Computer GraphicsComputer GraphicsCSC 630CSC 630

Computer GraphicsComputer GraphicsCSC 630CSC 630

Lecture 2- Linear AlgebraLecture 2- Linear Algebra

2

What is Computer Graphics?

• Using a computer to generate an image from a representation.

Model Imagecomputer

3

Model Representations• How do we represent an object?

– Points – Mathematical Functions

• X2 + Y2 = R2

– Polygons (most commonly used)• Points• Connectivity

4

Linear Algebra• Why study Linear Algebra?

– Deals with the representation and operations commonly used in Computer Graphics

5

What is a Matrix?• A matrix is a set of elements,

organized into rows and columns

1110

0100

aa

aa

n columns

m rows

m×n matrix

6

What is a Vector?• Vector: n×1

matrix

c

b

a

v

x

y

v

7

Representing Points and Vectors

• A 3D point– Represents a

location with respect to some coordinate system

• A 3D vector – Represents a

displacement from a position

c

b

a

p

c

b

a

v

8

Basic Operations• Transpose: Swap rows with columns

ihg

fed

cba

M

ifc

heb

gda

M T

z

y

x

V zyxV T

9

Vector Addition• Given v = [x y z]T and w = [a b c]T

v + w = [x+a y+b z+c]T

• Properties of Vector addition– Commutative: v + w = w + v– Associative (u + v) + w = u + (v + w)– Additive Identity: v + 0 = v– Additive Inverse: v + (-v) = 0

10

Parallelogram Rule• To visualize what a vector addition

is doing, here is a 2D example:

vw

v+w

11

Vector Multiplication• Given v = [x y z]T and a Scalar s and t

– sv = [sx sy sz]T and tv = [tx ty tz]T

• Properties of Vector multiplication– Associative: (st)v = s(tv)– Multiplicative Identity: 1v = v– Scalar Distribution: (s+t)v = sv+tv– Vector Distribution: s (v+w) = sv+sw

12

Vector Spaces• Consists of a set of elements,

called vectors.• The set is closed under vector

addition and vector multiplication.

13

Dot Product and Distances

• Given u = [x y z] T and v = [a b c] T

– v•u = vTu = ax+by+cz

• The Euclidean distance of u from the origin is denoted by ||u|| and called norm or length– ||u|| = sqrt(x2+y2+z2)– Notice that ||u|| = sqrt(u •u)– Unit vector ||u|| = 1, zero vector denoted 0

• The Euclidean distance between u and v is sqrt((x-a) 2+(y-b) 2+(z-c) 2)) and is denoted by ||u-v||

14

Properties of the Dot Product

• Given a vector u, v, w and scalar s– The result of a dot product is a

SCALAR value– Commutative: v•w = w•v– Non-degenerate: v•v=0 only when

v=0

15

Angles and Projection• Alternative view of the dot product• v•w=||v|| ||w|| cos() where is

the angle between v and w

v

w

v

w v=w

If v and w have length 1…

v·w = 0 v·w = 1 v·w = cos θ

θ

16

Angles and Projection• If v is a unit vector (||v|| = 1) then

if we perpendicularly project w onto v can call this newly projected vector u then

||u|| = v•ww

vu

17

Cross Product Properties

• The Cross Product c of v and w is denoted by vw

• c Is a VECTOR, perpendicular to the plane defined by v and w

• The magnitude of c is proportional to the sin of the angle between v and w

• The direction of c follows the right hand rule.• ||vw||=||v|| ||w|| |sin|

is the angle between v and w

• vw=-(wv)

v

wvw

18

Cross Product• Given 2 vectors v=[v1 v2 v3], w=[w1 w2 w3],

the cross product is defined to be the determinant of

where i,j,k are vectors

1221

3113

2332

321

321

wvwv

wvwv

wvwv

www

vvv

kji

19

Matrices• A compact way of representing operations on

points and vectors• 3x3 Matrix A looks like

aij refers to the element of matrix A in ith row and jth column

333231

232221

131211

aaa

aaa

aaa

20

Matrix Addition

hdgc

fbea

hg

fe

dc

ba

Just add elements

21

Matrix Multiplication• If A is an nk matrix and B is a kp

then AB is a np matrix with entries cij

where cij= aisbsj

• Alternatively, if we took the rows of A and columns of B as individual vectors then cij = Ai•Bj where the subscript refers to the row and column, respectively

dhcfdgce

bhafbgae

hg

fe

dc

ba Multiply each row by each column

22

Matrix Multiplication Properties

• Associative: (AB)C = A(BC)• Distributive: A(B+C) = AB+AC• Multiplicative Identity: I= diag(1)

