Description Written by an expert in the game industry, Christer Ericson's new book is a comprehensive guide to the components of efficient real-time
collision detection systems. The book provides the tools and know-how needed to implement industrial-strength collision detection for
the highly detailed dynamic environments of applications such as 3D games, virtual reality applications, and physical simulators.
Of
the many topics covered, a key focus is on spatial and object partitioning through a wide variety of grids, trees, and sorting methods.
The author also presents a large collection of intersection and distance tests for both simple and complex geometric shapes. Sections
on vector and matrix algebra provide the background for advanced topics such as Voronoi regions, Minkowski sums, and linear and quadratic
programming.
Of utmost importance to programmers but rarely discussed in this much detail in other books are the chapters covering
numerical and geometric robustness, both essential topics for collision detection systems. Also unique are the chapters discussing how
graphics hardware can assist in collision detection computations and on advanced optimization for modern computer architectures. All
in all, this comprehensive book will become the industry standard for years to come.
Audience
Professionals or students working in software engineering, game development, simulation, scientific visualization, or virtual reality.
Contents Preface
1 Introduction
1.1 Contents Overview
1.2 About the Code
2 Collision Detection Design Issues
2.1 Collision Algorithm Design Factors
2.2 Application Domain Representation
2.2.1 Object Representations
2.2.2 Collision versus Rendering
Geometry
2.2.3 Collision Algorithm Specialization
2.3 Different Types of Queries
2.4 Environment Simulation Parameters
2.4.1 Number of
Objects
2.4.2 Sequential versus Simultaneous Motion
2.4.3 Discrete versus Continuous Motion
2.5 Performance
2.5.1 Optimization Overview
2.6 Robustness
2.7 Ease of Implementation and Use
2.7.1 Debugging a Collision Detection System
2.8 Summary
3 A Math and Geometry
Primer
3.1 Matrices
3.1.1 Matrix Arithmetic
3.1.2 Algebraic Identities Involving Matrices
3.1.3 Determinants
3.1.4 Solving Small
Systems of Linear Equation using Cramer's Rule
3.1.5 Matrix Inverses for 2x2 and 3x3 Matrices
3.1.6 Determinant Predicates
3.1.6.1 ORIENT2D(A,
B, C)
3.1.6.2 ORIENT3D(A, B, C, D)
3.1.6.3 INCIRCLE2D(A, B, C, D)
3.1.6.4 INSPHERE(A, B, C, D, E)
3.2 Coordinate Systems and Points
3.3
Vectors
3.3.1 Vector Arithmetic
3.3.2 Algebraic Identities Involving Vectors
3.3.3 The Dot Product
3.3.4 Algebraic Identities Involving
Dot Products
3.3.5 The Cross Product
3.3.6 Algebraic Identities Involving Cross Products
3.3.7 The Scalar Triple Product
3.3.8 Algebraic
Identities Involving Scalar Triple Products
3.4 Barycentric Coordinates
3.5 Lines, Rays, and Segments
3.6 Planes and Halfspaces
3.7 Polygons
3.7.1 Testing Polygonal Convexity
3.8 Polyhedra
3.8.1 Testing Polyhedral Convexity
3.9 Computing Convex Hulls
3.9.1 Andrew's Algorithm
3.9.2 The Quickhull Algorithm
3.10 Voronoi Regions
3.11 Minkowski Sum and Difference
3.12 Summary
