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

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

3. Revision, Normals, Projections
Vectors
Revision, Normals, Projections

4. A 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 Can represent points in space
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. Definition, Application, Importance
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: Result
A (5, 6) B (-3, 2) = 5 * (-3) + 6 * 2 = = -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 Applications
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*
Dot Product Computation
Live Demo
(c) 2008 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

20. Definition, Features, Application
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*
2D Cross Product Computation
Live Demo
(c) 2008 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

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 Applications
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*
3D Cross Product Computation
Live Demo
(c) 2008 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

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

33. Basics, Methods, Problems, Optimization
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*
Sphere-sphere collision detection
Live Demo
(c) 2008 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

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*
Bounding sphere generation
Live Demo
(c) 2008 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

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*
Minimum bounding sphere generation
Live Demo
(c) 2008 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

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*
Axis-aligned bounding box generation
Live Demo
(c) 2008 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

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*
Axis-aligned bounding box collision detection
Live Demo
(c) 2008 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

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

51. Oriented bounding box updating*
Oriented bounding box updating
Live Demo
(c) 2008 National Academy for Software Development - All rights reserved. Unauthorized copying or re-distribution is strictly prohibited.*

52. Computational geometry & Collision detection? ? ? ? ?
Questions? ? ? ? ? ? ?