1. Chapter 4.1 Mathematical Concepts

2. 2 Applied Trigonometry Trigonometric functions Defined using right triangle  x y h

3. 3 Applied Trigonometry Angles measured in radians Full circle contains 2  radians

4. 4 Applied Trigonometry Sine and cosine used to decompose a point into horizontal and vertical components  r cos  r sin  r x y

5. 5 Applied Trigonometry Trigonometric identities

6. 6 Applied Trigonometry Inverse trigonometric functions Return angle for which sin, cos, or tan function produces a particular value If sin  = z, then  = sin -1 z If cos  = z, then  = cos -1 z If tan  = z, then  = tan -1 z

7. 7 Applied Trigonometry Law of sines Law of cosines Reduces to Pythagorean theorem when  = 90 degrees   b a c

8. 8 Vectors and Matrices Scalars represent quantities that can be described fully using one value Mass Time Distance Vectors describe a magnitude and direction together using multiple values

9. 9 Vectors and Matrices Examples of vectors Difference between two points Magnitude is the distance between the points Direction points from one point to the other Velocity of a projectile Magnitude is the speed of the projectile Direction is the direction in which it’s traveling A force is applied along a direction

10. 10 Vectors and Matrices Vectors can be visualized by an arrow The length represents the magnitude The arrowhead indicates the direction Multiplying a vector by a scalar changes the arrow’s length V 2V2V –V–V

11. 11 Vectors and Matrices Two vectors V and W are added by placing the beginning of W at the end of V Subtraction reverses the second vector V W V + W V W V V – W –W–W

12. 12 Vectors and Matrices An n-dimensional vector V is represented by n components In three dimensions, the components are named x, y, and z Individual components are expressed using the name as a subscript:

13. 13 Vectors and Matrices Vectors add and subtract componentwise

14. 14 Vectors and Matrices The magnitude of an n-dimensional vector V is given by In three dimensions, this is

15. 15 Vectors and Matrices A vector having a magnitude of 1 is called a unit vector Any vector V can be resized to unit length by dividing it by its magnitude: This process is called normalization

16. 16 Vectors and Matrices A matrix is a rectangular array of numbers arranged as rows and columns A matrix having n rows and m columns is an n  m matrix At the right, M is a 2  3 matrix If n = m, the matrix is a square matrix

17. 17 Vectors and Matrices The entry of a matrix M in the i-th row and j-th column is denoted M ij For example,

18. 18 Vectors and Matrices The transpose of a matrix M is denoted M T and has its rows and columns exchanged:

19. 19 Vectors and Matrices An n-dimensional vector V can be thought of as an n  1 column matrix: Or a 1  n row matrix:

20. 20 Vectors and Matrices Product of two matrices A and B Number of columns of A must equal number of rows of B Entries of the product are given by If A is a n  m matrix, and B is an m  p matrix, then AB is an n  p matrix

21. 21 Vectors and Matrices Example matrix product

22. 22 Vectors and Matrices Matrices are used to transform vectors from one coordinate system to another In three dimensions, the product of a matrix and a column vector looks like:

23. 23 Vectors and Matrices The n  n identity matrix is denoted I n For any n  n matrix M, the product with the identity matrix is M itself I n M = M MI n = M The identity matrix is the matrix analog of the number one I n has entries of 1 along the main diagonal and 0 everywhere else

24. 24 Vectors and Matrices An n  n matrix M is invertible if there exists another matrix G such that The inverse of M is denoted M -1

25. 25 Vectors and Matrices Not every matrix has an inverse A noninvertible matrix is called singular Whether a matrix is invertible can be determined by calculating a scalar quantity called the determinant

26. 26 Vectors and Matrices 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,

27. 27 Vectors and Matrices The determinant of a 3  3 matrix is

28. 28 Vectors and Matrices 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 are usually dealing with 2  2, 3  3, and a special form of 4  4 matrices

29. 29 Vectors and Matrices The inverse of a 2  2 matrix M is The inverse of a 3  3 matrix M is

30. 30 Vectors and Matrices A special type of 4  4 matrix used in computer graphics looks like R is a 3  3 rotation matrix, and T is a translation vector

31. 31 Vectors and Matrices The inverse of this 4  4 matrix is

32. 32 The Dot Product The dot product is a product between two vectors that produces a scalar The dot product between two n-dimensional vectors V and W is given by In three dimensions,