4 Bounding Volumes
4.1 Desired BV Characteristics
4.2 Axis-Aligned Bounding Boxes (AABBs)
4.2.1 AABB-AABB Intersection
4.2.2 Computing and Updating AABBs
4.2.3 AABB from the Object Bounding Sphere
4.2.4 AABB Reconstructed from Original Point Set
4.2.5 AABB from Hill-Climbing Vertices of
the Object Representation
4.2.6 AABB Recomputed from Rotated AABB
4.3 Spheres
4.3.1 Sphere-Sphere Intersection
4.3.2 Computing a Bounding
Sphere
4.3.3 Bounding Sphere from Direction of Maximum Spread
4.3.4 Bounding Sphere Through Iterative Refinement
4.3.5 The Minimum Bounding
Sphere
4.4 Oriented Bounding Boxes (OBBs)
4.4.1 OBB-OBB Intersection
4.4.2 Making the Separating-Axis Test Robust
4.4.3 Computing a Tight
OBB
4.4.4 Optimizing PCA-Based OBBs
4.4.5 Brute-Force OBB Fitting
4.5 Sphere-Swept Volumes
4.5.1 Sphere-Swept Volume Intersection
4.5.2
Computing Sphere-Swept Bounding Volumes
4.6 Halfspace Intersection Volumes
4.6.1 Kay-Kajiya Slab-Based Volumes
4.6.2 Discrete-Orientation
Polytopes (k-DOPs)
4.6.3 k-DOP-k-DOP Overlap Test
4.6.4 Computing and Realigning k-DOPs
4.6.5 Approximate Convex Hull Intersection Tests
4.7 Other Bounding Volumes
4.8 Summary
5 Basic Primitive Tests
5.1 Closest Point Computations
5.1.1 Closest Point
on Plane to Point
5.1.2 Closest Point on Line Segment to Point
5.1.2.1 Distance of Point to Segment
5.1.3 Closest Point on AABB to Point
5.1.3.1 Distance of Point to AABB
5.1.4 Closest Point on OBB to Point
5.1.4.1 Distance of Point to OBB
5.1.4.2 Closest Point on 3D Rectangle
to Point
5.1.5 Closest Point on Triangle to Point
5.1.6 Closest Point on Tetrahedron to Point
5.1.7 Closest Point on Convex Polyhedron
to Point
5.1.8 Closest Points of Two Lines
5.1.9 Closest Points of Two Line Segments
5.1.9.1 2D Segment Intersection
5.1.10 Closest Points
of a Line Segment and a Triangle
5.1.11 Closest Points of Two Triangles
5.2 Testing primitives
5.2.1 Separating Axis Test
5.2.1.1 Robustness
of the Separating Axis Test
5.2.2 Testing Sphere against Plane
5.2.3 Testing Box against Plane
5.2.4 Testing Cone against Plane
5.2.5
Testing Sphere against AABB
5.2.6 Testing Sphere against OBB
5.2.7 Testing Sphere against Triangle
5.2.8 Testing Sphere against Polygon
5.2.9 Testing AABB against Triangle
5.2.10 Testing Triangle against Triangle
5.3 Intersecting Lines, Rays, and (Directed) Segments
5.3.1
Intersecting Segment against Plane
5.3.2 Intersecting Ray or Segment against Sphere
5.3.3 Intersecting Ray or Segment against Box
5.3.4
Intersecting Line against Triangle
5.3.5 Intersecting Line against Quadrilateral
5.3.6 Intersecting Ray or Segment against Triangle
5.3.7
Intersecting Ray or Segment against Cylinder
5.3.8 Intersecting Ray or Segment against Convex Polyhedron
5.4 Additional Tests
5.4.1 Testing
Point in Polygon
5.4.2 Testing Point in Triangle
5.4.3 Testing Point in Polyhedron
5.4.4 Intersection of Two Planes
5.4.5 Intersection
of Three Planes
5.5 Dynamic Intersection Tests
5.5.1 Interval Halving for Intersecting Moving Objects
5.5.2 Separating Axis Test for
Moving Convex Objects
5.5.3 Intersecting Moving Sphere against Plane
5.5.4 Intersecting Moving AABB against Plane
5.5.5 Intersecting
