CSCE 552 Fall 2012

1. CSCE 552 Fall 2012 Math By Jijun Tang

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

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

4. arcs

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

6. 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

7. Vectors Add and Subtraction • 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 W –W V V – W V

8. Matrix Representation • The entry of a matrix M in the i-th row and j-th column is denoted Mij • For example,

9. Vectors and Matrices • Example matrix product

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

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

12. Inverse of 2x2 and 3x3 • The inverse of a 2  2 matrix M is • The inverse of a 3  3 matrix M is

13. Inverse of 4x4

14. Gauss-Jordan Elimination Gaussian Elimination

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

16. Dot Product Usage • The dot product can be used to project one vector onto another V a W

17. Dot Product Properties • The dot product satisfies the formula • a 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

18. Self 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

19. 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)

20. Illustration • The cross product satisfies the trigonometric relationship • This is the area ofthe parallelogramformed byV and W V ||V|| sin a a W

21. Area and Cross Product • The area A of a triangle with vertices P1, P2, and P3 is thus given by

22. Rules • 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

23. Rules in Figure

24. Formular • The cross product between V and W is • A helpful tool for remembering this formula is the pseudodeterminant (x, y, z) are unit vectors

25. Alternative • The cross product can also be expressed as the matrix-vector product • The perpendicularity property means

26. 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

27. Illustration

28. R S T • 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

29. 2D Example θ

30. 2D Rotations

31. Rotation Around Arbitrary Vector • Rotation about an arbitrary vector Counterclockwise rotation about an arbitrary vector (lx,ly,lz) normalised so that                    by an angle α is given by a matrix where

32. Transformation matrix • 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

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

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

35. Direction Vector and Point • 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

36. 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

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

38. Transformation Steps

39. Translation • Translation matrix • Translates the origin by the vector T

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

41. Rotation (Z) • Rotation matrix • Rotates points about the z-axis through the angle q

42. Rotations (X, Y) • Similar matrices for rotations about x, y

43. Transforming Normal Vectors • Normal vectors are transformed 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

44. Orderings

45. Orderings • Orderings of different type is important A rotation followed by a translation is different from a translation followed by a rotation • Orderings of the same type does not matter

46. Geometry -- Lines • 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

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

48. Plane Equation • 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

49. Properties of Plane • 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

50. Distance from Point to a Line • Distance d from a point P to a lineS + t V P d S V