Post on 16-Jan-2016
1
Modeling Transformations
Md. Tanvir Al Amin*Lecturer, Dept. of CSE, BUET
tanviralamin@gmail.com
CSE 409
*Special Thanks to Tanvir Parvez, Fredo Durand, James O Brien, Tomas Lozano-Perez, Jovan Popovic
Modeling Transformation - Class 1
2
Transformation Background
What is Transformation ?
3
What is a Transformation
Transformation:– An operation that changes one configuration
into another
For images, shapes, etc.– A geometric transformation maps positions that
define the object to other positions– Linear transformation means the transformation
is defined by a linear function... which is what matrices are good for.
4
5
What is a Transformation?
A function that maps points x to points x':Applications: animation, deformation, viewing,
projection, real-time shadows, …
6
Simple Transformations
Can be combinedAre these operations invertible?Yes, except scale = 0
7
Rigid-Body / Euclidean Transforms
Preserves distancesPreserves angles
TranslationRotation
Rigid / Euclidean
Identity
8
Similitudes / Similarity Transforms
Preserves angles
TranslationRotation
Rigid / Euclidean
Similitudes
Isotropic ScalingIdentity
9
Linear Transformations
TranslationRotation
Rigid / EuclideanLinear
Similitudes
Isotropic ScalingIdentity
Scaling
Shear
Reflection
10
Linear Transformations
L(p + q) = L(p) + L(q)L(ap) = a L(p)
TranslationRotation
Rigid / EuclideanLinear
Similitudes
Isotropic Scaling
Scaling
Shear
ReflectionIdentity
Translation is not linear because ap+t ≠ a(p+t)
11
Affine Transformations
preserves parallel lines
TranslationRotation
Rigid / EuclideanLinear
Similitudes
Isotropic Scaling
Scaling
Shear
ReflectionIdentity
Affine
6.837 Fall 06 – Durand 12
Projective Transformations
preserves lines
TranslationRotation
Rigid / EuclideanLinear
Affine
Projective
Similitudes
Isotropic Scaling
Scaling
Shear
Reflection
Perspective
Identity
13
Groups
Properties of a group (element, operation):– There exists a neutral element– There exists an inverse for each member– The elements are "closed under composition"– The composition operation is associative
Example: integers under addition0 is the neutral element-x is the inverse of xx+y is an integer(x+y)+z=x+(y+z)
14
Groups and Transformations
Properties of a group:– There exists an identity mapping– There exists an inverse mapping for each function– The functions are "closed under composition"– The composition operation is associative
These properties might seem trivial at first glance, but they are actually very important, because when these conditions are shown for any class of functions and their two-argument composition operation, then they form an algebraic group. – One of the consequences is that any series of translations can
be composed to a single translation. – Another consequence is that the inverse is unique.
15
How are Transforms Represented?
