CPCS 391 Computer Graphics 1

CPCS 391

Computer Graphics 1

Instructor: Dr. Sahar Shabanah Computer Graphics Math Review

Scalars

• Scalar Field

– Ex) Ordinary real numbers and operations on them

• Two Fundamental Operations

– Addition and Multiplication

– Commutative

– Associative

– Distributive

S S

S    



  ,  ,   ,  

a + b = b + a a × b = b × a

a + ( b + g ) = ( a + b ) + g

a × ( b × g ) = ( a × b ) × g

a × ( b + g ) = ( a × b ) + ( a × g )

Scalars

• Two Special Scalars

– Additive identity: 0,

– multiplicative identity: 1 – Additive inverse and – Multiplicative inverse

a + 0 = 0 + a = a





1





a + - ( ) a = 0

a × 1 = 1 × a = a

a × a

^-1

= 1

Vectors

• A vector is a directed line segment:

– starts at initial point (A) and – ends at terminal point (B)

• Components of v

• In general:

– v = n-tuples

• Special Vectors:

– Zero Vector

– Negative Vector

4

v = AB

 

⁰

0











u u

u u



^{v v v}_n



v  ₁, ₂,,

v = (x₂ - x₁,y₂ - y₁) =

(

v₁,v₂

)

^A(X1,Y1)

B(X2,Y2)

v

• Scalar-Vector Multiplication

u and v: vectors, α and β: scalars

Vectors Operations

a (

^u + v

)

⁼

a

^{u +}

a

^v

a

+

b

( )

^{u =}

a

^{u +}

b

^u

 v v v

_n



v   

 

₁

,

₂

,  ,

Vectors Operations

• Magnitude (length)

• Vector-Vector Addition

– Head-to-tail axiom:

– u+v=(u tial, v head)

   

 u u u v u u

ⁿ

v v u v

_n

v

_n

v 

ⁿ

v u















, ,

,

, ,

, ,

, ,

2 2

1 1

2 1 2

1







v = v

₁²

+ v

₂²

+ .. + v

_n²

Vectors Operations

• Vector-Vector Difference

u - v = ( u

₁

, u

₂

, … , u

_n

) - ( v

¹

, v

²

,… , v

ⁿ

)

= ( u

₁

- v

₁

, u

₂

- v

₂

,… , u

_n

- v

_n

)

V

U

U-V

Vectors Operations

• dot Product:

– combine two vectors to form a real number

– measure of the angle between two vectors, to what degree those two vectors are aligned

» cosθ = 1  parallel

» cosθ = 0  orthogonal

n nv u v

u v

u v

u

or v

u v

u











...

.

, cos .

2 2 1

1



v u

v cos  u.

• The cross product of vectors u and v is a vector uxv which is perpendicular to u and v

• The magnitude of uxv is proportional to the cosine of the angle between u and v

• The direction of uxv follows the right hand rule

Vectors: Cross Product

u ´ v = (u

₂

v

₃

- u

₃

v

₂

, u

₃

v

₁

- u

₁

v

₃

, u

₁

v

₂

- u

₂

v

₁

) or

u ´ v = u v sin q

Affine Space

• In 3D Space, a unit Vector, e3 , is Orthogonal to Given Two Nonparallel Vectors, e1 and e2

• Definition

• Consistent Orientation

– Ex) x-axis x y-axis = z-axis

0 2

3 1

3  e  e  e  e

 







 



















1 2 2

1

3 1 1

3

2 3 3

2

2 1

3

























e e

e



₁

,

₂

,

₃

 , 2 

₁

,

₂

,

₃



1

where e     e    

Affine Spaces

 Coordinate System

 Origin: a particular reference point

Arbitrary placement

of basis vectors Basis

vectors located at the origin

What is a Matrix?

• A matrix is a set of elements, organized into rows and columns

rows

columns

 

 





d c

b

a

Definitions

• dimension of a matrix

– n x m array of scalars (n Rows and m Columns)

• square matrix

– m = n  square matrix of dimension n

• Element

• Transpose:

– interchanging the rows and columns of a matrix

• Column and Row Matrices:

– Column matrix (n x 1 matrix):

– Row matrix (1 x n matrix):

  a

_ij

, i  1 ,  , n , j  1 ,  , m

 

aij



A

 

ji

T  a

A

 





















n i

b b b

b 

2 1

b

Basic Operations

• Addition

• Subtraction

• Multiplication



 











 



 



 



 





h d

g c

f b

e a

h g

f e

d c

b a



 











 



 



 



 





h d

g c

f b

e a

h g

f e

d c

b a



 











 



 







 





dh cf

dg ce

bh af

bg ae

h g

f e

d c

b a

Just add elements

Just subtract elements

Multiply each row by each column

Matrix Operations

• Scalar-Matrix Multiplication

• Matrix-Matrix Addition

• Matrix-Matrix Multiplication

– A: n x l matrix, – B: l x m 

– C: n x m matrix

   a

ij

 A 

 a

ij

 b

ij







 A B C

 











l

k

kj ik ij

ij

b a c

c

1

AB

C

Properties of Matrix Operations

• Scalar-Matrix Multiplication

• Matrix-Matrix Addition – Commutative:

– Associative:

• Identity Matrix I (Square Matrix)

   

A A

A A















A B

B

A   



^{B C}

 

^{A B}



^C

A     

 

 



 0 otherwise

if 1

, i j

a a_{ij ij}

I IB B

A AI





Matrix Multiplication:

• Associative:

• Is AB = BA? Maybe, but maybe not!

• multiplication is NOT commutative!

Properties of Matrix Operations

 

 



  

 

 



 



 





...

...

...

bg ae

h g

f e

d c

b

 a



 



  

 

 



 



 





...

...

...

fc ea

d c

b a

h g

f e

A(BC)=(AB)C

Row and Column Matrices

• Column Matrix

– p

^T

: row matrix

• Concatenations

– Associative

• By Row Matrix

 







 









z y x p

ABCp p

Ap p

 

 

 

T T

T T

T

T T T

A B

C p

p

A B

AB

 



 X Y Z 

T



p

Inverse of a Matrix

• Identity matrix:

AI = A

• Some matrices have an inverse:

AA

^-1

= I

• Inversion is tricky:

(ABC)

^-1

= C

^-1

B

^-1

A

^-1

Derived from non-commutativity property



















1 0

0

0 1

0

0 0

1 I

• Used for inversion

• If det(A) = 0, then A has no inverse

Determinant of a Matrix

 

 



 

d c

b

A a det( A )  ad  bc



 









 



a c

b d

bc A ₁ ad 1

Sum from left to right

Subtract from right to left Note: N! terms

Determinant of a Matrix

ceg bdi

afh cdh

bfg aei

i h

g

f e

d

c b

a













i h

g

f e

d

c b

a

i h

g

f e

d

c b

a

i h

g

f e

d

c b

a