Moving Sphere against Sphere
5.5.6 Intersecting Moving Sphere against Triangle (and Polygon)
5.5.7 Intersecting Moving Sphere against
AABB
5.5.8 Intersecting Moving AABB against AABB
5.6 Summary
6 Bounding Volume Hierarchies
6.1 Hierarchy Design
Issues
6.1.1 Desired BVH Characteristics
6.1.2 Cost Functions
6.1.3 Tree Degree
6.2 Building Strategies for Hierarchy Construction
6.2.1
Top-Down Construction
6.2.1.1 Partitioning Strategies
6.2.1.2 Choice of Partitioning Axis
6.2.1.3 Choice of Split Point
6.2.2 Bottom-Up
Construction
6.2.2.1 Improved Bottom-Up Construction
6.2.2.2 Other Bottom-Up Construction Strategies
6.2.2.3 Bottom-Up N-Ary Clustering
Trees
6.2.3 Incremental (Insertion) Construction
6.2.3.1 The Goldsmith-Salmon Incremental Construction Method
6.3 Hierarchy Traversal
6.3.1 Descent Rules
6.3.2 Generic Informed Depth-First Traversal
6.3.3 Simultaneous Depth-First Traversal
6.3.4 Optimized Leaf-Direct
Depth-First Traversal
6.4 Example Bounding Volume Hierarchies
6.4.1 OBB-Trees
6.4.2 AABB-Trees and BoxTrees
6.4.3 Sphere-Tree through
Octree Subdivision
6.4.4 Sphere-Tree from Sphere-Covered Surfaces
6.4.5 Generate-and-Prune Sphere Covering
6.4.6 k-DOP Trees
6.5 Merging
Bounding Volumes
6.5.1 Merging Two AABBs
6.5.2 Merging Two Spheres
6.5.3 Merging Two OBBs
6.5.4 Merging Two k-DOPs
6.6 Efficient Tree
Representation and Traversal
6.6.1 Array Representation
6.6.2 Preorder Traversal Order
6.6.3 Offsets Instead of Pointers
6.6.4 Cache-Friendlier
Structures (Non-Binary Trees)
6.6.5 Tree Node and Primitive Ordering
6.6.6 On Recursion
6.6.7 Grouping Queries
6.7 Improved Queries through
Caching
6.7.1 Surface Caching: Caching Intersecting Primitives
6.7.2 Front Tracking
6.8 Summary
7 Spatial Partitioning
7.1 Uniform Grids
7.1.1 Cell Size Issues
7.1.2 Grids as Arrays of Linked Lists
7.1.3 Hashed Storage and Infinite Grids
7.1.4 Storing
Static Data
7.1.5 Implicit Grids
7.1.6 Uniform Grid Object-Object Test
7.1.6.1 One Test at a Time
7.1.6.2 All Tests at a Time
7.1.7 Additional
Grid Considerations
7.2 Hierarchical Grids
7.2.1 Basic Hgrid Implementation
7.2.2 Alternative Hierarchical Grid Representations
7.2.3
Other Hierarchical Grids
7.3 Trees
7.3.1 Octrees (and Quadtrees)
7.3.2 Octree Object Assignment
7.3.3 Locational Codes and Finding the
Octant for a Point
7.3.4 Linear Octrees (Hash-Based)
7.3.5 Computing the Morton Key
7.3.6 Loose Octrees
7.3.7 k-d Trees
7.3.8 Hybrid
Schemes
7.4 Ray and Directed Line Segment Traversals
7.4.1 k-d Tree Intersection Test
7.4.2 Uniform Grid Intersection Test
7.5 Sort and
Sweep Methods
7.5.1 Sorted Linked List Implementation
7.5.2 Array-Based Sorting
7.6 Cells and Portals
7.7 Avoiding Retesting
7.7.1 Bit
Flags
7.7.2 Time Stamping
7.7.3 Amortized Time Stamp Clearing
7.8 Summary
8 BSP Tree Hierarchies
8.1 BSP Trees
8.2
Types of BSP Trees
8.2.1 Node-Storing BSP Trees
8.2.2 Leaf-Storing BSP Trees
8.2.3 Solid-Leaf BSP Trees
8.3 Building the BSP Tree
8.3.1
Selecting Dividing Planes
8.3.2 Evaluating Dividing Planes
8.3.3 Classifying Polygons with Respect to a Plane
8.3.4 Splitting Polygons
against a Plane
8.3.5 More on Polygon splitting Robustness
8.3.6 Tuning BSP Tree Performance
