1. Ntreev Soft (in-house training) 1 Collision Detection In 3D Environments
Collision Detection
In 3D Environments
이웅수
peras@ntreev.com
oiotoxt@gmail.com
Ntreev Soft Co. Ltd.
v0.2

2. Ntreev Soft (in-house training) 2 Collision Detection In 3D Environments
Notice
• 본 문서를 외부에 유포하지 말아주세요.
• 본 문서는 2004년 7월 23일 현재 v0.1 입니다.
• 본 문서는 2004년 7월 29일 현재 v0.2 입니다.
• 본 문서에 포함된 도안들 중 많은 부분은 여러 다른 저자(Bibliography & People)들의 paper를 참고하거나 그것으로부터 발췌한 것입니다. 그
들의 관용을 바랍니다.
Beauty is our business.
– Edsger W. Dijkstra

3. Ntreev Soft (in-house training) 3 Collision Detection In 3D Environments
Contents
• Concepts:
– Geometry
– Terminologies
– (Time, Response, Performance, Robustness)
• Spatial Data Structures
– Nonconvex Polyhedra (Convex Decomposition)
– Space Partitioning (Voxel Grid, Quadtree, Octree, k-d Tree, BSP)
– Model Partitioning (BVH)
– Broad Phase Collision Detection (Sweep and Prune)
• Scene Management
– Portals, PVS, Scene Graph, LOD
• Basic Primitives
– Sphere: Sphere – Sphere, Ray – Sphere, Line segment – Sphere
– AABB: Ray – Box, Sphere – Box
– SAT (Separating Axis Test) : Line segment – Box, Triangle – Box, Box – Box
– Polygons: Ray – Triangle, Line segment – Triangle, Ray – Polygon, Triangle – Triangle, Polygon – Polygon,
Triangle – Sphere, Polygon - Volume
• Convex Objects
– Finding a Common Point
– Finding a Separating Plane
– GJK (Gilbert-Johnson-Keerthi) Distance Algorithm & Enhanced GJK
– Lin-Canny Closest Feature Algorithm
– Voronoi Clip Algorithm (V-Clip)

4. Ntreev Soft (in-house training) 4 Collision Detection In 3D Environments
Geometry
• Classical Geometry
– Euclidean Geometry
– Non-Euclidean Geometry
– Affine Geometry
• Modern Geometry
– Topology
– Differential Geometry
– Complex Projective Geometry
– Finite Projective Geometry
• Reference: [KK88]

5. Ntreev Soft (in-house training) 5 Collision Detection In 3D Environments
Euclidean Geometry
• 유클레이데스 <원론 (Stoicheia)>
• 유클레이데스기하학에서의 변환 f
– 1대 1의 점대응
– 선분(또는 직선)은 선분(또는 직선)에 대응
– 길이를 바꾸지 않는 대응
• 즉, f 변환으로 바뀌지 않는 성질이 곧 유클레이데스기하학의 ‘기하학적 성질’
이다.

6. Ntreev Soft (in-house training) 6 Collision Detection In 3D Environments
Affine Geometry
• F. Klein이 <Erlangen Programm>에서 아핀변환에 의해 변하지 않는 기하학
의 이론을 ‘아핀기하학’이라고 명명.
• 아핀변환 f
– 1대 1의 점대응
– 직선 → 직선
• 즉, 아핀변환 f 는 평행투영에 의한 변환이며 직선, 교점, 평행관계 등은 그대로
유지되나, 길이, 각의 크기 등의 개념은 무의미 (모든 3각형은 같은 도형)

7. Ntreev Soft (in-house training) 7 Collision Detection In 3D Environments
Non-Euclidean Geometry
• 19세기에 이르러 확립
• 유클레이데스의 <원론> ‘제5공준(=평행선 공준)’
– 『직선 a 위에 없는 한 점 P를 지나 a와 만나지 않는 직선은 오직 한 개만
그을 수 있다 』
– 『P를 지나 a와 만나지 않는 직선을 무수히 그을 수 있다 』고 가정하여
도 전혀 모순이 일어나지 않는 기하학이 성립.
• 평행선의 개수를 이용한 기하학 분류
– 직선 바깥의 한 점을 지나는 평행선이
오직 한 개 존재 : 유클레이데스기하학
무수히 존재 : 로바체프스키-보야이의 비유클레이데스기하학
하나도 존재하지 않음 : 리만의 비유클레이데스기하학

