Physics

• physics  animation!!
• animations must at least partially bypass physics in order to run seamlessly
• cut-scenes and state-transition animations are detached from physics

General purpose simulators

• offer the dream of environment where everything is physically simulated
• Pinball series (1980+), Incredible Machines (1993), Little Big Planet (2008), Angry Birds (2009)
• for instance, in GTA V, the player can interact with less than 0.1% objects!

• Revolute - a hinge or pin, where the bodies rotate
  • wheels, chains, rotating doors, catapults, levers
• Distance - a point on each body will be kept at a fixed distance apart
• Rope - restricts the maximum distance between two points
• Prismatic - body's motion is restricted to a single degree of freedom
  • elevators, sliding doors, pistons
• Weld - holds the bodies at the same orientation
• Cone-Twist - adds a cone and twist axis limits (6 degrees of freedom)
• Gear - controls two other joints so that the movement of one affects the other
• Motor - joint with torque or angular impulses

common issues of simple platformers

Steps

• positions for the next frame are determined
• a spatial data structure finds collision candidates
• collision candidates are filtered out into a set of real collision pairs
• collisions are resolved by the collision resolver (by applying impulses or penalty forces)
• constraints are satisfied by the constraint resolver

Collidable entities

• we need to provide a collision representation for each object
• simple shapes are preferred (a car might be modelled as a rectangle)
• more-complex shapes should be used only when the simple ones provide inadequate information to achieve the desired behavior
• if collidable entities don't overlap, no more testing is required
• if they do overlap, more refined testing is required

Naive solution

• each pair
•  -> 10 objects result in 100 checks, 100 in 10 000 etc.

Quad-tree

• efficiency usually , since the index takes about   comparisons to traverse
• good for point-like small objects
• worst-case:

Sweep and Prune

• sorts the starts (lower bound) and ends (upper bound) of bounding volumes of each solid along a number of axes

BSP

• binary space partitioning, good for complex static geometries (facilities, buildings)

Grid, Oct-tree, R-Tree, R+tree, R*-tree, X-tree, M-tree,...

Sphere-sphere

•  where  is a central point

Sphere-ray

• we just test the distance between a single sphere center and a ray

AABB-AABB

• test the borders

AABB-Ray

• intervals  and  mustn't overlap

Capsules

• capsule-capsule - calculate the distance between two line segments
• capsule-ray - find the distance between a ray and a line segment

OBB-OBB

• axis-separating theorem by Gottschalk

Triangles

• triangle-triangle - compute the plane equation and test each point if it lies on the same side
• triangle-ray - Möller's affine combination

Explosion

• adding energy to a system of rigid bodies

Fracturing

• breaking objects apart

Restitution

• amount of bounce an object has when it hits something

Impulsive response

• when two bodies collide, the bodies compress slightly and then rebound, changing their velocities, losing energy

Friction

• force that arises between two bodies that are in continuous contact, resisting their movement relative to one another
• removes energy from a system of rigid bodies
• static, dynamic, rolling

Collision resolving

• LCP - Linear Complementarity Problem
• very complex numeric algorithms
• resolvers: PGS, Dantzig solver, Lemke method, Jacobi method

Coming to rest

• all objects lose energy, leading to eventual rest
• not simple (floating-point error, inaccuracies, numerical instability)
  • sleep criteria
    • momentum or energy are below a threshold
  • simulation islands
    • grouping objects that either are interacting or have a potential to interact

• 2D rigid body and fluid simulation library, Box2D extension
• Android, iOS, Windows, OS X
• phase 1: collision detection
  • find pairs of particles closer than the particle diameter
• phase 2: apply pressure
  • sum up the weight of contacts for each particle
  • calculate the pressure and apply repulsive forces
• phase 3: apply other forces
  • viscous force, spring force, elastic force,...
• phase 4: update positions
• phase 5: handle interaction with Box2D

• WIP game that uses a falling sand-style simulation
• each pixel is considered a particle having their own attributes
• flood-algorithm for ignition effects
• each screen is divided into 64x64 chunks processed in 4 stages from the bottom up