1. Physics simulation in a 3D environment Sylvain EUDIER MSCS Candidate Union College May 28 th, Spring 2004

2. Agenda  Why Physics Simulation?  Where am I, where am I going?  Start Coding  Entry point: The Spring Model  Extension to the Flag / Cloth simulation  Introduction to Collision Detection  Collision improvement: an example  Conclusion

3. Why Physics Simulation?  Getting more and more interest from the game industry  How does it work behind the scenes?  Combines physics and CS

4. Where am I?  Physics are used in many programs (CAD, games, simulators…)  Commercial physics libraries exist  As well as open source  Evolution of the simulation models up to now

5. Evolution : Quake 2 – Doom 3

6. Where am I going?  How precisely do we want to simulate the world  How do we want to represent it  For what expected quality / expense

7. Start Coding – Define the rules  Use of C++  Representation using the OpenGL API  Game-like precision  Find a model for this problem (classes)  Starting point: Write a stable and easy-to- use CVector3D class

8. The CVector3D class  3 constructors : Default, copy, by components  Overloading of operators: +, -, *, /, +=, -=, *=, /=  Methods: length, normalize, unit, crossProduct and dotProduct

9. What can I start with?  The spring model Simulate the behavior of a deformable object under certain constraints. Easy to implement (as a beginning) Gives convincing results rapidly  Allows me to test the architecture of my program

10. The Spring Model  To the basic formula, we add the inner friction (to stabilize it):

11. The Spring Model  These properties are the basics we can give to a mass.  Considered as a dot

12. The Spring Model  For the computation: L: steady length x: actual length of the spring u: unit vector between mass1 and mass2

13. Application to a rope  The rope is made of several masses that interact with each other  By changing the variables, the rope may be: stiffer, more / less extendable  We can create different kinds of extensible material

14. Demonstration  Rope simulation 1  Rope simulation 2

15. Spring Model : First impressions  (+) The result looks good enough for such a simple simulation.  (-) The rope behaved differently on different machines (different speeds)  (-) The rope cannot be very stiff

16. Spring Model : Speed problem  Need for a time regulation algorithm Why? How?  After the first try, I had a slow and fast behavior… Due to the GetTickCount() function Use of the QueryPerformanceCounter()

17. Spring Model : Stiffness issue  The stiffness problem: Due to the Euler’s approximation method

18. Spring Model : Stiffness issue – Why?

19. Euler function stability comparison

20. Spring Model : Extensions  The rope does not include any bending information: Can be solved using interleaved springs (explained later, cf. Flag)  Stiffness problem: Regarding the sources I found, the Runge-Kutta algorithm should solve the problem

21. The Runge-Kutta Algorithm Runge-Kutta 4 (55, 200000)

22. Spring Model : Flag simulation  A flag is just a patch of springs Create n*m masses Create (n-1)*(m-1) springs Connect the springs to the masses  Possibility to add a wind effect

23. Spring Model : Flag simulation  Flag simulation

24. Flag simulation : Results  (+) The mesh reacts well to the wind and gravity  (-) The flag is harder to simulate because of the stiffness problem and the lack of bending factor

25. Flag simulation : Extensions  Can simulate a flag flexibility with interleaved springs…  …and add a universal repulsive force to every node  More complex and realistic simulation

26. High quality flag simulation  Demonstration

27. Collision Detection  Why?  How?  Dependencies: A strong math library: vectors, matrices, plane-point collision, face-face collision… Possibility to work on predefined meshes

28. On the way to the collision  Math library: Matrices:  Overloading of arithmetic operators (+, -, *, +=…)  Overloading of input / output operators ([], <<)  Matrices functions : determinant, multiplications, inversions, transposition, identity  Matrix-related functions : rotate, scale, translate… Vectors Collision functions: PointToPlaneDistance, IsPointInPolygon…

29. Importing 3DS files  3DS is a standard in the industry  I already had an importing class for 3DS files .3DS files have several advantages: Face defined clockwise, Texture information, Normals information, And a lot more…

30. Into the collision  Brute force algorithm: CheckForCollisions(): MakePreselection(Scene, Collisions) For all objects in the Collision List if(this object collides with another one) Find the collision point Apply the physics on the objects, at that point  But this will never work!!!