8. Ntreev Soft (in-house training) 8 Collision Detection In 3D Environments
Projective Geometry (1)
• 대응(=寫像)에 의해서 도형의 성질을 다루는 기하학

9. Ntreev Soft (in-house training) 9 Collision Detection In 3D Environments
Projective Geometry (2)
• 『Projective geometry is all geometry 』
사영기하학
비유클레이데스기하학 아핀기하학
유클레이데스기하학쌍곡선기하학 타원기하학

10. Ntreev Soft (in-house training) 10 Collision Detection In 3D Environments
Topology
• ‘위’(=위치, Lage)와 ‘상’(=형상, Gestalt)에 관한 (정성적) 수학(기하학)
• 동상
– C ≡ I ≡ J ≡ L ≡ M ≡ N ≡ S ≡ U ≡ V ≡ W ≡ Z,
– E ≡ F ≡ G ≡ T ≡ Y, H ≡ K, A ≡ R, D ≡ O, P, B, Q, X
• 동형
• 위상공간(=Topological space)
– 위상적 구조가 주어진 (=집합의 원소 사이에 ‘근방’이 정의되어진) 집합.
• 위상 공간 ⊃ 거리 공간 ⊃ 유클레이데스 공간
• V – E + F = 2 (다면체), V – E + F = 0 (윤환면)

11. Ntreev Soft (in-house training) 11 Collision Detection In 3D Environments
Vector Space
• Let V be a set whose elements are called vectors.
• Assume that equality of vectors is defined.
• Suppose that there are two operations, vector addition (+) and scalar multiplication (•), such that
the following properties hold.
• 10 Axioms ( a, b, c ∈ R and v, u, w ∈ V )
1. ( closure under + ) u + v ∈ V
2. ( + is associative ) u + (v + w) = (u + v) + w
3. ( + is commutative ) u + v = v + u
4. ( additive identity ) There is a vector 0 ∈ V such that v + 0 = v for any v ∈ V
5. ( additive inverses ) for each v ∈ V there is a vector –v ∈ V such that v + (-v) = 0
6. ( closure under • ) c • v ∈ V, written simply as cv
7. ( • is distributive over real addition ) (a + b)v = av + bv
8. ( • is distributive over vector addition ) a(u + v) = au + av
9. ( • is associative ) a(bv) = (ab)v
10. ( multiplicative identity ) The number 1 ∈ R has the property 1•v = v
• The triple ( V, +, • ) is called a vector space over the real numbers.

12. Ntreev Soft (in-house training) 12 Collision Detection In 3D Environments
Computational Geometry
• Application Domain
– Computer Graphics
– Robotics
– GIS (Geographic Information Systems)
– CAD (Computer Aided Design) / CAM (Computer Aided Manufacturing)
– Molecular modeling
– Pattern recognition
• Reference: [BKOS00]

13. Ntreev Soft (in-house training) 13 Collision Detection In 3D Environments
Terminologies (1)
• Convex, Concave
• BV (Bounding Volume)
• Manifold & Non-Manifold
– Roughly, every edge has two faces joining it.
• k-DOP ⊃ OBB ⊃ AABB
• Half Space
• SSV (Swept Sphere Volumes, S-topes)
• Features
Convex Concave Manifold Non-Manifold

14. Ntreev Soft (in-house training) 14 Collision Detection In 3D Environments
Terminologies (2)
• Simplices
– Points, Line segments, Triangles, Tetrahedra
• Polygons
• Polyhedra
• Convex quadrics
– Spheres, Cones, Capped cones, Cylinders, Capped cylinder, …
• Convex objects
– Simplices, Convex polygons, Convex polyhedra, Convex quadrics, …
• Polytopes
– Simplices ∪ Convex polygons ∪ Convex polyhedra
• Polygon Soups