(defined only for square matrix) • Identity matrix: AI = A• NOT commutative: ABBA

100

010

001

I

23

Matrix Inverse• If A and B are nxn matrices and

AB=BA=I then B is the inverse of A, denoted by A-1

• (AB)-1=B-1A-1 same applies for transpose

• T

MM T 11

24

Determinant of a Matrix

• Defined on a square matrix (nxn)• Used for inversion

• If det A = 0, then A has no inverse

dc

baA

bcadA )det(

25

Determinant of a Matrix

• For a nxn matrix,

where A1i determinant of (n-1)x(n-1) submatrix A gotten by deleting the first row and the ith column

n

i ii

i AaAA1 1

11 )1(det

26

Transformations• Why use transformations?

Create object in convenient coordinates

Reuse basic shape multiple times Hierarchical modeling System independent Virtual cameras

27

Translation

z

y

x

z

y

x

t

t

t

= +

z

y

x

tttT zyx ),,(

z

y

x

z

y

x

t

t

t

= +

'

'

'

z

y

x

28

Properties of Translation

v=v)0,0,0(T

=v),,(),,( zyxzyx tttTsssT

=

=v),,(1zyx tttT

v),,(),,( zyxzyx tttTsssT v),,(),,( zyxzyx sssTtttT

v),,( zzyyxx tststsT

v),,( zyx tttT

29

Rotations (2D)

sin

cos

ry

rx

cos)sin(sin)cos('

sin)sin(cos)cos('

rry

rrx

)sin('

)cos('

ry

rx

cossinsincos)sin(

sinsincoscos)cos(

cossin'

sincos'

yxy

yxx

yx,

',' yx

x

y

30

Rotations 2D• So in matrix notation

y

x

y

x

cossin

sincos'

'

31

Rotations (3D)

100

0cossin

0sincos

)(

cos0sin

010

sin0cos

)(

cossin0

sincos0

001

)(

z

y

x

R

R

R

32

Properties of RotationsIRa )0(

)()()()( aaaa RRRR

)()()( aaa RRR

)()()(1 Taaa RRR

)()()()( abba RRRR order matters!

33

Combining Translation & Rotation

)1,1(T

)45( R

)45( R

)1,1(T

34

Combining Translation & Rotation

Tvv'

RTR

TR

R

vv

vv

vv

''

)(''

'''

vv R'

TR

T

vv

vv

''

'''

35

Scaling

zs

ys

xs

z

y

x

z

y

x

'

'

'

z

y

x

zyx

s

s

s

sssS

00

00

00

),,(

Uniform scaling iff zyx sss

36

Homogeneous Coordinates

z

y

x

w

Z

Y

X

can be represented as

wherew

Zz

w

Yy

w

Xx ,,

37

Translation Revisited

11000

100

010

001

),,(z

y

x

t

t

t

z

y

x

tttTz

y

x

zyx

38

Rotation & Scaling Revisited

11000

000

000

000

),,(z

y

x

s

s

s

z

y

x

sssSz

y

x

zyx

11000

0cossin0

0sincos0

0001

)(z

y

x

z

y

x

Rx

39

Combining Transformations

vv

vvvv

vvv

vv

M

TRSTRT

RSR

S

'''

''''''

'''

'

where TRSM

40

Transforming Tangents

t

qp

qp

qpt

qpt

M

M

MM

)(

'''

41

Transforming Normals

nnn

nn

nn

tntn

tn

tn

tn

TT

T

TT

TT

T

T

T

MM

M

M

M

M

11'

'

'

'

0'

0''

0