[ ] a a a ... a vector = offset / displacement 1 2 n - - PowerPoint PPT Presentation

University of British Columbia CPSC 314 Computer Graphics Jan-Apr 2013 Tamara Munzner http://www.ugrad.cs.ubc.ca/~cs314/Vjan2013

Math Review guest lecture: James Gregson Week 1, Wed Jan 2

2

News

• usual lecture order switched
• math review today (Jan 2)
• James Gregson, TA
• usually second lecture
• course intro and overview Friday (Jan 4)
• Tamara Munzner, instructor
• usually first lecture

3

Notation: Scalars, Vectors, Matrices

• scalar
• (lower case, italic)
• vector
• (lower case, bold)
• matrix
• (upper case, bold)

a

[ ]

n

a a a ...

2 1

= a ! ! ! " # $ $ $ % & =

33 32 31 23 22 21 13 12 11

a a a a a a a a a A

4

Vectors

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

5

Column vs. Row Vectors

• row vectors
• column vectors
• switch back and forth with transpose

! ! ! ! " # $ $ $ $ % & =

n col

a a a ...

2 1

a

row T col

a a =

[ ]

n row

a a a ...

2 1

= a

6

Vector-Vector Addition

• add: vector + vector = vector
• parallelogram rule
• tail to head, complete the triangle

! ! ! " # $ $ $ % & + + + = +

3 3 2 2 1 1

v u v u v u v u ) , 6 , 5 ( ) 1 , 1 , 3 ( ) 1 , 5 , 2 ( ) 6 , 9 ( ) 4 , 6 ( ) 2 , 3 ( = − + = + v u + u v

geometric algebraic examples:

7

Vector-Vector Subtraction

• subtract: vector - vector = vector

! ! ! " # $ $ $ % & − − − = −

3 3 2 2 1 1

v u v u v u v u ) 2 , 4 , 1 ( ) 1 , 1 , 3 ( ) 1 , 5 , 2 ( ) 2 , 3 ( ) 4 , 6 ( ) 2 , 3 ( − = − − − − = − = − v u u v v − ) ( v u − +

8

Vector-Vector Subtraction

• subtract: vector - vector = vector

! ! ! " # $ $ $ % & − − − = −

3 3 2 2 1 1

v u v u v u v u ) 2 , 4 , 1 ( ) 1 , 1 , 3 ( ) 1 , 5 , 2 ( ) 2 , 3 ( ) 4 , 6 ( ) 2 , 3 ( − = − − − − = − = − v u u v v − ) ( v u − + v u + v u u v − v u

argument reversal

9

Scalar-Vector Multiplication

• multiply: scalar * vector = vector
• vector is scaled

) * , * , * ( *

3 2 1

u a u a u a a = u ) 5 ,. 5 . 2 , 1 ( ) 1 , 5 , 2 ( * 5 . ) 4 , 6 ( ) 2 , 3 ( * 2 = = u * a u

10

Vector-Vector Multiplication

• multiply: vector * vector = scalar
• dot product, aka inner product

v u•

( ) ( ) ( )

3 3 2 2 1 1 3 2 1 3 2 1

v u v u v u v v v u u u ∗ + ∗ + ∗ = " " " # $ % % % & '

• "

" " # $ % % % & '

11

Vector-Vector Multiplication

• multiply: vector * vector = scalar
• dot product, aka inner product

v u•

( ) ( ) ( )

3 3 2 2 1 1 3 2 1 3 2 1

v u v u v u v v v u u u ∗ + ∗ + ∗ = " " " # $ % % % & '

• "

" " # $ % % % & '

( ) ( ) ( )

3 3 2 2 1 1 3 2 1 3 2 1

v u v u v u v v v u u u ∗ + ∗ + ∗ = " " " # $ % % % & '

• "

" " # $ % % % & '

12

Vector-Vector Multiplication

• multiply: vector * vector = scalar
• dot product, aka inner product

u1 u2 u3 " # $ $ $ % & ' ' '

• v1

v2 v3 " # $ $ $ % & ' ' ' = u1 ∗v1

( ) + u2 ∗v2 ( ) + u3 ∗v3 ( )

v u•

θ cos v u v u =

• u

v

θ

• geometric interpretation
• lengths, angles
• can find angle between two

vectors

13

Dot Product Geometry

• can find length of projection of u onto v
• as lines become perpendicular,

θ cos v u v u =

• u

v

θ

θ cos u

→

• v

u v v u u

• =

θ cos

14

Dot Product Example

19 6 7 6 ) 3 * 2 ( ) 7 * 1 ( ) 1 * 6 ( 3 7 1 2 1 6 = + + = + + = ! ! ! " # $ $ $ % &

• !