31. Buggy Collision  Demonstration

32. Into the collision (2)  New algorithm: Do ComputePhysics(NextTimeChunk); CheckForCollisions(Scene, Collisions); if(MaxPenetrationInAnObject < Limit) Problem is solved; if(Problem NOT solved) NextTimeChunk = PreviousTime / 2; CancelTheComputations(); else ValidateTheComputations(); While(Problem is NOT solved); proceed to the next time chunk; While(TimeChunknotSimulated);

33. The rollback function

34. Collision improvement  We can extend the sphere collision test to a more general one.  Add a real collision and motion behavior (friction, rotation…)  The MakePreselection function can improve a lot the computation time

35. Improvements and trade-offs  The vast majority of the program use an aggressive MakePreselection algorithm to be able to deal with a lot of objects  Optimization without loss of information AABB = Axis Aligned Bounding Box OBB = Oriented Bounding Box 6-dop = Discrete Orientation Polytope Convex Hull

36. Example of an approximation algorithm Approximation: Based on some assumptions over “insignificant” constraints of objects (=has to look good enough)  The Opposing Face Geometry algorithm: Algorithm in 8 steps, The pro… …And cons

37. Opposing Face Geometry algorithm  1. Preselection: check collision between object A's bounding sphere and object B's bounding sphere.  2. Find the closest k faces of object A relative to object B.

38. O.F.G. algorithm  3. Calculate the geometric center of the new selection and the bounding sphere radius.  4. Find the closest k faces of object B relative to object A's new selection of k faces.  5. Calculate the geometric center of object B's new selection of faces and their maximal bounding sphere radius.

39. The O.F.G. algorithm  6. PreSelection: check collision between spheres to determine if there is even a chance for face collisions.  7. Sort the two sets of faces by increasing distance  8. Test the two sets of faces against each other, starting with the closest pairs of faces.

40. Pro / Cons of such this algorithm  (+) This is a lot faster. Runtime of O(k 2 ) Where k is usually between 4 and 8 (k is a variable representing the number of faces we want to work on) Brute force approach would be O(n*m) n and m could be 1000 of faces  (-) Cannot really work on concave polygons This is TRUE for most of today’s engines

41. The discrete Time problem  Due to the intrinsic nature of the simulation : Time- discrete based  If the dt variation is too big, an object might be “teleported” through another one  Solution: Extrude the silhouette of the object. Test this polygon for collisions

42. Summary  Springs are the basis of a lot of models and can be used for powerful simulations (i.e. any kind of elastic models)  Collision detection needs a robust design and math support. There is a lot to do about optimization and trade-offs  Physics simulation is a vast field where a lot of techniques are to be discovered

43. Selection of References  “Computer Graphics with OpenGL”, third Edition by Hearn- Baker, Prentice Hall  Huge source of information for game programming: www.gamedev.net  Chris Hecker’s famous columns about physics: http://www.d6.com/users/checker/dynamics.htm#articles  Everything you need to know about geometry: http://astronomy.swin.edu.au/~pbourke/geometry/  A lot about everything, from physics to light computations: http://freespace.virgin.net/hugo.elias/

44. Conclusion - Discussion  Questions? Need Explanations? What kind of extensions could we add to a physics simulator?

45. References  Collision detection Advanced  Gamasutra - Features - "Advanced Collision Detection Techniques" [03.30.00] Gamasutra - Features - "Advanced Collision Detection Techniques" [03.30.00]  Advanced Collision Detection Techniques Advanced Collision Detection Techniques  Advanced Collision Detection Techniques Advanced Collision Detection Techniques  Chris Hecker's Rigid Body Dynamics Information Chris Hecker's Rigid Body Dynamics Information  DDJ DDJ GameDev.net - Opposing Face Geometry GameDev.net - Simple Bounding-Sphere Collision Detection GameDev.net - Practical Collision Detection GameDev.net - General Collision Detection for Games Using Ellipsoids Collision Response: Bouncy, Trouncy, Fun Gamasutra - Features - "Crashing into the New Year" [02.10.00] Snooker simulation (+Formulaes) Rotation computation  HyperPhysics HyperPhysics MODEL-BASED ANIMATION SIGGRAPH - Collision Detection ('88) AIWisdom.com - Game Articles & Research Geometry Cours de Mécanique - Index The good-looking textured light-sourced bouncy fun smart and stretchy page Deformation  GameDev.net - Real time deformation of solids, Part 1 GameDev.net - Real time deformation of solids, Part 1  GameDev.net - Real time deformation of solids, Part 2 GameDev.net - Real time deformation of solids, Part 2  Gamasutra - Features - "Exploring Spring Models" [10.05.01] Gamasutra - Features - "Exploring Spring Models" [10.05.01] Cloths  Awesome paper on cloth simulation Awesome paper on cloth simulation  ch06.pdf (application/pdf Object) ch06.pdf (application/pdf Object)  Rotational Motion Rotational Motion