Computational Geometry & Collision detection

1. Computational Geometry & Collision detection Vectors, Dot Product, Cross Product, Basic Collision Detection George Georgiev Telerik Corporation www.telerik.com

2. Table of Contents • Vectors • Extended revision • The vector dot product • The vector cross product • Collision detection • In Game programming • Sphere collision • Bounding volumes • AABBs

3. Vectors Revision, Normals, Projections

4. Vectors – revision • Ordered sequences of numbers • OA (6, 10, 18) – 3-dimensional • OA (6, 10) – 2-dimensional • OA (6, 10, 18, -5) – 4-dimensional • Have magnitude and direction A

5. Vectors – revision • No location • Wherever you need them • Can represent points in space • Points are vectors with a beginning at the coordinate system center • Example: • Point A(5, 10) describes the location (5, 10) • Vector U(5, 10), beginning at (0, 0), describes ‘the path’ to the location (5, 10)

6. Vectors – revision • All vectors on the same line are called collinear • Can be derived by scaling any vector on the line • E.g.: A(2, 1), B(3, 1.5), C(-1, -0.5) are collinear • Two vectors, which are not collinear, lie on a plane and are called coplanar • => Two non-collinear vectors define a plane • Three vectors, which are not coplanar, define a space

7. Vectors – revision • Collinear vectors: • Coplanar vectors:

8. Vectors – revision • Vectors defining a 3D vector space

9. Vectors – revision • Perpendicular vectors • Constitute a right angle • Deriving a vector, perpendicular to a given one: • Swap two of the coordinates of the given vector (one of the swapped coordinates can’t be zero) • Multiply ONE of the swapped coordinates by -1 • Example: • A (5, 10) given => A’(-10, 5) is perpendicular to A • V (3, 4, -1) given => V’(3, 1, 4) is perpendicular to V

10. Vectors – revision • Normal vectors to a surface • Constitute a right angle with flat surfaces • Perpendicular to at least two non-collinear vectors on the plane • Constitute a right angle with the tangent to curved surfaces

11. Vectors – revision • Projection of a vector on another vector

12. Vector Dot Product Definition, Application, Importance

13. Vector dot product • Dot Product (a.k.a. scalar product) • Take two equal-length sequences • e.g. sequence A (5, 6) and sequence B (-3, 2) • Multiply each element of A with each element of B • A [i] * B [i] • Add the products • Dot Product(A, B) = A[0] * B[0] + A[1] * B[1] + … + A[i] * B[i] + … + A[n-1] * B[n-1]

14. Vector dot product • Dot Product (2) • Example: • A (5, 6) B (-3, 2) = 5 * (-3) + 6 * 2 = -15 + 12 = -3 • Result • A scalar number

15. Vector dot product • Dot product of coordinate vectors • Take two vectors of equal dimensions • Apply the dot product to their coordinates • 2D Example: • A(1, 2) . B(-1, 1) = 1*(-1) + 2*1 = 1 • 3D Example: • A(1, 2, -1) . B(-1, 1, 5) = 1*(-1) + 2*1 + (-1) * 5 = -4 • Simple as that

16. Vector dot product • Meaning in Euclidean geometry • If A(x1, y1, …), B(x2, y2, …) are vectors • theta is the angle, in radians, between A and B • Dot Product (A, B) = A . B = = |A|*|B|*cos(theta) • Applies to all dimensions (1D, 2D, 3D, 4D, … nD)

17. Vector dot product • Meaning in Euclidean geometry (2) • If U and V are unit vectors, then U . V = • cosine of the angle between U and V • the oriented length of the projection of U on V • If U and V are non-unit vectors • ( U . V ) divided by |U|*|V| = cosine of the angle between U and V • ( U . V ) divided by |V| = the oriented length of the projection of U on V

18. Vector dot product • Consequences • If A . B > 0, A and B are in the same half-space • If A . B = 0, A and B are perpendicular • If A . B < 0, A and B are in different half-spaces • Applications • Calculating angles • Calculating projections • Calculating lights • Etc…

19. Dot Product Computation Live Demo

20. Vector Cross Product Definition, Features, Application

21. Vector cross product • Cross product • Operates on vectors with up to 3 dimensions • Forms a determinant of a matrix of the vectors • Result – depends on the dimension • In 2D – a scalar number (1D) • In 3D – a vector (3D) • Not defined for 1D and dimensions higher than 3

22. 2D Vector cross product • 2D Cross product • Take the vectors U(x1, y1) and V(x2, y2) • Multiply their coordinates across and subtract: • U(x1, y1) x V(x2, y2)= (x1 * y2) – (x2 * y1) • Result • A scalar number