! ! " # $ $ $ % & u1 u2 u3 " # $ $ $ % & ' ' '

• v1

v2 v3 " # $ $ $ % & ' ' ' = u1 ∗v1

( ) + u2 ∗v2 ( ) + u3 ∗v3 ( )

15

Vector-Vector Multiplication, The Sequel

• multiply: vector * vector = vector
• cross product
• algebraic

! ! ! " # $ $ $ % & − − − = ! ! ! " # $ $ $ % & × ! ! ! " # $ $ $ % &

1 2 2 1 3 1 1 3 2 3 3 2 3 2 1 3 2 1

v u v u v u v u v u v u v v v u u u

16

Vector-Vector Multiplication, The Sequel

• multiply: vector * vector = vector
• cross product
• algebraic

! ! ! " # $ $ $ % & − − − = ! ! ! " # $ $ $ % & × ! ! ! " # $ $ $ % &

1 2 2 1 3 1 1 3 2 3 3 2 3 2 1 3 2 1

v u v u v u v u v u v u v v v u u u

17

Vector-Vector Multiplication, The Sequel

• multiply: vector * vector = vector
• cross product
• algebraic

! ! ! " # $ $ $ % & − − − = ! ! ! " # $ $ $ % & × ! ! ! " # $ $ $ % &

1 2 2 1 3 1 1 3 2 3 3 2 3 2 1 3 2 1

v u v u v u v u v u v u v v v u u u

blah blah

3 1 2

18

Vector-Vector Multiplication, The Sequel

• multiply: vector * vector = vector
• cross product
• algebraic
• geometric
• parallelogram

area

• perpendicular

to parallelogram

! ! ! " # $ $ $ % & − − − = ! ! ! " # $ $ $ % & × ! ! ! " # $ $ $ % &

1 2 2 1 3 1 1 3 2 3 3 2 3 2 1 3 2 1

v u v u v u v u v u v u v v v u u u

b a× b a×

a×b = a b sinθ

a b

• a × b

θ

19

RHS vs. LHS Coordinate Systems

• right-handed coordinate system
• left-handed coordinate system

x y z x y z

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

y x z × = y x z × =

convention

20

Basis Vectors

• take any two vectors that are linearly

independent (nonzero and nonparallel)

• can use linear combination of these to define

any other vector:

b a c

2 1

w w + =

2a 0.5b c = 2a + 0.5b a c b

21

Orthonormal Basis Vectors

• if basis vectors are orthonormal (orthogonal

(mutually perpendicular) and unit length)

• we have Cartesian coordinate system
• familiar Pythagorean definition of distance

2x 0.5y c = 2 x + . 5 y x y

, 1 =

• =

= y x y x

• rthonormal algebraic properties

22

Basis Vectors and Origins j i

• p

y x + + =

• p

i j

• coordinate system: just basis vectors
• can only specify offset: vectors
• coordinate frame: basis vectors and origin
• can specify location as well as offset: points

23

Working with Frames

p

F1

F1

i j

• j

i

• p

y x + + =

24

Working with Frames

p

F1

F1 p = (3,-1)

i j

• j

i

• p

y x + + =

25

Working with Frames

p

F1

F1 p = (3,-1)

i j

• j

i

• p

y x + + =

F1

26

Working with Frames

p

F1

F1 p = (3,-1) F2

i j

• j

i

• p

y x + + =

F2

i j

• 27

Working with Frames

p

F1

F1 p = (3,-1) F2 p = (-1.5,2)

i j

• j

i

• p

y x + + =

F2

i j

• 28

Working with Frames

p

F1

F1 p = (3,-1) F2 p = (-1.5,2)

i j

F2

i j

• j

i

• p

y x + + =

F2

29

Working with Frames

p

F1

F1 p = (3,-1) F2 p = (-1.5,2) F3

i j

F2

i j

• F3

i j

• j

i

• p

y x + + =

30

Working with Frames

p

F1

F1 p = (3,-1) F2 p = (-1.5,2) F3 p = (1,2)

i j

F2

i j

• F3

i j

• j

i

• p

y x + + =

31

Working with Frames

F1 p = (3,-1) F2 p = (-1.5,2) F3 p = (1,2)

p

F1

i j

F2

i j

• F3

i j

• j

i

• p

y x + + =

F3

32

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, ...

) , , ( c b a = p z y x

• p

c b a + + + =

33

Lines

• slope-intercept form
• y = mx + b
• implicit form
• y – mx – b = 0
• Ax + By + C = 0
• f(x,y) = 0

x y f(x,y)=0 f(x,y) = y - mx - b m = -b/a y=b x=a

34

Implicit Functions

• find where function is 0
• plug in (x,y), check if
• 0: on line
• < 0: inside
• > 0: outside
• analogy: terrain
• sea level: f=0
• altitude: function value
• topo map: equal-value

contours (level sets)

y x f f(x,y)=0 x y f(x,y)=0

35

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 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

2 2 2

) ( ) ( ) , ( r y y x x y x f

c c

− − + − = ) ( ) ( : ) , ( ), , (

2 =

− −

• −

= = r y x c y x p

c c

c p c p

2 2

= − − r c p = − − r c p

36

Parametric Curves

• parameter: index that changes continuously
• (x,y): point on curve
• t: parameter
• vector form
• !