15. Ntreev Soft (in-house training) 15 Collision Detection In 3D Environments
Terminologies (3)
• CD (Collision Detection), PD (Penetration Depth)
• Proximity Query
– A procedure to compute the geometric contact (and distance) between
objects.
• Convex Hulls
• Minkowski Addition / Difference
• CSO (Configuration Space Obstacle)
• Voronoi Diagrams, Voronoi Sites, Voronoi Regions
• Frame Coherence, Geometric Coherence
• Interval Halving
• Ω-notation (asymptotic lower bound) & O-notation (asymptotic upper
bound)

16. Ntreev Soft (in-house training) 16 Collision Detection In 3D Environments
AABB and OBB

17. Ntreev Soft (in-house training) 17 Collision Detection In 3D Environments
k-DOP
• k-DOP (Discrete Orientation Polytope)
• Set of k / 2 infinite slabs.
• There is an even faster test if all objects have the same k and same
slab normals.
• OBB and AABB are 6-DOP

18. Ntreev Soft (in-house training) 18 Collision Detection In 3D Environments
SSV
PSS LSS RSS

19. Ntreev Soft (in-house training) 19 Collision Detection In 3D Environments
Boundary Representations
• Winged Edge Structure – by Baumgart
• DCEL (Doubly Connected Edge List) – by Muller and Preparata
• Hierarchical Representation – by Dobkin and Kirkpatrick

20. Ntreev Soft (in-house training) 20 Collision Detection In 3D Environments
Convex Hulls (1)
• The smallest convex shape that contains a given collection of convex
shapes[vdB04].
• The convex hull of a set of points S in n dimensions is the intersection
of all convex sets containing S. For N points P1, ..., PN, the convex hull
C is then given by the expression[Mw].

21. Ntreev Soft (in-house training) 21 Collision Detection In 3D Environments
Convex Hulls (2)
• The incremental construction algorithm
– Let PS be a set of n points in 3-space. CH(PS) is the convex hull of PS.
– Choose four points in PS that do not lie in a common plane, so that their convex
hull is a tetrahedron.
– Let PSr := { p1, …, pr }.
– If pr lies inside CH(PSr-1), or on its boundary, then CH(PSr) = CH(PSr-1)
– Now suppose that pr lies outside CH(PSr-1). Imagine that you are standing at pr,
and that you are looking at CH(PSr-1). You will be able to see some facets of
CH(PSr-1). The visible facets form a connected region on the surface of CH(PSr-1),
called the visible region of pr on CH(PSr-1), which is enclosed by a closed curve
consisting of edges of CH(PSr-1). We call this curve the horizon of pr on CH(PSr-
1). The horizon of pr forms the border between the part of the boundary that can
be kept and the part of the boundary that must be replaced.
– Suppose that we knew all facets of CH(PSr-1) visible from pr. Then it would be
easy to remove all the information stored for these facets from the DCEL,
compute the new facets connecting pr to the horizon, and store the information
for the new facets in the DCEL.
• Reference: [BKOS00]

22. Ntreev Soft (in-house training) 22 Collision Detection In 3D Environments
Minkowski Addition / Difference
• The convex shape that is the result of “adding” two convex shapes,
that is, sweeping one shape along the point set of another.
• Minkowski Sum (The sum of sets A and B in a vector space)
– A + B = { a + b : a ∈ A, b ∈ B}
• The negation operation
– -B = { -b : b ∈ B }

23. Ntreev Soft (in-house training) 23 Collision Detection In 3D Environments
CSO
• Configuration Space Obstacle
– A + (-B) = A – B
• The CSO contains all the information needed to answer a number of
queries on pairs of objects in the context of collision detection.
– A ∩ B ≠ Ø ≡ 0 ∈ A – B
– The distance between two objects A and B, denoted by d(A,B), is
defined as
d(A,B) = min{ ║a - b║ : a ∈ A, b ∈ B }
d(A,B) = min{ ║x║ : x ∈ A - B }
– The penetration depth of a pair of intersecting objects, denoted by
p(A,B), is the length of the shortest vector over which one of the objects
needs to be translated in order to bring the pair in touching contact.
p(A,B) = inf { ║x║ : x ∈ A – B } (infimum: greatest lower bound)

