1 Vector Multiplication Dot (scalar) product Vector Multiplication - PDF document

To Do To Do Foundations of Computer Graphics Foundations of Computer Graphics  Complete Assignment 0 (a due 26, b due 31) (Spring 2012) (Spring 2012)  Get help if issues with compiling, programming CS 184, Lecture 2: Review of Basic Math  Textbooks: access to OpenGL references http://inst.eecs.berkeley.edu/~cs184  About first few lectures  Somewhat technical: core math ideas in graphics  HW1 is simple (only few lines of code): Lets you see how to use some ideas discussed in lecture, create images Motivation and Outline Vectors Motivation and Outline Vectors  Many graphics concepts need basic math like linear algebra  Vectors (dot products, cross products, …) =  Matrices (matrix-matrix, matrix-vector mult., …)  E.g: a point is a vector, and an operation like translating or rotating points on object can be matrix-vector multiply  Length and direction. Absolute position not important    Chapters 2.4 (vectors) and 5.2 (matrices) Usually written as or in bold. Magnitude written as a a  Worthwhile to read all of chapters 2 and 5  Use to store offsets, displacements, locations  Should be refresher on very basic material for most of you  But strictly speaking, positions are not vectors and cannot be added:  If you don’t understand, talk to me (review in office hours) a location implicitly involves an origin, while an offset does not. Vector Addition Vector Addition Cartesian Coordinates Cartesian Coordinates A = 4 X + 3 Y a+b = b+a b a X  Geometrically: Parallelogram rule  X and Y can be any (usually orthogonal unit ) vectors  In cartesian coordinates (next), simply add coords   x       T 2 2 A   A x y A x y   y 1

Vector Multiplication Dot (scalar) product Vector Multiplication Dot (scalar) product b  Dot product (2.4.3)  Cross product (2.4.4)   Orthonormal bases and coordinate frames (2.4.5,6) a        a b b a ? a b a b cos         a b ( c ) a b a c   Note: book talks about right and left-handed a b    1   cos   coordinate systems. We always use right-handed      ( ka b ) a kb ( ) k a b ( ) a b   Projections (of b on a) Dot product: some applications in CG Dot product: some applications in CG Projections (of b on a) b  Find angle between two vectors (e.g. cosine of angle between light source and surface for shading)   Finding projection of one vector on another (e.g. a coordinates of point in arbitrary coordinate system)  a b     b a b cos   b a ?  Advantage: computed easily in cartesian components a   b a ?  a a b     b a b a a 2 a a Dot product in Cartesian components Vector Multiplication Vector Multiplication Dot product in Cartesian components  Dot product (2.4.3)     x x     a b a b     ?  Cross product (2.4.4)  y   y  a b  Orthonormal bases and coordinate frames (2.4.5,6)     x x      a b a b     x x y y a b a b  y   y  a b  Note: book talks about right and left-handed coordinate systems. We always use right-handed 2

Cross (vector) product Cross product: Properties Cross (vector) product Cross product: Properties     a b b a       a b a b sin x y z        a b b a y x z b      a a 0 y z x           a ( b c ) a b a c z y x       a ( kb ) k a ( b ) z x y a     Cross product orthogonal to two initial vectors x z y  Direction determined by right-hand rule  Useful in constructing coordinate systems (later) Vector Multiplication Cross product: Cartesian formula? Cross product: Cartesian formula? Vector Multiplication  Dot product (2.4.3)    x y z y z y z a b b a    Cross product (2.4.4)     a b x y z z x x z   a a a a b a b    Orthonormal bases and coordinate frames (2.4.5,6)  x y z  x y y x  b b b a b a b     0 z y x a a b        * a b A b z 0 x y     Note: book talks about right and left-handed a a b     y x 0 z    coordinate systems. We always use right-handed a a b Dual matrix of vector a Orthonormal bases/coordinate frames bases/coordinate frames Coordinate Frames Coordinate Frames Orthonormal  Important for representing points, positions, locations  Any set of 3 vectors (in 3D) so that    u v w 1  Often, many sets of coordinate systems (not just X, Y, Z)     Global, local, world, model, parts of model (head, hands, …)    u v v w u w 0   w u v  Critical issue is transforming between these systems/bases  Topic of next 3 lectures       p ( p u u ) ( p v v ) ( p w w ) 3

Constructing a coordinate frame Constructing a coordinate frame? Constructing a coordinate frame Constructing a coordinate frame?  Often, given a vector a (viewing direction in HW1), want to We want to associate w with a , and v with b  But a and b are neither orthogonal nor unit norm construct an orthonormal basis  And we also need to find u  Need a second vector b (up direction of camera in HW1) a   Construct an orthonormal basis (for instance, camera w a coordinate frame to transform world objects into in HW1)  b w  u  b w   v w u Matrices What is a matrix Matrices What is a matrix  Can be used to transform points (vectors)  Array of numbers (m × n = m rows, n columns)  Translation, rotation, shear, scale (more detail next lecture)   1 3    Section 5.2.1 and 5.2.2 of text 5 2    Instructive to read all of 5 but not that relevant to course    0 4   Addition, multiplication by a scalar simple: element by element Matrix Matrix- -matrix multiplication matrix multiplication Matrix- Matrix -matrix multiplication matrix multiplication  Number of columns in first must = rows in second  Number of columns in first must = rows in second   1 3     1 3 9 2 7 3 3 13    3 6 9 4       3 6 9 4  5 2    5 2   19 44 61 26     2 7 8 3     2 7 8 3       0 4      0 4 8 28 3 2 1 2  Element (i,j) in product is dot product of row i of first  Element (i,j) in product is dot product of row i of first matrix and column j of second matrix matrix and column j of second matrix 4

Matrix- -matrix multiplication matrix multiplication Matrix- -matrix multiplication matrix multiplication Matrix Matrix  Number of columns in first must = rows in second  Number of columns in first must = rows in second         1 3 9 2 7 3 3 13 1 3 9 2 7 3 3 13             3 6 9 4 3 6 9 4       5 2 19 44 61 26 5 2 19 44 61 26               2 7 8 3     2 7 8 3    0 4   8 28 3 2 1 2   0 4   8 28 3 2 1 2   Element (i,j) in product is dot product of row i of first  Element (i,j) in product is dot product of row i of first matrix and column j of second matrix matrix and column j of second matrix Matrix- -matrix multiplication matrix multiplication Matrix- -Vector Multiplication Vector Multiplication Matrix Matrix  Number of columns in first must = rows in second  Key for transforming points (next lecture)  Treat vector as a column matrix (m × 1)   1 3    3 6 9 4   5 2 NOT EVEN LEGAL!!    2 7 8 3   0 4   E.g. 2D reflection about y-axis (from textbook)  Non-commutative (AB and BA are different in general)        1 0 x x        Associative and distributive      0 1 y y  A(B+C) = AB + AC  (A+B)C = AC + BC Transpose of a Matrix (or vector?) Transpose of a Matrix (or vector?) Identity Matrix and Inverses Identity Matrix and Inverses   T 1 0 0   1 2       1 3 5   3 4      I  0 1 0  2 4 6     3 3  5 6     0 0 1   ) T T T ( AB B A     1 1 AA A A I     1 1 1 ( AB ) B A 5

Vector multiplication in Matrix form Vector multiplication in Matrix form  Dot product?   T a b a b   x b          x y z y x x y y z z   a a a b a b a b a b   z   b  Cross product?     0 z y x a a b        * a b A b z 0 x y    a a b      y x 0  z  a a b Dual matrix of vector a 6