Collidable Representation

• The intersection shapes/primitives for an object in the scene are stored via its own geometry and transformation (its 'collision representation')
• Some common intersectable shapes used include
• sphere
• capsule
• Axis Aligned Bounding Box (AABB)
• Discrete Oriented Polytopes (DOP)
• Polygon Soup (Poly Soup)
• These are also called 'collision volumes'
• Scene objects can contain multiple collision primitives representing them (e.g. a chair can be represented by a cube and a triangular prism)
• This can also be used for early-termination checking: a coarse check of a larger,wrapping primitive can be used as an easy first-check for intersection
• Usually the collision engine contains a 'collision/physics world' in which all collidable objects are represented
• benefit of cache coherency

Convex/Concave

• Usually convex shapes are used as a line can only enter and exit a convex shape once (concave adds more complexity)
• If a concave shape is shrink-wrapped, there will be air pockets; convex shapes will have NO air pockets when shrink wrapped

Poly Soup

• A collection of non-convex shapes, not necessarily enclosed or in a 'shape' form
• Used to represent complex static geometry like terrain and buildings
• Front-face is used to represent 'outside' (via stored winding order) and back-face represents 'inside'

Sphere intersection test

• s = c - p
• if |s| <= r, then p is inside the sphere
• c = sphere center, p = test point, r = sphere radius

Sphere intersecting other sphere test

• s = c1 - c2
• if |s|2 <= (r1 + r2)2, then the spheres intersect
• Difference squared is used to prevent square root calculation (|s|) for performance

Separating axis theorem

• If there is a line that can separate the projection of two convex shapes, then they do not intersect

GJK Algorithm

• Works for intersection testing between arbitrary polytopes (convex polygons in 2d, convex polyhedra in 3d)
• http://mollyrocket.com/849
• Uses the 'minkowski difference'
• Takes difference between all points within shape A and B
• If the set contains the origin point, then the shapes intersect
• since any point minus itself is 0 or the origin
• The minkowski difference between two convex shapes is also a convex shape
• Attempts to create a tetrahedron that surrounds the origin from the minkowski difference set
• If this tetrahdron cannot be created, then the shapes do not intersect

Paper Bullet Problem (Tunneling)

• If an object is moving faster than its size each, it can "skip over" an intersection
• Can be solved via
• Swept Shapes: create a new shape as the combination of the previous and next frame
• Only works well for moving in a straight line
• Continuous Collision Detection (CCD)
• Finds the earliest 'Time of Impact' (TOI) between two objects over a time interval

Performance Optimizations

• Collision detection can be expensive, esp. if you check every possible object with each other (O(n2))
• Can use techniques we've seen before in the render such as spacial partitioning
• Octtrees
• Binary Space Parititioning Trees (BSPs)
• etc.
• Other techniques:
• Temporal Coherency (Save calculated values that haven't changed from the previous frame)
• Broad Phase, Midphase, and Narrow Phase
• Broad Phase: Use AABBs to determine which collidables are potentially intersecting
• Midphase: Use coarse bounding volumes, such as spheres for another early termination step
• Narrow Phase: Use the individual primitives for the closest/specific possible collision detection
• Sweep and Prune algorithm
• sort the minimum and maximum dimensions of the AABBs along the x,y,z axes then check for intersection along the sorted list
• Can reduce O(n log n) to O(n)

Collision Queries

• Most physics engines support checking things such as "What is the earliest intersection if I move this character in this direction" or "find all enemy objects within this radius"

Raycasting

• Have paramtetric line segment p(t) = p0 + td, 0<=t<=1
• Check for intersection along this line
• Example ray casting usage: check for line of sight from an AI to the character by ray casting from the AI point of view to the main characters chest, and seeing if any intersections get in the way

Shape Casting

• Move a collision volume along a parametric line segment and check for intersections along the way
• This is more complex as you can have multiple collision points at the same time; usually returns an array of collisions
• Can be used to move 3d objects, and do things like automatically move up a curb or move left/right against a wall

Phantoms

• Collision objects which can intersect with regular objects, but regular objects cannot collide with them
• regular collisions with a phantom are ignored, but a phantom can still collide with regular objects

Collision Filtering

• Ability to group collision types together to only check between collisions of the same types
• Also ability to turn collision on/off for an object

Collision Materials

• Holds information such as friction, sound effects, particle effects, etc. for an object types such as skin, wood, metal, etc.

