1. CSCE 590E Spring 2007 Collision Detection By Jijun Tang

2. Announcements Final game demo will be held at 2:00pm, Tuesday, May 8 th Homework 4 will be given today Second presentation will be held on April 16 th and 18 th  April 16 th : Space Banditos, Slackers, Psychosoft  April 18 th : Project Gnosis, Cheeze Puffs!, Team Swampus

3. Needs to do Two groups need to send me the first presentation after the class Some people didn’t send me the members in their small project. Please send the membership of the small project to stahlj@cse.sc.edu

4. Real-time Physics in Game at Runtime: Enables the emergent behavior that provides player a richer game experience Potential to provide full cost savings to developer/publisher Difficult  May require significant upgrade of game engine  May require significant update of asset creation pipelines  May require special training for modelers, animators, and level designers Licensing an existing engine may significantly increase third party middleware costs

5. Engines Commercial  Game Dynamics SDK (Havok.com)  Renderware Physics (renderware.com)  NovodeX SDK (novodex.com) Free  Open Dynamic Engine (ODE) (ode.org)  Tokamak Game Physics SDK (tokamakphysics.com)  Newton Game Dynamics SDK (newtondynamics.com)

6. Particle Physics What is a Particle?  A sphere of finite radius with a perfectly smooth, frictionless surface  Experiences no rotational motion (or assume the sphere has no size) Particle Kinematics  Defines the basic properties of particle motion  Position, Velocity, Acceleration

7. Location of Particle in World Space  SI Units: meters (m)  Changes over time when object moves Particle Position

8. Particle Velocity and Acceleration Velocity (SI units: m/s)  First time derivative of position: Acceleration (SI units: m/s 2 )  First time derivative of velocity  Second time derivative of position

9. Newton ’ s 2 nd Law of Motion Paraphrased – “ An object ’ s change in velocity is proportional to an applied force ” The Classic Equation:  m = mass (SI units: kilograms, kg)  F(t) = force (SI units: Newtons)

10. What is Physics Simulation? The Cycle of Motion:  Force, F(t), causes acceleration  Acceleration, a(t), causes a change in velocity  Velocity, V(t) causes a change in position Physics Simulation:  Solving variations of the above equations over time to emulate the cycle of motion

11. Concrete Example: Target Practice Target Projectile Launch Position, p init

12. Choose V init to Hit a Stationary Target  p target is the stationary target location  We would like to choose the initial velocity, V init, required to hit the target at some future time, t hit.  Here is our equation of motion at time t hit :  Solution in general is a bit tedious to derive …  Infinite number of solutions!  Hint: Specify the magnitude of V init, solve for its direction Target Practice

13. 969.31 Example 1 V init = 25 m/s Value of Radicand of tan  equation: Launch angle  : 19.4 deg or 70.6 deg

14. Finite Difference Methods The Explicit Euler Integrator: Properties of object are stored in a state vector, S Use the above integrator equation to incrementally update S over time as game progresses Must keep track of prior value of S in order to compute the new For Explicit Euler, one choice of state and state derivative for particle:

15. Finite Difference Methods The Verlet Integrator: Must store state at two prior time steps, S(t) and S(t-  t) Uses second derivative of state instead of the first Valid for constant time step only (as shown above) For Verlet, choice of state and state derivative for a particle:

16. Generalized Rigid Bodies Key Differences from Particles  Not necessarily spherical in shape  Position, p, represents object ’ s center-of-mass location  Surface may not be perfectly smooth Friction forces may be present  Experience rotational motion in addition to translational (position only) motion

17. Additional forces Linear Spring Viscous Damping Aerodynamic Drag Friction …

18. Linear Springs

19. Viscous Damping

20. Aerodynamic Drag S: projected front area C D : drag coefficient

21. Friction

22. Collision Detection and Resolution

23. What is Collision Detection A fundamental problem in computer games, computer animation, physically-based modeling, geometric modeling, and robotics. Including algorithms:  To check for collision, i.e. intersection, of two given objects  To calculate trajectories, impact times and impact points in a physical simulation.