33. 33 The Dot Product The dot product satisfies the formula  is the angle between the two vectors Dot product is always 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

34. 34 The Dot Product The dot product of a vector with itself produces the squared magnitude Often, the notation V 2 is used as shorthand for V  V

35. 35 The Dot Product The dot product can be used to project one vector onto another  V W

36. 36 The Cross Product The cross product is a product between two vectors the produces a vector The cross product only applies in three dimensions The cross product is perpendicular to both vectors being multiplied together The cross product between two parallel vectors is the zero vector (0, 0, 0)

37. 37 The Cross Product The cross product between V and W is A helpful tool for remembering this formula is the pseudodeterminant

38. 38 The Cross Product The cross product can also be expressed as the matrix-vector product The perpendicularity property means

39. 39 The Cross Product The cross product satisfies the trigonometric relationship This is the area of the parallelogram formed by V and W  V W ||V|| sin 

40. 40 The Cross Product The area A of a triangle with vertices P 1, P 2, and P 3 is thus given by

41. 41 The Cross Product Cross products obey the right hand rule If first vector points along right thumb, and second vector points along right fingers, Then cross product points out of right palm Reversing order of vectors negates the cross product: Cross product is anticommutative

42. 42 Transformations Calculations are often carried out in many different coordinate systems We must be able to transform information from one coordinate system to another easily Matrix multiplication allows us to do this

43. 43 Transformations Suppose that the coordinate axes in one coordinate system correspond to the directions R, S, and T in another Then we transform a vector V to the RST system as follows

44. 44 Transformations We transform back to the original system by inverting the matrix: Often, the matrix’s inverse is equal to its transpose—such a matrix is called orthogonal

45. 45 Transformations A 3  3 matrix can reorient the coordinate axes in any way, but it leaves the origin fixed We must at a translation component D to move the origin:

46. 46 Transformations Homogeneous coordinates Four-dimensional space Combines 3  3 matrix and translation into one 4  4 matrix

47. 47 Transformations V is now a four-dimensional vector The w-coordinate of V determines whether V is a point or a direction vector If w = 0, then V is a direction vector and the fourth column of the transformation matrix has no effect If w  0, then V is a point and the fourth column of the matrix translates the origin Normally, w = 1 for points

48. 48 Transformations The three-dimensional counterpart of a four-dimensional homogeneous vector V is given by Scaling a homogeneous vector thus has no effect on its actual 3D value

49. 49 Transformations Transformation matrices are often the result of combining several simple transformations Translations Scales Rotations Transformations are combined by multiplying their matrices together

50. 50 Transformations Translation matrix Translates the origin by the vector T

51. 51 Transformations Scale matrix Scales coordinate axes by a, b, and c If a = b = c, the scale is uniform

52. 52 Transformations Rotation matrix Rotates points about the z-axis through the angle 

53. 53 Transformations Similar matrices for rotations about x, y

54. 54 Transformations Normal vectors transform differently than do ordinary points and directions A normal vector represents the direction pointing out of a surface A normal vector is perpendicular to the tangent plane If a matrix M transforms points from one coordinate system to another, then normal vectors must be transformed by (M -1 ) T

55. 55 Geometry A line in 3D space is represented by S is a point on the line, and V is the direction along which the line runs Any point P on the line corresponds to a value of the parameter t Two lines are parallel if their direction vectors are parallel

56. 56 Geometry A plane in 3D space can be defined by a normal direction N and a point P Other points in the plane satisfy P Q N

57. 57 Geometry A plane equation is commonly written A, B, and C are the components of the normal direction N, and D is given by for any point P in the plane

58. 58 Geometry A plane is often represented by the 4D vector (A, B, C, D) If a 4D homogeneous point P lies in the plane, then (A, B, C, D)  P = 0 If a point does not lie in the plane, then the dot product tells us which side of the plane the point lies on

59. 59 Geometry Distance d from a point P to a line S + t V P VS d

60. 60 Geometry Use Pythagorean theorem: Taking square root, If V is unit length, then V 2 = 1

61. 61 Geometry Intersection of a line and a plane Let P(t) = S + t V be the line Let L = (N, D) be the plane We want to find t such that L  P(t) = 0 Careful, S has w-coordinate of 1, and V has w-coordinate of 0

62. 62 Geometry If L  V = 0, the line is parallel to the plane and no intersection occurs Otherwise, the point of intersection is