Download Note 3 (self study)

Computer Graphics
CSC 630
Lecture 2- Linear Algebra
What is Computer
Graphics?
• Using a computer to generate an
image from a representation.
Model
computer
Image
2
Model Representations
• How do we represent an object?
– Points
– Mathematical Functions
• X2 + Y2 = R2
– Polygons (most commonly used)
• Points
• Connectivity
3
Linear Algebra
• Why study Linear Algebra?
– Deals with the representation and
operations commonly used in Computer
Graphics
4
What is a Matrix?
• A matrix is a set of elements,
organized into rows and columns
m×n matrix
n columns
m rows
a00
a
 10
a01 

a11 
5
What is a Vector?
• Vector: n×1 matrix
a 


v  b 
 c 
y
v
x
6
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
a 


p  b 
 c 
a 


v  b 
 c 
7
Basic Operations
• Transpose: Swap rows with columns
a
M   d
 g
 x
V   y 
 z 
b
e
h
c
f 
i 
V T  x
a d
M T  b e
 c f
y
g
h 
i 
z
8
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
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Parallelogram Rule
• To visualize what a vector addition is
doing, here is a 2D example:
v
v+w
w
10
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
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Vector Spaces
• Consists of a set of elements, called
vectors.
• The set is closed under vector
addition and vector multiplication.
12
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||
13
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
14
Angles and Projection
• Alternative view of the dot product
• v•w=||v|| ||w|| cos() where  is the
angle between v and w
If v and w have length 1…
v
v
w
v·w = 0
v=w
v·w = 1
θ
w
v·w = cos θ
15
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•w
w
u
v
16
Cross Product Properties
• The Cross Product c of v and w is denoted
by vw
• 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.
vw
w
• ||vw||=||v|| ||w|| |sin|
–  is the angle between v and w
• vw=-(wv)
v
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Cross Product
• Given 2 vectors v=[v1 v2 v3], w=[w1 w2
w3], the cross product is defined to
be the determinant of
i
j
v1 v2
w1 w2
k  v2 w3  v3w2 


v3   v3w1  v1w3 
w3  v1w2  v2 w1 
where i,j,k are vectors
18
Matrices
• A compact way of representing operations
on points and vectors
• 3x3 Matrix A looks like
 a11 a12 a13 


 a21 a22 a23 
a

 31 a32 a33 
aij refers to the element of matrix A in ith
row and jth column
19
Matrix Addition
a
c

b e


d  g
f  a  e


h  c  g
b f 

d  h
Just add elements
20
Matrix Multiplication
• If A is an nk matrix and B is a kp
then AB is a np 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
a b   e
c d   g


f  ae  bg


h  ce  dg
af  bh
cf  dh 
Multiply each row by
21
each column
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: ABBA
1 0 0


I  0 1 0 
0 0 221
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
• M T 1  M 1T
23
Determinant of a Matrix
• Defined on a square matrix (nxn)
• Used for inversion
• If det A = 0, then A has no inverse
a b 
A

c
d


det( A)  ad  bc
24
Determinant of a Matrix
• For a nxn matrix,
det A  A  i 1 a1i (1) A1i
n
1i
where A1i determinant of (n-1)x(n-1)
submatrix A gotten by deleting the first
row and the ith column
25
Transformations
• Why use transformations?





Create object in convenient coordinates
Reuse basic shape multiple times
Hierarchical modeling
System independent
Virtual cameras
26
Translation
 x '
 y ' =
 
 z ' 
 x


T (tx, ty, tz )  y 
 z 
=
 x
 y
 
 z 
t x 
+ ty 
 
tz 
 x
 y
 
 z 
t x 
+ ty 
 
tz 
27
Properties of Translation
T (0,0,0) v
= v
T ( sx, sy, sz ) T (tx, ty, tz ) v
= T ( sx  tx, sy  ty, sz  tz ) v
T ( sx, sy, sz ) T (tx, ty, tz ) v
= T (tx, ty, tz ) T ( sx, sy, sz ) v
T 1 (tx, ty, tz ) v
= T (tx,ty ,tz ) v
28
Rotations (2D)
y
x  r cos 
y  r sin 
x' , y '

x, y

x
x'  x cos  y sin 
y '  x sin   y cos
x'  r cos(   )
y '  r sin(    )
cos(   )  cos  cos  sin  sin 
sin(    )  cos  sin   sin  cos
x'  (r cos  ) cos  (r sin  ) sin 
y '  (r cos  ) sin   (r sin  ) cos
29
Rotations 2D
• So in matrix notation
 x   cos
 '   
 y   sin 
'
 sin   x 
 
cos  y 
30
Rotations (3D)
0
1
Rx ( )  0 cos
0 sin 
0 
 sin  
cos 
 cos  0 sin  
Ry ( )   0
1
0 
 sin  0 cos 
cos   sin  0
Rz ( )   sin  cos  0
 0
0
1
31
Properties of Rotations
Ra ( 0)  I
Ra ( ) Ra ( )  Ra (   )
Ra ( ) Ra ( )  Ra ( ) Ra ( )
Ra 1 ( )  Ra( )  Ra T ( )
Ra ( ) Rb( )  Rb( ) Ra ( )
order matters!
32
Combining Translation &
Rotation
T (1,1)
R(45)
R(45)
T (1,1)
33
Combining Translation &
Rotation
v'  v  T
v' '  Rv'
v ' '  R( v  T )
v' '  Rv  RT
v'  Rv
v' '  v'T
v' '  Rv  T
34
Scaling
 x'  sx x 
 y '   sy y 
  

 z '   sz z 
 sx 0 0 
S ( sx, sy, sz )   0 sy 0 
 0 0 sz 
Uniform scaling iff sx  sy  sz
35
Homogeneous
Coordinates
 x
 y
  can be represented as
 z 
where
X
x ,
w
X 
Y 
 
Z 
 
w
Y
Z
y , z
w
w
36
Translation Revisited
1
 x 
0


T (tx, ty, tz )  y   
0
 z  
0
0
1
0
0
0
0
1
0
tx   x 
ty   y 
tz   z 
 
1 1 
37
Rotation & Scaling
Revisited
0
1
 x 
0
cos



Rx( )  y   
0 sin 
 z  
0
0
 sx 0 0
 x 
0 sy 0



S ( sx, sy, sz )  y  
 0 0 sz
 z  
0 0 0
0
 sin 
cos
0
0  x 
0  y 
0  z 
 
1  1 
0  x 
0  y 
0  z 
 
1  1 
38
Combining
Transformations
v '  Sv
v ' '  Rv '  RS v
v ' ' '  Tv ' '  TRv '  TRS v
v ' ' '  Mv
where
M  TRS
39
Transforming Tangents
t  pq
t '  p 'q '
 Mp  Mq
 M (p  q )
 Mt
40
Transforming Normals
nT t  0
n'T t '  0
n'T Mt  0
n'T Mt  nT t
n'T M  nT
M T n'  n
n'  M
T 1
nM
1T
n
41