24. Ntreev Soft (in-house training) 24 Collision Detection In 3D Environments
Voronoi Diagram
• The partitioning of a plane with n points into n convex polygons such
that each polygon contains exactly one point and every point in a
given polygon is closer to its central point than to any other[Mw].
Voronoi Site
Voronoi Region
Voronoi Diagram

25. Ntreev Soft (in-house training) 25 Collision Detection In 3D Environments
Coherences
• Frame Coherence
– The measure of reusability of computations from earlier frames.
• Geometric Coherence
– “Geometric coherence has to do with the degree to which a location in
space can be associated with one designated object, rather than the
degree to which objects cohere.”
• Reference: [vdB04]

26. Ntreev Soft (in-house training) 26 Collision Detection In 3D Environments
Asymptotic Complexity
log n
n
n log n (fastest sorting)n22n
n

27. Ntreev Soft (in-house training) 27 Collision Detection In 3D Environments
Spatial Data Structures
• Convex Decomposition (Nonconvex Polyhedra)
– Joe’s Algorithm
• Space Partitioning (SP)
– Voxel Grid
– Quadtree
– Octree
– k-d Tree
– BSP (Axis Aligned, Polygon Aligned)
• Model Partitioning
– Bounding Volume Hierarchies (BVH)
• Broad Phase Collision Detection
– Sweep and Prune

28. Ntreev Soft (in-house training) 28 Collision Detection In 3D Environments
Space Partitioning
• Algorithms:
– Voxel Grid
– Quadtree
– Octree
– k-d Tree
– BSP (Axis Aligned, Polygon Aligned)
Voxel Grid Quadtree & Octree k-d Tree BSP

29. Ntreev Soft (in-house training) 29 Collision Detection In 3D Environments
Octree

30. Ntreev Soft (in-house training) 30 Collision Detection In 3D Environments
k-d Tree
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
8 9 10 11 12 13 14 15
4 5 6 7
2 3
1

31. Ntreev Soft (in-house training) 31 Collision Detection In 3D Environments
Bounding Volume Hierarchies (1)
• Bounding Volumes
– OBB, AABB, k-DOP, Sphere, SSV, Convex hull, Ellipsoid, …
AABB OBBSphere Convex Hull6-DOP
Cost of (Overlap Tests + BV Update), Complexity, Tightness of Fit
Decreasing Increasing

32. Ntreev Soft (in-house training) 32 Collision Detection In 3D Environments
Bounding Volume Hierarchies (2)
• A binary bounding volume hierarchy

33. Ntreev Soft (in-house training) 33 Collision Detection In 3D Environments
Bounding Volume Hierarchies (3)
• Building an OBB Tree
– Recursive top-down construction (Partition and Refit)
A
B
C
D
E
A
B C
D E

34. Ntreev Soft (in-house training) 34 Collision Detection In 3D Environments
Bounding Volume Hierarchies (4)
• Fitting OBBs

35. Ntreev Soft (in-house training) 35 Collision Detection In 3D Environments
Bounding Volume Hierarchies (5)
• Basic Hierarchy Traversal

36. Ntreev Soft (in-house training) 36 Collision Detection In 3D Environments
Bounding Volume Hierarchies (6)
• Dual Hierarchy Traversal

37. Ntreev Soft (in-house training) 37 Collision Detection In 3D Environments
Bounding Volume Hierarchies (7)
• AABB Trees vs OBB Trees (Approximation of a Torus)

38. Ntreev Soft (in-house training) 38 Collision Detection In 3D Environments
Bounding Volume Hierarchies (8)
• AABB Trees and Deformable Models

39. Ntreev Soft (in-house training) 39 Collision Detection In 3D Environments
SP vs BVH
BVH: Object centric, Spatial redundancy
SP: Space centric, Object redundancy