x' = ax + by + c
y' = dx + ey + f
x'
y'
a b
d e
c
f=
x
y+
p' = M p + t
16
Translation in homogenous coordinates
x' = ax + by + c
y' = dx + ey + f
x'
y‘
1
a b
d e
0 0
c
f
1
=x
y
1
p' = M p
x'
y'
a b
d e
c
f=
x
y+
p' = M p + t
Affine formulation Homogeneous formulation
17
Homogeneous Co-ordinates
Translation, scaling and rotation are expressed (non-homogeneously) as:– translation: P = P + T– Scale: P = S · P– Rotate: P = R · P
Composition is difficult to express, since translation not expressed as a matrix multiplication
Homogeneous coordinates allow all three to be expressed homogeneously, using multiplication by 3 matrices
W is 1 for affine transformations in graphics
18
Homogeneous Coordinates
Add an extra dimension• in 2D, we use 3 x 3 matrices
• In 3D, we use 4 x 4 matrices
Each point has an extra value, wx'
y'
z'
w'
=
x
y
z
w
a
e
i
m
b
f
j
n
c
g
k
o
d
h
l
p
p' = M p
19
Homogeneous Coordinates
Most of the time w = 1, and we can ignore it
If we multiply a homogeneous coordinate by an affine matrix, w is unchanged
x'
y'
z'
1
=
x
y
z
1
a
e
i
0
b
f
j
0
c
g
k
0
d
h
l
1
20
Homogeneous Co-ordinates
P2d is a projection of Ph onto the w = 1 plane
So an infinite number of points correspond to : they constitute the whole line (tx, ty, tw)
x
y
w Ph(x,y,w)
P2d(x,y,1)
w=1
Homogeneous Visualization
Mechanics of Rigid Transformations
Translate
Rotate
Scale
22
23
Translation – 2D
(4,5) (7,5)
Y
XBefore Translation
1
*
100
10
01
1
y
x
d
d
y
x
TPPd
dT
y
xP
y
xP
y
x
y
x
Form sHomogeniou
x’ = x + dx y’ = y + dy
(7,1) (10,1)
X
Y
Translation by (3,-4)
24
Translation – 3D
PPdddT
dz
dy
dx
z
y
x
d
d
d
zyx
z
y
x
z
y
x
*),,(
11
*
1000
100
010
001
z
y
x
dzz
dyy
dxx
25
Scaling – 2D
(4,5) (7,5)
Y
X(2,5/4) (7/2,5/4)
X
Y
Before Scaling Scaling by (1/2, 1/4)
y
x
y
x
y
x
sy
sx
y
x
s
s
PPS
ysy
xsx
*
**
0
0
*
*
*
Types of Scaling:
Differential ( sx != sy )Uniform ( sx = sy )
1
*
100
00
00
1
Form sHomogeniou
y
x
s
s
y
x
y
x
26
Scaling – 3D
1
*
*
*
1
*
1000
000
000
000
*),,(
z
y
x
z
y
x
zyx
sz
sy
sx
z
y
x
s
s
s
PPsssS
Original
scale all axes
scale Y axiszsz
ysy
xsx
z
y
x
*
*
*
Rotation – 2D
sin
cos
r
rv
rota
ted
cossinsincos
sinsincoscos expand
rry
rrx
cossin
sincos
sin
cosbut
yxy
yxx
ry
rx
original
sin
cos
r
rv
28
Rotation – 2D
(5,2) (9,2)
Y
X
(2.1,4.9)
(4.9,7.8)
X
YBefore Rotation Rotation of 45 deg. w.r.t. origin
1
*
100
0cossin
0sincos
1
Form sHomogeniou
y
x
y
x
cos*sin*
sin*cos**
cossin
sincos
*
yx
yx
y
x
PPR
yyx
xyx
cos*sin*
sin*cos*
Modeling Transformation - Class 2
29
30
Rotation – 3D
1
cos*sin*
sin*cos*
1
*
1000
0100
00cossin
00sincos
*,
z
yx
yx
z
y
x
PPR k
For 3D-Rotation 2 parameters are needed
Angle of rotation
Axis of rotation
Rotation about z-axis:
31
Rotation about Y-axis & X-axis
1
cos*sin*
sin*cos*
1
*
1000
0cos0sin
0010
0sin0cos
*,
zx
y
zx
z
y
x
PPR j
1
cos*sin*
sin*cos*
1
*
1000
0cossin0
0sincos0
0001
*,
zy
zy
x
z
y
x
PPR iAbout x-axis
About y-axis
32
Rotation about Z axis
About z axis
x'
y'
z'
1
=
x
y
z
1
cos θ
sin θ
0
0
-sin θ
cos θ
0
0
0
0
1
0
0
0
0
1
ZRotate(θ)
x
y
z
p
p'
θ
Rotation
How to rotate around (kx, ky, kz), a unit vector on an arbitrary axis …Example : Rotate 30 degree around vector
2i+1j+3k For now, solution is Rodrigues Formula– Can it be found from some rotation around x
axis, then some rotation around y axis, then z axis ?
– We will munch it later ….
