CIS 350 – I

Game Programming

Instructor: Rolf Lakaemper

Parts of these slides are based onwww2.informatik.uni-wuerzburg.de/ mitarbeiter/holger/lehre/osss02/schmidt/vortrag.pdf

by Jakob Schmidt

Introduction To

Collision Detection

What ?

The problem:

The search for intersecting planes of different 3D models in a scene.

Collision Detection is an important problem in fields like computer

animation, virtual reality and game programming.

Intro

The problem can be defined as

if, where and when

two objects intersect.

Intro

This introduction will deal with the basic problem:

IF

two (stationary) objects intersect.

Intro

The simple solution:

Pairwise collision check of all polygons the objects are made

of.

Intro

Problem: • complexity O(n²)

• not acceptable for reasonable number n of polygons

• not applicable for realtime application

Bounding Volumes

Part 1: Bounding Volumes

Reduce complexity of collision computation by substitution of the (complex) original object

with a simpler object containing the original one.

Bounding Volumes

The original objects can only intersect if the simpler ones do. Or better: if the simpler

objects do NOT intersect, the original objects won’t either.

Bounding Volumes

How to choose BVs ?

• Object approximation behavior (‘Fill efficiency’)• Computational simplicity• Behavior on (non linear !) transformation (incl. deformation) • Memory efficiency

Bounding Volumes

Different BVs used in game programming:

• Axes Aligned Bounding Boxes (AABB)• Oriented Bounding Boxes (OBB)

•Spheres• k-Discrete Oriented Polytopes (k DOP)

AABB OBBSphere k-DOP

Bounding Volumes

Axes Aligned Bounding Box (AABB)

• Align axes to the coordinate system

• Simple to create

• Computationally efficient

• Unsatisfying fill efficiency

• Not invariant to basic transformations, e.g. rotation

Bounding Volumes

Axes Aligned Bounding Box (AABB)

Collision test: project BBs onto coordinate axes. If they overlap on each axis, the objects collide.

Bounding Volumes

Oriented Bounding Box (OBB)

Align box to object such that it fits optimally in terms of fill efficiency

Computationally expensiveInvariant to rotationComplex intersection check

Bounding Volumes

The overlap test is based on the

Separating Axes Theorem(S. Gottschalk. Separating axis theorem. Technical Report TR96-024,Department

of Computer Science, UNC Chapel Hill, 1996)

Two convex polytopes are disjoint iff there exists a separating axis orthogonal to a face of either polytope or orthogonal to an edge

from each polytope.

Bounding Volumes

Each box has 3 unique face orientations, and 3 unique edge directions. This leads to 15

potential separating axes to test (3 faces from one box, 3 faces from the other box, and 9

pairwise combinations of edges).

Bounding Volumes

Sphere

• Relatively complex to compute• Bad fill efficiency• Simple overlap test• invariant to rotation

Bounding Volumes

K-DOP

•Easy to compute•Good fill efficiency•Simple overlap test

•Not invariant to rotation

Bounding Volumes

k-DOP is considered to be a trade off between AABBs and OBBs.

Its collision check is a general version of the AABB collision check, having k/2

directions

Bounding Volumes

k-DOPs are used e.g. in the game

‘Cell Damage’ (XBOX, Pseudo

Interactive, 2002)

How to Compute and Store k-DOPs:

k-directions

k-directions Bi define halfplanes (they are the normals to the halfplanes) , the

intersection of these halfplanes defines the k-DOP bounding volume.

Halfplane Hi = {x | Bi x – di <= 0}

Bounding Volumes

Bounding Volumes

3D Example: UNREAL-Engine

Bounding Volumes

2D Example for halfplanes defining a k-DOP

Normal vector Bi

Bounding Volumes

Again: halfplane Hi = {x | Bi x – di <= 0}

•If the directions Bi are predefined, only the distances di must be stored to specify the halfplane Hi . This is one scalar value per direction.

•If the directions are NOT predefined, Bi

and di must be stored (3D case: 4 values)

Bounding Volumes

How to compute di :

Hessian Normal Form ( Bx – d = 0 with d = Bp) with

unit vectorof a plane automatically gives

distance d if a single point p on the plane is known.

Bounding Volumes

Compute distances of all vertices to plane Bx = 0, i.e. multiply (dot product) each

vertex with the unit normal vector B

Bx = 0

Unit vector B

Bounding Volumes

di is the minimum distance of the object to the plane Bx = 0

Bx = 0di

Bounding Volumes

Collision

Given: k non kollinear directions Bi andV = set of vertices of object.

Compute di = min{Bi v| v in V } and Di = max{Bi v| v in V}. di and Di define an interval on the axis given by Bi .

This is the interval needed for the collision detection !

Bounding Volumes

Collision

B1

Plane defined by B1 Interval [d1, D1] defined by B1

d1

D1

Bounding Volumes

Part 2: Collision on different scales:

Hierarchies

Hierarchies

Idea:

To achieve higher exactness in collision detection, build a

multiscale BV representation of the object

Hierarchies

Hierarchies

Use the hierarchy from coarse to fine resolution to exclude non intersecting

objects

Hierarchies

The hierarchy is stored in a tree, named by the underlying BV scheme:

AABB – treeOBB – tree

Sphere – treekDOP – tree

Hierarchies

Sphere Trees are used for example in

“Gran Tourismo”

Hierarchies

Simple example:

• Binary tree• Each node contains all primitives of its subtree• Leaves contain single primitive

Hierarchies

Hierarchies

Hierarchies

Recursive Collision Detection

)

Returns TRUE if BBs overlap.How could this be improvedto give a precise overlap test ?

Hierarchies

Hierarchies

How to create a hierarchy tree

Top down:

• Use single BV covering whole object• Split BV

• Continue recursively until each BV contains a single primitive

Hierarchies

Bottom up:

• Start with BV for each primitive• Merge

Hierarchies

Example for top down using OBBs :

Hierarchies

Comparison AABB / OBB

Multiple Objects

Part 3: Collision between

Multiple Objects

Multiple Objects

Virtual environment usually consists of more than 2 objects. Pairwise detailed collision

between all objects is too slow. Solution again:

1. Exclude non colliding objects2. Check collision between remaining

objects

Multiple Objects

Methods to exclude non colliding objects:

1.Grid MethodOr

2. Sort and Sweep (AABB)

Multiple Objects

Grid Method:

Create 3d grid volume overlay

Only check collision between objects sharing at least one cell

Multiple Objects

2D example

Multiple Objects

Sort and Sweep

• Create single AABB for each object

• Project BVs onto coordinate axes

• Create a sorted list of start and endpoints for each coordinate axis, hence store the

intervals created by each object

(Cont’d)

Multiple Objects

• Traverse each list• If startpoint of object i is hit, insert i into

‘active list’• If endpoint of object i is hit, remove i from

‘active list’• If 2 objects i1,i2 are active at the same time

they overlap in the dimension processed• Objects overlapping in all single dimensions

overlap in world

Multiple Objects

S3S1E3S2E1E2

1

2

3

s3

e3

e1

s1

s2

e2

e2e1s2 e3s1 s3

S1S2E1S3E2E3

X Y

OVERLAP 1,2

Multiple Objects

Note: sort and sweep for a single step is relatively expensive.

Since not all objects are transformed for the next frame, the list is not created newly for each frame, but updated.