24. Collision Detection Complicated for two reasons  Geometry is typically very complex, potentially requiring expensive testing  Naïve solution is O(n2) time complexity, since every object can potentially collide with every other object Two basic techniques  Overlap testing: Detects whether a collision has already occurred  Intersection testing: Predicts whether a collision will occur in the future

25. Overlap Testing (a posteriori) Overlap testing: Detects whether a collision has already occurred, sometime is referred as a posteriori Facts  Most common technique used in games  Exhibits more error than intersection testing Concept  For every (small) simulation step, test every pair of objects to see if they overlap  Easy for simple volumes like spheres, harder for polygonal models

26. Overlap Testing Results Useful results of detected collision  Pairs of objects will have collision  Time of collision to take place  Collision normal vector Collision time calculated by moving object back in time until right before collision  Bisection is an effective technique

27. Bisect Testing: collision detected

28. Bisect Testing: Iteration I

29. Bisect Testing: Iteration II

30. Bisect Testing: Iteration III

31. Bisect Testing: Iteration IV

32. Bisect Testing: Iteration V Time right before the collision

33. Overlap Testing: Limitations Fails with objects that move too fast (compared to the object)  Thin glass vs. bulltes  Unlikely to catch time slice during overlap

34. Solution for This Limitation Speed of the fastest object multiplies the time step should be smaller than the smallest objects in the scene Possible solutions  Design constraint on speed of objects Hard to apply without affecting the play  Reduce simulation step size Too expensive

35. Intersection Testing (a priori) Predict future collisions When predicted:  Move simulation to time of collision  Resolve collision  Simulate remaining time step

36. Intersection Testing: Swept Geometry Extrude geometry in direction of movement Swept sphere turns into a “ capsule ” shape

37. Intersection Testing: Sphere-Sphere Collision d

38. Special Cases No collision: B2 = 0: both objects are stationary, or they are traveling at parallel When will collision occur?

39. Intersection Testing: Sphere-Sphere Collision Smallest distance ever separating two spheres: If there is a collision

40. Intersection Testing: Limitations Issue with networked games  Future predictions rely on exact state of world at present time  Due to packet latency, current state not always coherent Assumes constant velocity and zero acceleration over simulation step  Has implications for physics model and choice of integrator

41. Dealing with Complexity Two issues 1. Complex geometry must be simplified 2. Reduce number of object pair tests

42. Simplified Geometry Approximate complex objects with simpler geometry, like this ellipsoid or bounding boxes

43. Minkowski Sum By taking the Minkowski Sum of two complex volumes and creating a new volume, overlap can be found by testing if a single point is within the new volume

44. Minkowski Sum

45. Using Minkowski Sum

46. Bounding Volumes Bounding volume is a simple geometric shape  Completely encapsulates object  If no collision with bounding volume, no more testing is required Common bounding volumes  Sphere  Box

47. Box Bounding Volumes

48. More Examples

49. Using Bounding Box in Game Complex objects can have multiple bounding boxes  Human object can have one big bounding box for the whole body  Human object can have one bounding box per limb, head, etc Bounding box can be hierarchical:  Test the big first  if possible collision, test the smaller ones

50. Reduce Number of Detections O(n) Time Complexity can be achieved. One solution is to partition space

51. Achieving O(n) Time Complexity Another solution is the plane sweep algorithm Requires (re-)sorting in x (y) coordinate

52. Terrain Collision Detection: Height Field Landscape

53. Locate Triangle on Height Field Essentially a 2D problem

54. Locate Point on Triangle Plane equation: A, B, C are the x, y, z components of the triangle plane ’ s normal vector Where with one of the triangles vertices being Giving:

55. Terrain Collision Detection: Locate Point on Triangle The normal can be constructed by taking the cross product of two sides: Solve for y and insert the x and z components of Q, giving the final equation for point within triangle:

56. Treating Nonuniform Polygon Mesh Hard to detect the triangle where the point lies in Using Triangulated Irregular Networks (TINs) Barycentric Coordinates

