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Graphics
Mathematics for
Computer Graphics
CSE 581 – Roger Crawfis
(slides developed from Korea University slides)
CSE 581 – Interactive Computer Graphics
Spaces

Scalars
 (Linear) Vector Space


Affine Space


Scalars, vectors, and points
Euclidean Space



Scalars and vectors
Scalars, vectors, points
Concept of distance
Projections
CSE 581 – Interactive Computer Graphics
Scalars (1/2)

Scalar Field


Ex) Ordinary real numbers and operations on them
Two Fundamental Operations

Addition and multiplication
 ,   S ,     S ,     S
     
     
            
            
               
Commutative
Associative
Distributive
CSE 581 – Interactive Computer Graphics
Scalars (2/2)

Two Special Scalars

Additive identity: 0, multiplicative identity: 1
  0  0   
 1  1    

1
Additive inverse   and multiplicative inverse 
of 
      0
   1  1
CSE 581 – Interactive Computer Graphics
Vector Spaces (1/4)

Two Entities: Scalars and Vectors
 Vectors




Directed line segments
n-tuples of numbers
Two operations: vector-vector addition, scalarvector multiplication
Special Vector: Zero Vector
u0u
u   u   0

Let u denote a vector
Directed line segments
CSE 581 – Interactive Computer Graphics
Vector Spaces (2/4)

Scalar-Vector Multiplication
 u  v   u  v
   u  u  u


u and v: vectors, α and β: scalars
Scalar-vector multi.
Vector-Vector Addition

Head-to-tail axiom: to visualize easily
Head-to-tail axiom
CSE 581 – Interactive Computer Graphics
Vector Spaces (3/4)

Vectors = n-tuples v  v1 , v2 ,, vn 

Vector-vector addition
u  v  u1 , u2 , , un   v1 , v2 , , vn 
 u1  v1 , u2  v2 , , un  vn 


Scalar-vector multiplication

Vector space: R n
v  v1 , v2 ,, vn 
Linear Independence





Linear combination: u  1u1   2u2     nun
Vectors
are
if the only set of scalars

 linear independent

1u1   2u2     nun  0 is 1   2     n  0
CSE 581 – Interactive Computer Graphics
Vector Spaces (4/4)

Dimension




The greatest number of linearly independent vectors
Basis

n linearly independent vectors (n: dimension)

Unique expression in terms of the basis vectors
Representation  i 




v  1v1   2v2     n vn
Change of Basis: Matrix M

 

Other basis v1 , v2 , , vn




 v  1v1   2v2     nvn
 1 
 1 
  
 
 2   M 2 
  
  
 
 

 n 
 n 
CSE 581 – Interactive Computer Graphics
Affine Spaces (1/2)

No Geometric Concept in Vector Space!!



Ex) location and distance
Vectors: magnitude and direction,
no position
Coordinate System

Origin: a particular reference point
Basis
vectors
located
at the
origin
Identical vectors
Arbitrary
placement
of basis
vectors
CSE 581 – Interactive Computer Graphics
Affine Spaces (2/2)

Third Type of Entity: Points
 New Operation: Point-Point Subtraction

P and Q : any two points

v  P Q
 vector-point addition

P  v Q
P  Q  Q  R  P  R Head-to-tail axiom for points
 

 Frame: a Point P0 and a Set of Vectors v1 , v2 , , vn

Representations of the vector and point: n scalars
Vector
v  1v1   2 v2     n vn
Point
P  P0  1v1   2 v2     n vn
CSE 581 – Interactive Computer Graphics
Euclidean Spaces (1/2)

No Concept of How Far Apart Two Points in
Affine Spaces!!
 New Operation: Inner (dot) Product


Combine two vectors to form a real
α, β, γ, …: scalars, u, v, w, …:vectors
u v  v u
u  v   w  u  w  v  w
v  v  0 if v  0
00  0

Orthogonal: u  v  0
CSE 581 – Interactive Computer Graphics
Euclidean Spaces (2/2)