33
34
Rotation
About (kx, ky, kz), a unit vector on an arbitrary axis(Rodrigues Formula)
x'
y'
z'
1
=
x
y
z
1
kxkx(1-c)+c
kykx(1-c)+kzs
kzkx(1-c)-kys
0
0
0
0
1
kykx(1-c)-kzs
kyky(1-c)+c
kzky(1-c)+kxs
0
kxkz(1-c)+kys
kykz(1-c)-kxs
kzkz(1-c)+c
0
Rotate(k, θ)
x
y
z
θ
k
We will return to this case of arbitrary rotation later …
where c = cos θ & s = sin θ
Non Rigid Transforms
35
Non Rigid Transforms
We just studied Translation, Scale, Rotate All these are Rigid Transformations Is Shear rigid ?
– No– Shear is Linear Transform
However, Translate, Scale, Rotate, Shear, Reflect all are member of general class “Affine Transformation”
36
37
Mirror Reflection
100
010
001
axis-Xabout Reflection
xM
yyxx
100
010
001
axis-Yabout Reflection
yM
yyxx
(1,1)
(1,-1)
Y
X
(-1,1) (1,1)
X
Y
38
Shearing Transformation
100
01
01
100
01
001
100
010
01
b
a
SHbSH
a
SH xyyx
unit cubeSheared in X
directionSheared in Y
directionSheared in both X
and Y direction
39
Shear along Z-axis
1
*
*
1
*
1000
0100
010
001
*),(
z
shzy
shzx
z
y
x
sh
sh
PPshshSH
y
x
y
x
yxxy
y
x
z
Inverse Transforms
40
41
Inverse Transformations
y-y
x-x
),(-(sx,sy)
T(-θ
-(θ
(-dx,-dy)-
(dx,dy)
MM
MM
SS
RRR
TT
sysx
1
1
1
)()1)
1
:RefMirror
: Sclaing
: Rotation
:nTranslaito
11
Composing Transformations
42
43
How are transforms combined?
(0,0)
(1,1)(2,2)
(0,0)
(5,3)
(3,1)Scale(2,2) Translate(3,1)
TS =2
0
0
2
0
0
1
0
0
1
3
1
2
0
0
2
3
1=
Scale then Translate
Use matrix multiplication: p' = T ( S p ) = TS p
Caution: matrix multiplication is NOT commutative!
0 0 1 0 0 1 0 0 1
44
Non-commutative Composition
Scale then Translate: p' = T ( S p ) = TS p
Translate then Scale: p' = S ( T p ) = ST p
(0,0)
(1,1)(4,2)
(3,1)
(8,4)
(6,2)
(0,0)
(1,1)(2,2)
(0,0)
(5,3)
(3,1)Scale(2,2) Translate(3,1)
Translate(3,1) Scale(2,2)
45
TS =
2
0
0
0
2
0
0
0
1
1
0
0
0
1
0
3
1
1
ST =
2
0
0
2
0
0
1
0
0
1
3
1
Non-commutative Composition
Scale then Translate: p' = T ( S p ) = TS p2
0
0
0
2
0
3
1
1
2
0
0
2
6
2
=
=
Translate then Scale: p' = S ( T p ) = ST p
0 0 1 0 0 1 0 0 1
46
Combining Translations, Rotations
Order matters!! TR is not the same as RT (demo)General form for rigid body transformsWe show rotation first, then translation
(commonly used to position objects) on next slide. Slide after that works it out the other way
47
Rotate then Translate
' ( )P TR P MP RP T
11 12 13 11 12 13
21 22 23 21 22 23
31 32 33 31 32 33
1 0 0 0
0 1 0 0
0 0 1 0 0 1
0 0 0 1 0 0 0 1 0 0 0 1
x x
y y
z z
T R R R R R R T
T R R R R R R T R TM
T R R R R R R T
48
Translate then Rotate
' ( ) ( )P RT P MP R P T RP RT
11 12 13
3 3 3 3 3 121 22 23
1 331 32 33
0 1 0 0
0 0 1 0
0 10 0 0 1
0 0 0 1 0 0 0 1
x
y
z
R R R T
R R TR R R TM
R R R T
49
Classification of Transformations Again
1. Rigid-body Transformation Preserves parallelism of lines Preserves angle and length e.g. any sequence of R() and T(dx,dy)