Ordinary Differential Equations (ODEs)

• Used in physics engines for calculations
• Utilize derivatives and linear integration
• An ODE has a single independant variable, such as time, as the input to a function, where the ODE is a function containing derivatives up to the n-1th order:
• For example, the classic force = mass * acceleration (F=ma) is more accurately represented as an ODE:
• r**(t) is acceleration (2nd derivative of position), r*(t) is velocity (first derivative of position)
• Used to approximate 'Taylor Series' (1+x+x2+x3 ... inf.)
• The error of the ODE is represented via big O notation as O(nx), where x is the number after the highest power in the equation
• The error is Δt2 because the highest power the approximation goes is 1; t2 and above do not effect the calculation

Common ODE methods

• Explicit Euler method: good for when velocity is nearly constant, otherwise inaccurate
• Verlet Integration
• Regular Verlet
• Velocity Verlet

Plastic vs. Elastic Collisions

• Elastic collisions have 'bounce' - opposite reactions to a force
• Plastic collisions have no bounce - no reactive force when a collision happens

Newton's law of restitutaion for instantaneous collisions with no friction

• Assumes the following when a collision happens:
• An instantaneous force; it is an 'impulse'
• There is no friction at the contact point
• There is a coefficient of restitutaion describing how much energy is lost.
• 1 = no energy lost; is an elastic collision
• 0 = all enery lost; is a plastic collision

Penalty Forces

• Manually introduce penalty forces to force interpenatrating objects apart
• Optimization compared to resolving all collisions via the physical simulation
• SONY PS3 Rubber Duck demo used this

Coming to Rest

• Forcing an object to stop as otherwise it's forces will never completely reach zero due to floating point error and non-exact algorithms
• Can be achieved by putting the physics on an object 'to sleep' if its close enough to zero force and meeting required criteria

Gravity

• Usually set globally in the physics engine

Torque and Force

• A torque is a force off-center from the center of mass on the object, causing a rotational force
• Force: applied to the center of mass, thus causing pure movement and not rotation

Impulse

• Instantanius change in velocity/momentum; duration too short for time step Δt to handle properly

The collision/physics step

1. Forces and Torques integrated by time step Δt
2. Collision Detection library checks for new contact
3. Resolve/Handle Collisions
4. Constraints Resolved by the Constraint Solver

Game-Driven vs. Physics-Driven

• Game Driven: the game manually controls object movement
• Physics Driven: The physics/collision system controls object movement
• A single object can switch between these two properties as required

Constraint

• A restriction on the movement of an object (reduces degrees of freedom from normal 6)
• For example, a hinge forcing a door to rotate only around a single axis

Powered Constraint

• Constraints where the translation/orientation can be controller externally, such as from the animation system
• For example, an elbow constraint on a human character can be forced to limit its rotation if the forearm further up the skeleton is hitting another object blocking its path

Rag Dolls

• Are implemented via rigid bodies connected via constraints
• Uses "Constraint chains" to improve the stability of the simulation

Fixed Bodies

• Collision only bodies that do not take part in dynamics simulation
• Provide an optimization/performance boost

Physics Simulation Update Frame Steps

• Update Game-Driven Rigid Bodies
• Update phantoms
• Update forces, apply impulses, adjust constraints
• Step the Simlulation
• Numerically integrate equations
• Collision Detection
• Apply Constraints
• Update physics-driven game objects
• Query Phantoms
• Perform collision cast queries

Bullet Ray Casting / Bullet Traces

• Ray casts are good if the bullet is moving in a straight line
• to account for gravity causing the bullet to drop or similar, the bullet can be modeled as a real rigid body with gravity applied
• Throwing bricks/bottles in The Last of Us does this

Collision vs. Visible Geometry

• To avoid case where it looks like the bullet hit the target but not hit was registered, due to the collision object not exactly matching the rendered object, can instead check against a rendered texture which a pixel represents which object it is, and check against that

Explosions

• Can have the explosion apply impulses to objects within its radius
• Can have animations for characters/objects to react to the explosion
• Can have animation system apply a force to foliage, such as swaying/bending plants, grass, trees, etc. due to the shock wave

Destructable Objects

• Can be made up of many smaller rigid body objects
• Possibility to have two versions of the object: a non-desructable, single-piece object and then the desructabl, many-piece version. Swap between the two as necessary

Camera collision

Some tricks/things to keep in mind when implementing a third person camera, to avoid the camera hitting things or going inside things

• Zooming in to avoid collisions
• Be careful with horizontal angle adjustment: too much can mess up the player's aim or focus, for example.
• Good to limit depending if the player is in combat or not
• be careful to limit vertical movement, e.g. so the camera doesn't end up pointing straight down at the character's head
• Avoid objects between the camera and the player by either swinging the camera around the object or making the object transparent

Current Areas of research for physics in games

• Deformable Bodies - DMM Engine
• Cloth - Havok Engine
• Hair - can be implemented via cloth with a transparent hair texture
• Water Surface simulations/buoyancy
• General Fluid Dynamics Simulations
• Physically Based Audio Synthesis
• GPGPU - Naughty Dog Cloth Engine runs on the PS4 GPU