Contents of This Lecture. Tunneling. Time is Discrete. Collision Response. Ball to Wall Collision 9/3/2013

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1 Contents of This Lecture Introduction to Game Physics with Box2D 2. Mathematics for Game Physics Lecture 2.3: Collision Detection and Response Ian Parberry Dept. of Computer Science & Engineering University of North Texas Collision Detection and Response 1. Discrete time 2. Ball to wall collision 3. Ball to line collision 4. Ball to ball collision Dot product Chapter 2 Introduction to Game Physics with Box2D 2 Time is Discrete Tunneling POI TOI Chapter 2 Introduction to Game Physics with Box2D 3 Chapter 2 Introduction to Game Physics with Box2D 4 Collision Response For the rest of this lecture we ll focus on collision response: What to do when a collision is detected. Ball to wall (easy, to get us started) Ball to line (ball to wall is a subproblem) Ball to ball (ball to line in disguise?) Ball to Wall Collision Chapter 2 Introduction to Game Physics with Box2D 5 Chapter 2 Introduction to Game Physics with Box2D 6 1

2 Ball to Wall: POI Ball to Wall: POI POI, From previous slide: POI, tan / tan / tan / POI, tan / tan / tan / Therefore, tan /tan /. Similarly, tan / /. POI, /,. Chapter 2 Introduction to Game Physics with Box2D 7 Chapter 2 Introduction to Game Physics with Box2D 8 Ball to Wall:, POI, 2,2 Chapter 2 Introduction to Game Physics with Box2D 9 Chapter 2 Introduction to Game Physics with Box2D 10 Dot Product: Algebra World Dot Product: Geometry World Before we get to Ball to Line collision detection and response, we need a new concept: dot product. Suppose, and,. Then the dot product of and is:,, is the length of the perpendicular projection of onto times the length of (left), and equivalently, the length of the perpendicular projection of onto times the length of (right). Chapter 2 Introduction to Game Physics with Box2D 11 Chapter 2 Introduction to Game Physics with Box2D 12 2

3 cos arccos float Angle( const D3DXVECTOR2& u, const D3DXVECTOR2& v){ return acos(d3dxvec2dot(&u, &v)/ (D3DXVec2Length(u) * D3DXVec2Length(v))); } //Angle Suppose is reflected off a line to give result. The line is specified by its unit normal. Chapter 2 Introduction to Game Physics with Box2D 13 Chapter 2 Introduction to Game Physics with Box2D 14 The perpendicular projection of onto has length Therefore, it is vector Therefore, 2 void Reflect(const D3DXVECTOR2& u, const D3DXVECTOR2& n, D3DXVECTOR2& v){ v = u - 2.0f * D3DXVec2Dot(&u, &n)* n; } //Reflect Chapter 2 Introduction to Game Physics with Box2D 15 Chapter 2 Introduction to Game Physics with Box2D 16 Chapter 2 Introduction to Game Physics with Box2D 17 Chapter 2 Introduction to Game Physics with Box2D 18 3

4 By the Law of Cosines: 2 cos Using dot product: cos Therefore Chapter 2 Introduction to Game Physics with Box2D 19 Chapter 2 Introduction to Game Physics with Box2D 20 2 cos Frames of Reference What if both balls are moving? Take the perspective of a bug sitting on one of the balls. Provided it is not accelerating. This is called using the bug s frame of reference. Add in the bug s velocity afterwards. It s a little more complicated than it sounds Chapter 2 Introduction to Game Physics with Box2D 21 Chapter 2 Introduction to Game Physics with Box2D 22 A Bugs Life I am at the center of the Universe. All things revolve around me. Computing Velocity POI?? Chapter 2 Introduction to Game Physics with Box2D 23 Chapter 2 Introduction to Game Physics with Box2D 24 4

5 Tangent and Normal x2 Therefore, 2 Therefore, 2 Chapter 2 Introduction to Game Physics with Box2D 25 Chapter 2 Introduction to Game Physics with Box2D 26 Claim: and. We can prove this for frictionless, perfectly elastic collisions in 1 dimension using conservation of kinetic energy and conservation of momentum. Conservation of kinetic energy: /2 /2 /2 /2 Conservation of momentum: Chapter 2 Introduction to Game Physics with Box2D 27 Chapter 2 Introduction to Game Physics with Box2D 28 Conservation of kinetic energy: Conservation of momentum: Dividing the first by the second: (1)(2) tells us that 2 2 (2)(1) tells us that 2 2 (1) (2) Chapter 2 Introduction to Game Physics with Box2D 29 Chapter 2 Introduction to Game Physics with Box2D 30 5

6 Fixing Up Component Along Normal We need to swap the components of velocity along the normal. That is: What s Real? Of course, we don t have to make it real, because frictionless, perfectly elastic 1D collisions aren t real either. There s friction and spin to take into account too. We could share the sum of the components of the velocities along the normal, perhaps dividing them evenly between the two balls. We could also solve it mathematically, but that is a little beyond this class. Chapter 2 Introduction to Game Physics with Box2D 31 Chapter 2 Introduction to Game Physics with Box2D 32 Reflection Code //Assume n is a unit vector void Reflect(D3DXVECTOR2& u, D3DXVECTOR2& v, const D3DXVECTOR2& n) { D3DXVECTOR2 vdiff = u v; vdiff = D3DXVec2Dot(&vDiff, &n) * n; u -= vdiff; v += vdiff; } //Reflect Putting it all Together Once we discover that two balls have collided (their centers are closer than the sum of their radiuses): 1. Move the balls back to the point of impact. 2. Reflect their velocity vectors in the tangent between them at the point of impact, sharing the perpendicular components equally. 3. Move the balls forward in that direction a distance equal to that moved back in Step 1. Read the code in the book, function BallBounce. Chapter 2 Introduction to Game Physics with Box2D 33 Chapter 2 Introduction to Game Physics with Box2D 34 Suggested Reading Section 2.2 Conclusion Suggested Activities Problems 5 6 from Section 2.5. Chapter 2 Introduction to Game Physics with Box2D 35 Chapter 2 Introduction to Game Physics with Box2D 36 6