23. 2D Vector cross product • Scalar meaning in Euclidean geometry • If U(x1, y1) and V(x2, y2) are 2D vectors • theta is the angle between U and V • Cross Product (U, V) = U x V = = |U| * |V| * sin(theta) • |U| and |V| denote the length of U and V • Applies to 2D and 3D

24. 2D Vector cross product • Scalar meaning in Euclidean geometry (2) • For every two 2D vectors U and V • U x V = the oriented face of the parallelogram, defined by U and V • For every three 2D points A, B and C • If U x V = 0, then A, B and C are collinear • If U x V > 0, then A, B and C constitute a ‘left turn’ • If U x V < 0, then A, B and C constitute a ‘right turn’

25. 2D Vector cross product • Applications • Graham scan (2D convex hull) • Easy polygon area computation • Cross product divided by two equals oriented (signed) triangle area • 2D orientation • ‘left’ and ‘right’ turns

26. 2D Cross Product Computation Live Demo

27. 3D Vector cross product • 3D Cross product • Take two 3D vectors U(x1, y1, z1) and V(x2, y2, z2) • Calculate the following 3 coordinates • x3 = y1*z2 – y2*z1 • y3 = z1*x2 – z2*x1 • z3 = x1*y2 – x2*y1 • Result • A 3D vector with coordinates (x3, y3, z3)

28. 3D Vector cross product • Meaning in Euclidean geometry • The magnitude • Always positive (length of the vector) • Has the unsigned properties of the 2D dot product • The vector • Perpendicular to the initial vectors U and V • Normal to the plane defined by U and V • Direction determined by the right-hand rule

29. 3D Vector cross product • The right-hand rule • Index finger points in direction of first vector (a) • Middle finger points in direction of second vector (b) • Thumb points up in direction of the result of a x b

30. 3D Vector cross product • Unpredictable results occur with • Cross product of two collinear vectors • Cross product with a zero-vector • Applications • Calculating normals to surfaces • Calculating torque (physics)

31. 3D Cross Product Computation Live Demo

32. Computational geometry ? ? Questions? ? ? ? ? ? ? ? ? ?

33. Collision detection Basics, Methods, Problems, Optimization

34. Collision detection • Collisions in Game programming • Any intersection of two objects’ geometry • Raise events in some form • Usually the main part in games • Collision response – deals with collision events

35. Collision detection • Collision objects • Can raise collision events • Types • Spheres • Cylinders • Boxes • Cones • Height fields • Triangle meshes

36. Collision detection • Sphere-sphere collision • Easiest to detect • Used in • particle systems • low-accuracy collision detection • Collision occurrence: • Center-center distance less than sum of radiuses • Optimization • Avoid computation of square root

37. Sphere-sphere collision detection Live Demo

38. Collision detection • Triangle meshes collision • Very accurate • Programmatically heavy • Computation heavy (n2) • Rarely needed

39. Collision detection • Collision detection in Game programming • Combines several collision models • Uses bounding volumes • Uses optimizations • Axis-sweep • Lower accuracy in favor of speed

40. Collision detection • Bounding volumes • Easy to check for collisions • Spheres • Boxes • Cylinders, etc. • Contain high-triangle-count meshes • Tested for collision before the contained objects • If the bounding volume doesn’t collide, then the mesh doesn’t collide

41. Collision detection • Bounding sphere • Orientation-independent • Center – mesh’s center • Radius • distance from mesh center to farthest vertex • Effective for • convex, oval bodies • mesh center equally distant from surface vertices • rotating bodies

42. Bounding sphere generation Live Demo

43. Collision detection • Minimum bounding sphere • Center – the center of the segment, connecting the two farthest mesh vertices • Radius – the half-length of the segment, connecting the two farthest mesh vertices • Efficient with • convex, oval bodies • rotating bodies • Sphere center rotated with the other mesh vertices

44. Minimum bounding sphere generation Live Demo

45. Collision detection • Axis-aligned bounding box (AABB) • Very fast to check for collisions • Usually smaller volume than bounding spheres • Edges parallel to coordinate axes • Minimum corner • coordinates – lowest coordinate ends of mesh • Maximum corner • coordinates – highest coordinate ends of mesh

46. Collision detection • Axis-aligned bounding box (2) • Efficient with • non-rotating bodies • convex bodies • oblong bodies • If the body rotates, the AABB needs to be recomputed

47. Axis-aligned bounding box generation Live Demo

48. Collision detection • Checking AABBs for collision • Treat the minimum and maximum corners’ coordinates as interval edges • 3D case • If the x intervals overlap • And the y intervals overlap • And the z intervals overlap • Then the AABBs intersect / collide

49. Axis-aligned bounding box collision detection Live Demo

50. Collision detection • Oriented bounding box (OBB) • Generated as AABB • Rotates along with the object’s geometry • Advantage: • Rotating it is much faster than creating an new AABB • Usually less volume than AABB • Disadvantage: • Much slower collision check