Magnitude (length) of a vector
v 

Distance between two points
P Q 

vv
P  Q   P  Q 
Measure of the angle between two vectors
u  v  u v cos


cosθ = 0  orthogonal
cosθ = 1  parallel
CSE 581 – Interactive Computer Graphics
Projections

Problem of Finding the Shortest Distance from a
Point to a Line of Plane
 Given Two Vectors,

Divide one into two parts: one
parallel and one orthogonal to the other
w  v  u
w  v  v  v  u  v  v  v
wv
 
vv
wv
 u  w  v  w 
v
vv
Projection of one
vector onto another
CSE 581 – Interactive Computer Graphics
Matrices






Definitions
Matrix Operations
Row and Column Matrices
Rank
Change of Representation
Cross Product
CSE 581 – Interactive Computer Graphics
What is a Matrix?

A matrix is a set of elements, organized into
rows and columns
rows
columns
a b 
c d 


6.837 Linear Algebra Review
CSE 581 – Interactive Computer Graphics
Definitions

n x m Array of Scalars (n Rows and m Columns)



n: row dimension of a matrix, m: column dimension
m = n  square matrix of dimension n
Element aij , i  1, , n, j  1, , m
 
 
A  aij

Transpose: interchanging the rows and columns of a
matrix
T
 
A  a ji
 b1 
 Column Matrices and Row Matrices  
b2 

 Column matrix (n x 1 matrix): b  bi  

 Row matrix (1 x n matrix): b T
 
bn 
CSE 581 – Interactive Computer Graphics
Basic Operations

Addition, Subtraction, Multiplication
a b   e
c d    g

 
f  a  e b  f 


h  c  g d  h 
Just add elements
a b   e
c d    g

 
f  a  e b  f 


h  c  g d  h 
Just subtract elements
a b   e
c d   g


f  ae  bg


h  ce  dg
6.837 Linear Algebra Review
af  bh
cf  dh 
Multiply each
row by each
column
CSE 581 – Interactive Computer Graphics
Matrix Operations (1/2)

Scalar-Matrix Multiplication
A  aij 

Matrix-Matrix Addition
C  A  B  aij  bij 

Matrix-Matrix Multiplication

A: n x l matrix, B: l x m  C: n x m matrix
 
C  AB  cij
l
cij   aik bkj
k 1
CSE 581 – Interactive Computer Graphics
Matrix Operations (2/2)

Properties of Scalar-Matrix Multiplication

Properties of Matrix-Matrix Addition
  A    A
A  A


AB BA
Commutative:
Associative: A  B  C  A  B

 
 C

Properties of Matrix-Matrix Multiplication

Identity Matrix I (Square Matrix)
ABC  AB C
AB  BA
1 if i  j
I  aij , aij  
0 otherwise
 
AI  A
IB  B
CSE 581 – Interactive Computer Graphics
Multiplication

Is AB = BA? Maybe, but maybe not!
a b   e
c d   g



f  ae  bg ...


h   ...
...
e
g

f  a b  ea  fc ...




h   c d   ...
...
Heads up: multiplication is NOT commutative!
6.837 Linear Algebra Review
CSE 581 – Interactive Computer Graphics
Row and Column Matrices



pT: row matrix
Concatenations


 x
p   y 
 z 
Column Matrix
p  Ap
p  ABCp
Associative
By Row Matrix
AB T
 BT A T
pT  pT CT BT AT
CSE 581 – Interactive Computer Graphics
Vector Operations

Vector: 1 x N matrix
 Interpretation: a line
in N dimensional
space
 Dot Product, Cross
Product, and
Magnitude defined on
vectors only
a 
  
v  b 
 c 
y
v
x
6.837 Linear Algebra Review
CSE 581 – Interactive Computer Graphics
Vector Interpretation

Think of a vector as a line in 2D or 3D
 Think of a matrix as a transformation on a line
or set of lines
 x  a b   x' 
 y   c d    y '
 
