Animation and Games Development

1. Animation and Games Development 242-515, Semester 1, 2014-2015 • Objective • to give some background on the course 7. Basic Maths

2. Overview • Vectors • Matricies • The Dot Product • The Cross Product • Transformations as Matricies • Multiple Coordinate Spaces • Models in a Scene Graph

3. 1. Vectors • A vector is a magnitude(size) and a direction. • drawn as an arrow

4. Two vectors V and W are added by placing the beginning of W at the end of V • Subtraction reverses the second vector W V V + W W –W V V – W V

5. Triangle Rule for Subtraction • Place c and d tail-to-tail. • d – c is the vector from the head of c to the head of d

8. Vectors are added and subtracted component-wise:

9. Vector Magnitude • The magnitude of a vector is a scalar (a number):

11. Normalization • A normalized vector always has unit length. • To normalize a non-zero vector, divide by its magnitude.

12. Normalize [12, -5]:

13. Normalization as a Diagram unit circle

14. 2. Matrices • A matrix is an array of numbers arranged in rows and columns • A matrix with n rows and m columns is an nm matrix • M is a 2  3 matrix • If n = m, the matrix is a square matrix

15. The transpose of a matrix M is denoted MT and has its rows and columns exchanged:

16. An n-dimensional vector V can be thought of as an n 1 columnmatrix: • Or a 1 nrow matrix:

17. Matrix Multiplication • The entries of the product of two matricies A and B are calculated by: • The number of columns in Amust equal the number of rows in B. • If A is a nm matrix, and B is an mp matrix, then AB is an np matrix.

18. Example:

19. In three dimensions, the product of a matrix and a column vector looks like:

20. Identity Matrix In For any nn matrix M, the product with the identity matrix is M itself • InM = M • MIn = M

21. Invertible • An nn matrix M is invertible if there exists another matrix G such that • The inverse of M is denoted M-1

22. Determinant • The determinant of a square matrix M is denoted det M or |M| • A matrix is invertible if its determinant is not zero. • For a 2  2 matrix,

23. The determinant of a 3  3 matrix is

24. Inverse • Explicit formulas exist for matrix inverses • These are good for small matrices, but other methods are generally used for larger matrices • In computer graphics, we usually deal with 2  2, 3  3, and a special form of 4  4 matrices

25. 3. The Dot Product • The dot product is a product between two vectors that produces a magnitude (a number). • The dot product between twon-dimensional vectors V and W is given by • In three dimensions,

26. Dot product a· b is the magnitude of the projection of the b vector onto the a vector.

27. Sign of the Dot Product

28. The dot product equation: • a is the angle between the two vectors • That's why the dot product is 0 between perpendicular vectors • If V and W are unit vectors, the dot product is 1 for parallel vectors pointing in the same direction, -1 for opposite

29. Dot Product for Finding Angles • The dot product can be used to find the angle between two vertors a and b. • First normalize a and b. • The angle between them is cos-1â . b ^

30. A Line as a Vector • A line in 3D space can be represented by • S is a point on the line, and V is the direction along which the line runs. R is a point on the line, t is a magnitude. R(t) = S + t V V t R S

31. Distance to a Line • Distance d from a point P to the line R(t)= S + t V P d S V

32. Use Pythagorean theorem: • Taking square root, • If V is unit length, then V 2 = 1

33. 4. The Cross Product • The cross product between two vectors a and b produces a vector (a x b) which is perpendicular to both.

34. The cross product between V and W is • One way of remembering this equation is as the determinant:

35. It can also be expressed as a matrix-vector product • The perpendicularity property means that:

36. The cross product equation is: • This is equal to the area of the parallelogram formed by V and W: V ||V|| sin a a W

37. Here's why:

38. b a

39. What about the axb Direction? • Does the vector a x bpoint up or down from the plane of a and b? • Place the tail of b at the head of a. • Using your right hand, curl your fingers in the direction of the vectors. • Your thumb points in the direction of a x b

40. Reversing the order of the vectors negates the cross product: • the cross product is anti-commutative • If a x b = 0, then a is parallel to b.

41. 5. Transformations as Matricies • A complex transformation can be built by combining simple transformations, of the types: • translations • scaling • rotations • Transformations are combined by multiplying their matrices together.

42. 2D Translation (x+dx, y+dy) y y (x, y) dy x x dx (0,0)

43. 2D Scaling • Resizes an object in each dimension about origin y y (sxx, syy) (x, y) x x

44. 2D Rotation • Rotate counter-clockwise about the origin by angle  y y (x', y') (x, y)  x x

45. Homogenous Transformation Matricies • An 4x4 homogenous matrix can store 3D transformations. • This makes matrix inversion and combinations of transformations simpler. • It also makes it possible to define perspective transformations.

46. 3D Translation Translation by (Δx, Δy, Δz):

47. 3D Scaling Scaling about the origin

48. 3D Rotations • z-axis rotation • Rotates about the z-axis through the angle q

49. Rotations about the x-, y- axes

50. Combining Transformations • The combined 4  4 homogenenous matrix is very common in computer graphics: • R is a 3  3 rotation matrix, and T is a translation vector