" # $ % & = ! " # $ % & ) ( ) ( t h t g y x ) (t f = p

37

2D Parametric Lines

• start at point p0,

go towards p1, according to parameter t

• p(0) = p0, p(1) = p1

x y " # $ % & ' = x0 + t(x1 − x0) y0 + t(y1 − y0) " # $ % & ' ) ( ) (

1

p p p p − + = t t

p(0.0) p0 p1 p1-p0 p(1.0) p(0.25) p(0.5) p(1.5) p(-0.5) p(-1.0)

) ( ) ( d

• p

t t + =

38

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

) ( ) (

1

p p p p − + = t t

39

Matrix-Matrix Addition

• add: matrix + matrix = matrix
• example

! " # $ % & + + + + = ! " # $ % & + ! " # $ % &

22 22 21 21 12 12 11 11 22 21 12 11 22 21 12 11

m n m n m n m n n n n n m m m m ! " # $ % &− = ! " # $ % & + + + − + = ! " # $ % &− + ! " # $ % & 5 9 8 1 1 4 7 2 5 3 ) 2 ( 1 1 7 5 2 4 2 3 1

40

Scalar-Matrix Multiplication

• multiply: scalar * matrix = matrix
• example

! " # $ % & = ! " # $ % &

22 21 12 11 22 21 12 11

* * * * m a m a m a m a m m m m a ! " # $ % & = ! " # $ % & = ! " # $ % & 15 3 12 6 5 * 3 1 * 3 4 * 3 2 * 3 5 1 4 2 3

41

Matrix-Matrix Multiplication

• can only multiply (n,k) by (k,m):

number of left cols = number of right rows

• legal
• undefined

! ! ! " # $ $ $ % & ! " # $ % & m l k j i h g f e c b a ! " # $ % & ! ! ! " # $ $ $ % & k j i h q g p f

• e

c b a

42

Matrix-Matrix Multiplication

• row by column

! " # $ % & = ! " # $ % & ! " # $ % &

22 21 12 11 22 21 12 11 22 21 12 11

p p p p n n n n m m m m

21 12 11 11 11

n m n m p + =

43

Matrix-Matrix Multiplication

• row by column

! " # $ % & = ! " # $ % & ! " # $ % &

22 21 12 11 22 21 12 11 22 21 12 11

p p p p n n n n m m m m

21 12 11 11 11

n m n m p + =

21 22 11 21 21

n m n m p + =

44

Matrix-Matrix Multiplication

• row by column

! " # $ % & = ! " # $ % & ! " # $ % &

22 21 12 11 22 21 12 11 22 21 12 11

p p p p n n n n m m m m

21 12 11 11 11

n m n m p + =

22 12 12 11 12

n m n m p + =

21 22 11 21 21

n m n m p + =

45

Matrix-Matrix Multiplication

• row by column

! " # $ % & = ! " # $ % & ! " # $ % &

22 21 12 11 22 21 12 11 22 21 12 11

p p p p n n n n m m m m

21 12 11 11 11

n m n m p + =

22 12 12 11 12

n m n m p + =

21 22 11 21 21

n m n m p + =

22 22 12 21 22

n m n m p + =

46

Matrix-Matrix Multiplication

• row by column
• noncommutative: AB != BA

! " # $ % & = ! " # $ % & ! " # $ % &

22 21 12 11 22 21 12 11 22 21 12 11

p p p p n n n n m m m m

21 12 11 11 11

n m n m p + =

22 12 12 11 12

n m n m p + =

21 22 11 21 21

n m n m p + =

22 22 12 21 22

n m n m p + =

47

Matrix-Vector Multiplication

• points as column vectors: postmultiply
• points as row vectors: premultiply

! ! ! ! " # $ $ $ $ % & ! ! ! ! ! " # $ $ $ $ $ % & = ! ! ! ! " # $ $ $ $ % & h z y x m m m m m m m m m m m m m m m m h z y x

44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11

' ' ' '

Mp p'=

[ ] [ ]

T

m m m m m m m m m m m m m m m m h z y x h z y x ! ! ! ! ! " # $ $ $ $ $ % & =

44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11

' ' ' '

T T T

M p p' =

48

Matrices

• transpose
• identity
• inverse
• not all matrices are invertible

! ! ! ! ! " # $ $ $ $ $ % & = ! ! ! ! ! " # $ $ $ $ $ % &

44 34 24 14 43 33 23 13 42 32 22 12 41 31 21 11 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11

m m m m m m m m m m m m m m m m T m m m m m m m m m m m m m m m m ! ! ! ! " # $ $ $ $ % & 1 1 1 1

I AA =

−1

49

Matrices and Linear Systems

• linear system of n equations, n unknowns
• matrix form Ax=b

1 2 5 1 3 4 2 4 2 7 3 = + + − = − − = + + z y x z y x z y x ! ! ! " # $ $ $ % & − = ! ! ! " # $ $ $ % & ! ! ! " # $ $ $ % & − − 1 1 4 1 2 5 3 4 2 2 7 3 z y x

50

Readings For Lecture

• FCG Chapter 2: Miscellaneous Math
• except 2.7 (2.11 in 2nd edition)
• FCG Chapter 5: Linear Algebra
• except 5.4 (not in 2nd edition)