  
6.837 Linear Algebra Review
V
V’
CSE 581 – Interactive Computer Graphics
Vectors: Dot Product

Interpretation: the dot product measures to what
degree two vectors are aligned
A
B
C
B
A+B = C
(use the head-to-tail
method to combine
vectors)
A
6.837 Linear Algebra Review
CSE 581 – Interactive Computer Graphics
Vectors: Dot Product
d 
a  b  abT  a b c e   ad  be  cf
 f 
a  aa T  aa  bb  cc
2
a  b  a b cos( )
6.837 Linear Algebra Review
Think of the dot
product as a matrix
multiplication
The magnitude is the dot
product of a vector with
itself
The dot product is also
related to the angle between
the two vectors – but it
doesn’t tell us the angle
CSE 581 – Interactive Computer Graphics
Vectors: Cross Product

The cross product of vectors A and B is a vector
C which is perpendicular to A and B
 The magnitude of C is proportional to the cosine
of the angle between A and B
 The direction of C follows the right hand rule –
this why we call it a “right-handed coordinate
system”
a  b  a b sin(  )
6.837 Linear Algebra Review
CSE 581 – Interactive Computer Graphics
Inverse of a Matrix

Identity matrix:
AI = A
 Some matrices have
an inverse, such that:
AA-1 = I
 Inversion is tricky:
(ABC)-1 = C-1B-1A-1
Derived from noncommutativity property
6.837 Linear Algebra Review
1 0 0


I  0 1 0 
0 0 1
CSE 581 – Interactive Computer Graphics
Determinant of a Matrix

Used for inversion
 If det(A) = 0, then A
has no inverse
 Can be found using
factorials, pivots, and
cofactors!
 Lots of interpretations
– for more info, take
18.06
6.837 Linear Algebra Review
a b 
A

c
d


det( A)  ad  bc
1  d  b
A 
ad  bc  c a 
1
CSE 581 – Interactive Computer Graphics
Determinant of a Matrix
a b
d e
g h
c
f  aei  bfg  cdh  afh  bdi  ceg
i
a b
d e
g h
c a b
f d e
i g h
c a b
f d e
i g h
6.837 Linear Algebra Review
c
f
i
Sum from left to right
Subtract from right to lef
Note: N! terms
CSE 581 – Interactive Computer Graphics
Change of Representation (1/2)

Matrix Representation of the Change between
the Two Bases

Ex) two bases u1 , u2 , , un  and v1 , v2 ,, vn 
v  1u1   2u2     nun
or
v  1v1   2v2     n vn

Representations of v
a  1  2   n 
T

or
b  1  2   n 
T
Expression of u1 , u2 , , un  in the basis v1 , v2 ,, vn 
ui   i1v1   i 2v2     invn ,
i  1,, n
CSE 581 – Interactive Computer Graphics
Change of Representation (2/2)
 


 u1 
 v1 
A 
:
n
x
n
matrix
ij
u 
v 
 2   A 2 
 
 
 
 
un 
v n 
 u1 
a 
and
b


i
i
u 
v  aT  2  or
 
 
un 
 By direct substitution
 v1 
 v1 
 u1 
 v1 
v 
v 
u 
v 
aT A  2   b T  2 
aT  2   bT  2 
 
 
 
 
 
 
 
 
 vn 
v n 
u n 
vn 
 
 
 v1 
v 
v  bT  2 
 
 
 vn 
 b T  aT A
CSE 581 – Interactive Computer Graphics
Cross Product

In 3D Space, a unit Vector, w, is Orthogonal to
Given Two Nonparallel Vectors, u and v
wu  w v  0

Definition

Consistent Orientation
 2  3   3  2 
w  u  v    3 1  1 3 
1 2   2 1 
where u  1 ,  2 ,  3 , v  1 ,  2 , 3 

Ex) x-axis x y-axis = z-axis
CSE 581 – Interactive Computer Graphics