40. Ntreev Soft (in-house training) 40 Collision Detection In 3D Environments
Sweep and Prune (1)
• Compute the AABB (fixed vs dynamic) for each object.
• Dimension reduction by projecting AABB onto each x, y, z – axis.
• Sort the endpoints and find overlapping intervals.
• Possible collision
– Only if projected intervals overlap in all 3 dimensions.
• Sorting Methods
– Initial sort
[Quick sort] runs in O(n log n) just as in any ordinary situation.
– Updating
[Insertion sort] runs in O(n + k) due to coherence.
– We sort an almost sorted list from last stimulation step.
– In fact, we look for “swap” of positions in all 3 dimension.
• Reference: [Bar92][vdB04]

41. Ntreev Soft (in-house training) 41 Collision Detection In 3D Environments
Sweep and Prune (2)

42. Ntreev Soft (in-house training) 42 Collision Detection In 3D Environments
Convex Objects
• Proximity Queries
• Overview of Algorithms for Polytopes
– Finding a Common Point
– Finding a Separating Plane
– Distance and Penetration Depth Computation
• Simplex-based Algorithm
– GJK Distance Algorithm & Enhanced GJK
Johnson’s Distance Algorithm
Support Mappings
Penetration Depth
• Feature-based Algorithm
– Lin-Canny Closest Feature Algorithm
– Voronoi Clip Algorithm

43. Ntreev Soft (in-house training) 43 Collision Detection In 3D Environments
Finding a Separating Plane (1)
• Separating Axis Theorem
– Two polytopes A and B are disjoint iff there exists a separating axis
which is: perpendicular to a face from either or perpendicular to an edge
from each.
• Implementations
– Given two generic polytopes, the number of candidate axes to test
F1 + F2 + ( E1 * E2 )
– OBBs have only E = 3 distinct edge directions, and only F = 3 distinct
face normals. OBBs need at most 15 axis tests. (2D: 4 axes to test)
• Used in:
– Line segment ↔ Box
– Triangle ↔ Box
– OBB ↔ OBB
– k-DOP ↔ k-DOP

44. Ntreev Soft (in-house training) 44 Collision Detection In 3D Environments
Finding a Separating Plane (2)
Polytope Polytope Number of axes to test
Line segment Triangle 0 + 1 + ( 1 * 3 ) = 4
Line segment Box 0 + 3 + ( 1 * 3 ) = 6
Triangle Triangle 1 + 1 + ( 3 * 3 ) = 11
Triangle Box 1 + 3 + ( 3 * 3 ) = 13
Box Box 3 + 3 + ( 3 * 3 ) = 15
• The SAT works well for simple polytopes, such as line
segments (rays), triangles, and boxes.

45. Ntreev Soft (in-house training) 45 Collision Detection In 3D Environments
GJK Distance Algorithm & Enhanced GJK
• Johnson’s Distance Algorithm
• Support Mappings
• Penetration Depth

46. Ntreev Soft (in-house training) 46 Collision Detection In 3D Environments
Voronoi Clip Algorithm (1)
• Voronoi Region
• Basic Algorithm
– Given one feature from each polyhedron, find the nearest points of the
two features.
– If each nearest point is in the Voronoi region of the other feature, closest
features have been found.
– Else, walk to one or both of their neighbors or some other feature.
• Running Time Analysis
– Distance strictly decreases with each change of feature pair, and no pair
of features can be selected twice.
– Convergence to closest pair typically much better for dynamic
environments.
O(1) achievable in simulations with coherence.
Closer to sub-linear time performance even without coherence.

