1. University of British Columbia CPSC 314 Computer Graphics Jan-Apr 2016 Tamara Munzner http://www.ugrad.cs.ubc.ca/~cs314/Vjan2016 Math Basics Week 1, Fri Jan 8

2. 2 Readings For Lecture Shirley/Marschner (3 rd edition) Ch 2: Miscellaneous Math, Sec 2.1-2.4 Ch 5: Linear Algebra, Sec 5.1-5.3 Gortler Ch 2: Linear, Sec 2.1 – 2.4

3. Vectors and Matrices 3

4. 4 Notation: Scalars, Vectors, Matrices scalar (lower case, italic) vector (lower case, bold) matrix (upper case, bold)

5. 5 Vectors arrow: length and direction oriented segment in nD space offset / displacement location if given origin

6. 6 Column vs. Row Vectors row vectors column vectors switch back and forth with transpose

7. 7 Vector-Vector Addition add: vector + vector = vector parallelogram rule tail to head, complete the triangle geometricalgebraic examples:

8. 8 Vector-Vector Subtraction subtract: vector - vector = vector

9. 9 Vector-Vector Subtraction subtract: vector - vector = vector argument reversal

10. 10 Scalar-Vector Multiplication multiply: scalar * vector = vector vector is scaled

11. 11 Vector-Vector Multiplication: Dot multiply v1: vector * vector = scalar dot product, aka inner product

12. 12 Vector-Vector Multiplication: Dot multiply v1: vector * vector = scalar dot product, aka inner product

13. 13 Vector-Vector Multiplication: Dot multiply v1: vector * vector = scalar dot product, aka inner product geometric interpretation lengths, angles can find angle between two vectors

14. 14 Dot Product Geometry can find length of projection of u onto v as lines become perpendicular,

15. 15 Dot Product Example

16. 16 Vector-Vector Multiplication, Cross multiply v2: vector * vector = vector cross product algebraic

17. 17 Vector-Vector Multiplication, Cross multiply v2: vector * vector = vector cross product algebraic 1 2 3

18. 18 Vector-Vector Multiplication, Cross multiply v2: vector * vector = vector cross product algebraic geometric parallelogram area perpendicular to parallelogram

19. 19 RHS vs. LHS Coordinate Systems right-handed coordinate system left-handed coordinate system right hand rule: index finger x, second finger y; right thumb points up left hand rule: index finger x, second finger y; left thumb points down convention

20. 20 Matrix-Matrix Addition add: matrix + matrix = matrix example

21. 21 Scalar-Matrix Multiplication multiply: scalar * matrix = matrix example

22. 22 Matrix-Matrix Multiplication can only multiply (n,k) by (k,m): number of left cols = number of right rows legal undefined

23. 23 Matrix-Matrix Multiplication row by column

24. 24 Matrix-Matrix Multiplication row by column

25. 25 Matrix-Matrix Multiplication row by column

26. 26 Matrix-Matrix Multiplication row by column

27. 27 Matrix-Matrix Multiplication row by column noncommutative: AB != BA

28. 28 Matrix-Vector Multiplication points as column vectors: postmultiply points as row vectors: premultiply

29. 29 Matrices transpose identity inverse not all matrices are invertible

30. 30 Matrices and Linear Systems linear system of n equations, n unknowns matrix form Ax=b

31. Basis Vectors and Frames 31

32. 32 Basis Vectors take any two vectors that are linearly independent (nonzero and nonparallel) can use linear combination of these to define any other vector:

33. 33 Orthonormal Basis Vectors if basis vectors are orthonormal: orthogonal (mutually perpendicular) and unit length we have Cartesian coordinate system familiar Pythagorean definition of distance orthonormal algebraic properties

34. 34 Basis Vectors and Origins coordinate system: just basis vectors can only specify offset: vectors coordinate frame: basis vectors and origin can specify location as well as offset: points

35. 35 Working with Frames F1F1F1F1 F1F1F1F1

36. 36 Working with Frames F1F1F1F1 F1F1F1F1 p = (3,-1)

37. 37 Working with Frames F1F1F1F1 F1F1F1F1 p = (3,-1) F1F1F1F1

38. 38 Working with Frames F1F1F1F1 F1F1F1F1 p = (3,-1) F2F2F2F2 F2F2F2F2

39. 39 Working with Frames F1F1F1F1 F1F1F1F1 p = (3,-1) F2F2F2F2 p = (-1.5,2) F2F2F2F2

40. 40 Working with Frames F1F1F1F1 F1F1F1F1 p = (3,-1) F2F2F2F2 p = (-1.5,2) F2F2F2F2 F2F2F2F2

41. 41 Working with Frames F1F1F1F1 F1F1F1F1 p = (3,-1) F2F2F2F2 p = (-1.5,2) F3F3F3F3 F2F2F2F2 F3F3F3F3

42. 42 Working with Frames F1F1F1F1 F1F1F1F1 p = (3,-1) F2F2F2F2 p = (-1.5,2) F3F3F3F3 p = (1,2) F2F2F2F2 F3F3F3F3

43. 43 Working with Frames F1F1F1F1 p = (3,-1) F2F2F2F2 p = (-1.5,2) F3F3F3F3 p = (1,2) F1F1F1F1 F2F2F2F2 F3F3F3F3 F3F3F3F3

44. 44 Named Coordinate Frames origin and basis vectors pick canonical frame of reference then don’t have to store origin, basis vectors just convention: Cartesian orthonormal one on previous slide handy to specify others as needed airplane nose, looking over your shoulder,... really common ones given names in CG object, world, camera, screen,...