Linear Algebra (Mathematics for CG) Reading: HB Appendix A

1. Computer Graphics Linear Algebra (Mathematics for CG) Reading: HB Appendix A Last Updated: 13-Jan-12

2. Notation: Scalars, Vectors, Matrices • Scalar • (lower case, italic) • Vector • (lower case, bold) • Matrix • (upper case, bold)

3. Vectors • Arrow: Length and Direction Oriented segment in 2D or 3D space

4. Column vs. Row Vectors Row vectors Column vectors Switch back and forth with transpose

5. Vector-Vector Addition • Add: vector + vector = vector • Parallelogram rule tail to head, complete the triangle geometric algebraic examples:

6. Vector-Vector Subtraction • Subtract: vector - vector = vector

7. Vector-Vector Subtraction • Subtract: vector - vector = vector argument reversal

8. Scalar-Vector Multiplication • Multiply: scalar * vector = vector vector is scaled

9. Vector-Vector Multiplication • Multiply: vector * vector = scalar • Dot product or Inner product

10. Vector-Vector Multiplication • Multiply: vector * vector = scalar • dot product, aka inner product

11. Vector-Vector Multiplication • Multiply: vector * vector = scalar • dot product or inner product • geometric interpretation • lengths, angles • can find angle between two vectors

12. Dot Product Geometry • Can find length of projection of u onto v • as lines become perpendicular,

13. Dot Product Example

14. Vector-Vector Multiplication • Multiply: vector * vector = vector • cross product

15. Vector-Vector Multiplication • multiply: vector * vector = vector • cross product algebraic geometric • parallelogram area • perpendicular to parallelogram

16. 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

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

18. Orthonormal Basis Vectors • If basis vectors are orthonormal(orthogonal (mutually perpendicular) and unit length) • we have Cartesian coordinate system • familiar Pythagorean definition of distance Orthonormalalgebraic properties

19. Basis Vectors and Origins • Coordinate system: just basis vectors can only specify offset: vectors • Coordinate frame (or Frame of Reference): basis vectors and origin can specify location as well as offset: points It helps in analyzing the vectors and solving the kinematics.

20. Working with Frames F1 F1

21. Working with Frames F1 F1 p = (3,-1)

22. F2 Working with Frames F1 F1 p = (3,-1) F2

23. F2 Working with Frames F1 F1 p = (3,-1) F2 p = (-1.5,2)

24. F2 F3 Working with Frames F1 F1 p = (3,-1) F2 p = (-1.5,2) F3

25. F2 F3 Working with Frames F1 F1 p = (3,-1) F2 p = (-1.5,2) F3 p = (1,2)

26. 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, ...

27. Lines • Slope-intercept form y = mx + b • Implicit form y – mx – b = 0 Ax + By + C = 0 f(x,y) = 0

28. Implicit Functions • An implicit function f(x,y) = 0 can be thought of as a height field where f is the height (top). A path where the height is zero is the implicit curve (bottom). • Find where function is 0 • plug in (x,y), check if • = 0 on line • < 0 inside • > 0 outside

29. Implicit Circles circle is points (x,y) where f(x,y) = 0 points p on circle have property that vector from c to p dotted with itself has value r2 points p on the circle have property that squared distance from c to p is r2 points p on circle are those a distance r from center point c

30. Parametric Curves • Parameter: index that changes continuously (x,y): point on curve t: parameter • Vector form

31. 2D Parametric Lines • start at point p0,go towards p1,according to parameter t p(0) = p0, p(1) = p1

32. Linear Interpolation • Parametric line is example of general concept Interpolation p goes through a at t = 0 p goes through b at t = 1 Linear weights t, (1-t) are linear polynomials in t

33. Matrix-Matrix Addition • Add: matrix + matrix = matrix • Example

34. Scalar-Matrix Multiplication • Multiply: scalar * matrix = matrix • Example

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

36. Matrix-Matrix Multiplication • row by column

37. Matrix-Matrix Multiplication • row by column

38. Matrix-Matrix Multiplication • row by column

39. Matrix-Matrix Multiplication • row by column

40. Matrix-Matrix Multiplication • row by column • Non commutative: AB != BA

41. Matrix-Vector Multiplication • points as column vectors: post-multiply

42. Matrices • Transpose • Identity • Inverse

43. Matrices and Linear Systems • Linear system of n equations, n unknowns • Matrix form Ax=b

44. Exercise 1 • Q 1: If the vectors, u and v in figure are coplanar with the sheet of paper, does the cross product v x u extend towards the reader or awayassuming right-handed coordinate system? • For u = [1, 0, 0] & v = [2, 2, 1]. Find • Q 2: The length of v • Q 3: length of projection of u onto v • Q 4: v x u • Q 5: Cos t, if t is the angle between u & v

45. Exercise 2 F1 F3 F2 F1 p = (?, ?) F2 p = (?, ?) F3 p = (?, ?)

46. Exercise 3 • Given P1(3,6,9) and P2(5,5,5). Find a point on the line with end points P1 & P2 at t = 2/3 using parametric equation of line, where t is the parameter.

47. Things to do Reading for OpenGL practice: HB 2.9 Ready for the Quiz

48. References • http://faculty.cs.tamu.edu/schaefer/teaching/441_Spring2012/index.html • http://www.ugrad.cs.ubc.ca/~cs314/Vjan2007/