air resistance, while angular damping affects rotational motion. Damping prevents infinite oscillations in constrained systems.
- Coordinate Systems: Using a consistent unit system (meters, kilograms, seconds) avoids scaling errors. Many game engines use arbitrary units but maintain internal consistency through conversion factors.
- Collision Detection: Broadphase algorithms like sweep and prune quickly rule out noncolliding pairs, while narrowphase methods such as separating axis theorem compute exact contact points and normals. Efficient collision detection is critical for maintaining performance in scenes with many objects.
- Constraint Solvers: During stacking and jointed assemblies, constraints such as nonpenetration and fixed distance are enforced via iterative solvers. Sequential impulse and projected Gauss-Seidel methods converge toward a valid configuration each frame, balancing stability and speed.
- Timestep Selection: A fixed timestep decouples simulation from rendering, preventing instability from variable frame rates. When physics must match real time, substepping divides the frame into smaller intervals, ensuring that fast moving objects do not tunnel through thin geometry.
Design Principle
Principle: Simulation Accuracy vs. Performance Trade-off
Definition: Higher accuracy requires smaller timesteps or more complex solvers, increasing computational cost. Real-time applications often sacrifice accuracy for speed by using fixed timesteps and simpler integration.
Application: In a game, use a fixed timestep of 1/60 second and semi-implicit Euler for rigid bodies. For a scientific simulation, employ Runge-Kutta 4th order with adaptive timestep to maintain precision where needed.
Quick Revision
- Newton's three laws are the basis for motion, force, and collision response.
- Euler integration is simple but can be unstable; more advanced methods improve accuracy.
- Force accumulation and mass are key parameters affecting simulation behavior.
- Damping prevents energy build-up and mimics real-world dissipation.
- Consistent units and coordinate systems are essential for reliable results.