47. Ntreev Soft (in-house training) 47 Collision Detection In 3D Environments
Voronoi Clip Algorithm (2)
Face
Edge 1 Edge 2
Edge 3
Vert 1
Vert 2
Vert 3

48. Ntreev Soft (in-house training) 48 Collision Detection In 3D Environments
Voronoi Clip Algorithm (3)
• Closest Feature Tracking Using Voronoi Regions

49. Ntreev Soft (in-house training) 49 Collision Detection In 3D Environments
Appendix: Collision Cases
• Vertex, Edge, Face 6 cases

50. Ntreev Soft (in-house training) 50 Collision Detection In 3D Environments
Appendix: The Elements of Simulator
Broad Phase Collision Detection
Narrow Phase Collision Detection
Contact Determination
Contact Analysis
Collision Solver
Time Control
Motion Solver
Constraint Solver
Mass Property Analysis

51. Ntreev Soft (in-house training) 51 Collision Detection In 3D Environments
Appendix: System Architecture of Simulator
Analysis & Response
Broad Phase
Collision Detection
Simulation
Collision
Transform Overlap
Parameters
Narrow Phase
Collision Detection

52. Ntreev Soft (in-house training) 52 Collision Detection In 3D Environments
Appendix: Public Domain Packages (1)
• Collision Detection / Proximity Query Packages
– SOLID
GJK
OBB tree
– OPCODE
Collision library for triangle meshes.
AABB tree
– RAPID
Part of V-COLLIDE
OBB tree
– CULLIDE
Interactive Collision Detection Between Complex Models in Large Environments using Graphics Hardware
Potentially Colliding Set (PCS)
– V-COLLIDE
More general than I-COLLIDE
The system uses OBB tree as defined in RAPID but uses temporal coherence to speed up
– I-COLLIDE
Lin-Canny Algorithm
N-body overlap tests (sweep and prune).
Distance calculation between convex polytopes.
– SWIFT
Variant of the LC feature-walking algorithm
More powerful than I-COLLIDE
Speedy Walking via Improved Feature Testing
– SWIFT++
An evolution of SWIFT
Supports nonconvex polyhedra

53. Ntreev Soft (in-house training) 53 Collision Detection In 3D Environments
Appendix: Public Domain Packages (2)
• Collision Detection / Proximity Query Packages
– DEEP
Dual-space Expansion for Estimating Penetration depth between convex polytopes
– H-COLLIDE
– PIVOT
Proximity Information from VOronoi Techniques
The system uses graphics hardware acceleration to support the queries
– PQP
Proximity Query Package
The system SSV and supports overlap testing, distance computation, and tolerance verification
Good for any proximity query
Released in July’99
Strong interest from Sony
– IMMAPCT
Interactive Massive Model Proximity And Collision Testing
Built on top of PQP
Use of METIS
– Q-COLLIDE
A non-UNC
A variation of I-COLLIDE and V-COLLIDE
The system uses separating Axes to determine intersections rather than tracking closest features but still uses spatial and
temporal coherence
• Tools
– QHull
Convex Hull Generation

54. Ntreev Soft (in-house training) 54 Collision Detection In 3D Environments
Appendix: Demos

55. Ntreev Soft (in-house training) 55 Collision Detection In 3D Environments
Bibliography
• [Bar92] David Baraff. Dynamic Simulation of Non-Penetrating Rigid Bodies. PhD thesis, Computer Science Department,
Cornell University, 1992. Technical Report 92-1275.
• [BKOS00] M. de Berg, M. van Kreveld, M. Overmars and O. Schwarzkopf. Computational Geometry 2nd Edition. Springer, 2000.
• [KK88] 김용운, 김용국. 토폴로지 入門. 祐成文化社, 1988.
• [Mir98] Brian Mirtich. V-Clip: Fast and robust polyhedral collision detection. ACM Transactions on Graphics, 1998.
• [Mw] http://mathworld.wolfram.com
• [vdB04] Gino van den Bergen. Collision Detection in Interactive 3D Environments. Morgan Kaufmann Publishers, 2004.
• [Wis03] CS679 - Fall 2003 - Copyright Univ. of Wisconsin.

56. Ntreev Soft (in-house training) 56 Collision Detection In 3D Environments
People
• Ming C. Lin
• Brian Mirtich
• Gino van den Bergen
• David H. Eberly

57. Ntreev Soft (in-house training) 57 Collision Detection In 3D Environments
Note