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Math Basics Week 1, Fri Jan 8

http:// www.ugrad.cs.ubc.ca /~cs314/Vjan2016. Math Basics Week 1, Fri Jan 8. 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. Vectors and Matrices.

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Math Basics Week 1, Fri Jan 8

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  1. http://www.ugrad.cs.ubc.ca/~cs314/Vjan2016 Math BasicsWeek 1, Fri Jan 8

  2. Readings For Lecture • Shirley/Marschner (3rd 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

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

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

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

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

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

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

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

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

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

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

  15. Dot Product Example

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

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

  18. Vector-Vector Multiplication, Cross • multiply v2: vector * vector = vector • cross product • algebraic • geometric • parallelogramarea • perpendicularto parallelogram

  19. RHS vs. LHS Coordinate Systems • right-handed coordinate system • left-handed coordinate system convention 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

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

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

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

  23. Matrix-Matrix Multiplication • row by column

  24. Matrix-Matrix Multiplication • row by column

  25. Matrix-Matrix Multiplication • row by column

  26. Matrix-Matrix Multiplication • row by column

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

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

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

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

  31. Basis Vectors and Frames

  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. 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. 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. Working with Frames F1 F1

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

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

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

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

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

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

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

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

  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, ...

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