57. Even with complex data structure, we still have to test each triangle (in a sub region) to see if the point lies in it Using Barycentric Coordinates to do the test

58. Terrain Collision Detection: Locate Point on Triangle Calculate barycentric coordinates for point Q in a triangle ’ s plane If any of the weights (w 0, w 1, w 2 ) are negative, then the point Q does not lie in the triangle

59. Collision Resolution: Examples Two billiard balls strike  Calculate ball positions at time of impact  Impart new velocities on balls  Play “ clinking ” sound effect Rocket slams into wall  Rocket disappears  Explosion spawned and explosion sound effect  Wall charred and area damage inflicted on nearby characters Character walks through wall  Magical sound effect triggered  No trajectories or velocities affected

60. Collision Resolution: Parts Resolution has three parts 1. Prologue 2. Collision 3. Epilogue

61. Prologue Collision known to have occurred Check if collision should be ignored Other events might be triggered  Sound effects  Send collision notification messages

62. Collision Place objects at point of impact Assign new velocities  Using physics or  Using some other decision logic

63. Epilogue Propagate post-collision effects Possible effects  Destroy one or both objects  Play sound effect  Inflict damage Many effects can be done either in the prologue or epilogue

64. Collision Resolution: Resolving Overlap Testing 1. Extract collision normal 2. Extract penetration depth 3. Move the two objects apart 4. Compute new velocities

65. Extract Collision Normal Find position of objects before impact Use two closest points to construct the collision normal vector

66. Extract Collision Normal for Sphere Collision Sphere collision normal vector  Difference between centers at point of collision

67. Depth Gilbert-Johnson-keerthi (GJK) bisection to extract the depth of convex objects Minkowski supporting lines

68. Collision Resolution: Resolving Intersection Testing Simpler than resolving overlap testing  No need to find penetration depth or move objects apart Simply 1. Extract collision normal 2. Compute new velocities

69. Collision Response Why?  Performed to keep objects from interpenetrating  To ensure behavior similar to real-world objects Two Basic Approaches  Approach 1: Instantaneous change of velocity at time of collision  Approach 2: Gradual change of velocity and position over time, following collision

70. Approach 1 Approach 1: Instantaneous change of velocity at time of collision  Benefits: Visually the objects never interpenetrate Result is generated via closed-form equations, and is perfectly stable  Difficulties: Precise detection of time and location of collision can be prohibitively expensive (frame rate killer) Logic to manage state is complex

71. Approaches 2 Approach 2: Gradual change of velocity and position over time, following collision  Benefits Does not require precise detection of time and location of collision State management is easy Potential to be more realistic, if meshes are adjusted to deform according to predicted interpenetration  Difficulties Object interpenetration is likely, and parameters must be tweaked to manage this Simulation can be subject to numerical instabilities, often requiring the use of implicit finite difference methods

72. Final Comments Instantaneous Collision Response  Classical approach: Impulse-momentum equations See text for full details Gradual Collision Response  Classical approach: Penalty force methods Resolve interpenetration over the course of a few integration steps Penalty forces can wreak havoc on numerical integration  Instabilities galore Implicit finite difference equations can handle it  But more difficult to code  Geometric approach: Ignore physical response equations Enforce purely geometric constraints once interpenetration has occurred

73. Final Comments Simple Games  Closed-form particle equations may be all you need  Numerical particle simulation adds flexibility without much coding effort  Collision detection is probably the most difficult part of this Generalized Rigid Body Simulation  Includes rotational effects and interesting (non- constant) forces  See text for details on how to get started

74. Final Comments Full-Up Simulation  The text and this presentation just barely touch the surface  Additional considerations Multiple simultaneous collision points Articulating rigid body chains, with joints Friction, rolling friction, friction during collision Mechanically applied forces (motors, etc.) Resting contact/stacking Breakable objects Soft bodies Smoke, clouds, and other gases Water, oil, and other fluids

75. Homework 4 Page 390 problem 1 and 5 Handwriting or printing using Word/Latex  Turn in before class  No need to upload to dropbox 5 points total (2.5 each) Due on this Wednesday