8.4 using the BSP Tree
8.4.1 Testing Point
against a Solid-Leaf BSP Tree
8.4.2 Intersecting Ray against a Solid-Leaf BSP Tree
8.4.3 Polytope Queries on Solid-Leaf BSP Trees
8.5
Summary
9 Convexity-Based Methods
9.1 Boundary-Based Collision Detection
9.2 Closest Features Algorithms
9.2.1 The
V-Clip Algorithm
9.3 Hierarchical Polyhedron Representations
9.3.1 The Dobkin-Kirkpatrick Hierarchy
9.4 Linear and Quadratic Programming
9.4.1 Linear Programming
9.4.1.1 Fourier-Motzkin Elimination
9.4.1.2 Seidel's Algorithm
9.4.2 Quadratic Programming
9.5 The Gilbert-Johnson-Keerthi
Algorithm
9.5.1 The Gilbert-Johnson-Keerthi Algorithm
9.5.2 Finding the Point of Minimum Norm in a Simplex
9.5.3 GJK, Closest Points
and Contact Manifolds
9.5.4 Hill-Climbing for Extreme Vertices
9.5.5 Exploiting Coherence by Vertex Caching
9.5.6 Rotated Objects Optimization
9.5.7 GJK for Moving Objects
9.6 The Chung-Wang Separating Vector Algorithm
9.7 Summary
11 Numerical Robustness
11.1 Robustness Problem Types
11.2 Representing
Real Numbers
11.2.1 The IEEE-754 Floating-Point Formats
11.2.2 Infinity Arithmetic
11.2.3 Floating-Point Error Sources
11.3 Robust Floating-Point
Usage
11.3.1 Tolerances Comparisons for Floating-Point Values
11.3.2 Robustness through Thick Planes
11.3.3 Robustness through Sharing
of Calculations
11.3.4 Robustness of Fat Objects
11.4 Interval Arithmetic
11.4.1 Interval Arithmetic Examples
11.4.2 Interval Arithmetic
in Collision Detection
11.5 Exact and Semi-Exact Computation
11.5.1 Exact Arithmetic using Integers
11.5.2 On Integer Division
11.5.3
Segment Intersection using Integer Arithmetic
11.6 Further Suggestions for Improving Robustness
11.7 Summary
12 Geometrical
Robustness
12.1 Vertex Welding
12.2 Computing Adjacency Information
12.2.1 Computing a Vertex-to-Face Table
12.2.2 Computing
an Edge-to-Face Table
12.2.3 Testing Connectedness
12.3 Holes, Cracks, Gaps, and T-Junctions
12.4 Merging Coplanar Faces
12.4.1 Testing
Coplanarity of Two Polygons
12.4.2 Testing Polygon Planarity
12.5 Triangulation and Convex Partitioning
12.5.1 Triangulation by Ear Cutting
12.5.1.1 Triangulating Polygons with Holes
12.5.2 Convex Decomposition of Polygons
12.5.3 Convex Decomposition of Polyhedra
12.5.4 Dealing
with "Nondecomposable" Concave Geometry
12.6 Consistency Testing using Euler's Formula
12.7 Summary
13 Optimization
13.1 CPU Caches
13.2 Instruction Cache Optimizations
13.3 Data Cache Optimizations
13.3.1 Structure Optimizations
13.3.2 Quantized and
Compressed Vertex Data
13.3.3 Prefetching and Preloading
13.4 Cache-Aware Data Structures and Algorithms
13.4.1 A Compact Static k-d
Tree
13.4.2 A Compact AABB Tree
13.4.3 Cache-Obliviousness
13.5 Software Caching
13.5.1 Cached Linearization Example
13.5.2 Amortized
Predictive Linearization Caching
13.6 Aliasing
13.6.1 Type-Based Alias Analysis
13.6.2 Restricted Pointers
13.6.3 Avoiding Aliasing
13.7
Parallelism through SIMD Optimizations
13.7.1 4 Spheres versus 4 Spheres SIMD Test
13.7.2 4 Spheres versus 4 AABBs SIMD Test
13.7.3 4
AABBs versus 4 AABBs SIMD Test
13.8 Branching
13.9 Summary
References
Index
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