2. Affine Transformation Preserves parallelism of lines Doesn’t preserve angle and length e.g. any sequence of R(), S(sx,sy) and T(dx,dy)
unit cube 45 deg rotaton Scale in X not in Y
Example Affine Transforms
50
51
Properties of rigid-body transformation
1002221
1211
y
x
trr
trr
The following Matrix is Orthogonal if the upper left 2X2 matrix has the following properties
1.A) Each row are unit vector. sqrt(r11* r11 + r12* r12) = 1
B) Each column are unit vector. sqrt(c11* c11 + c12* c12) = 1
2.A) Rows will be perpendicular to each other (r11 , r12 ) . ( r21 , r22) = 0
B) Columns will be perpendicular to each other (c11 , c12 ) . (c21 ,c22) = 0
e.g. Rotation matrix is orthogonal
100
0cossin
0sincos
• Orthogonal Transformation Rigid-Body Transformation
• For any orthogonal matrix B B-1 = BT
52
Associativity of Matrix Multiplication
Create new affine transformations by multiplying sequences of the above basic transformations.
q = CBAp
q = ( (CB) A) p = (C (B A))p = C (B (Ap) ) etc.
matrix multiplication is associative.
But to transform many points, best to do
M = CBA
then do q = Mp for any point p to be rendered.
To transform just a point, better to do q = C(B(Ap))
For geometric pipeline transformation, define M and set it up with the model-view matrix and apply it to any vertex subsequently defined to its setting.
Example Composite Transforms
53
2D Case
54
Rotation of about P(h,k): R,P
R,P=
Q(x,y)
P(h,k)
Step 1: Translate P(h,k) to origin
T(-h ,-k)
Q1(x’,y’
)
P1 (0,0)
Step 2: Rotate w.r.t to origin
R*
Q2(x’,y’
)
P2 (0,0)
Step 3: Translate (0,0) to P(h,k0)
T(h ,k) *P3(h,k)
Q3(x’+h, y’
+k)
55
Reflection about line L, ML
Step 1: Translate (0,b) to origin
T(0 ,-b)ML =
Step 2: Rotate - degrees
Step 3: Mirror reflect about X-axis
R(-) *T(0 ,b) *
Step 4: Rotate degrees
Step 5: Translate origin to (0,b)
M x*R() *
(0,b)
Y
X
t
O
Y
XO
Y
XO
Y
XO
Y
XO
(0,b)
Y
X
t
O
Scale about Arbitrary Axis
56
Scale about arbitrary axis
57
Scale about arbitrary axis
58
Scale about arbitrary axis
59
Scale about arbitrary axis
60
Shear
Shears are not really primitive transformsRelated to non-axis-aligned scales Express shear in terms Euclidean
transformation and non uniform scale– Its your homework – Hints on next slide– ??
61
Singular Value Decomposition
62
Decomposing Matrices
63
64
Problems to be solved:
Schaum’s outline series:
Problems: 4.1 4.2
4.3, 4.4, 4.5 => R,P 4.6, 4.7, 4.8 => Ssx,sy,,P
4.9, 4.10, 4.11, 4.21 => ML
4.12 => Shearing Pg-281(1.24), Pg-320(5.19)
=> Circular view-port
Modeling Transformation - Class 3, 4
65
Exploring rotations in 3D
66
Recap : Rotation
67
Recap : Rotations
68
Recap : Axis Aligned Rotations
69
Recap : Rotation around Arbitrary Axisa=(ax,ay,az) is a normalized vector
A point x is to be rotated by angle theta around this vectorFind the Rotation Matrix
70
Arsenal
That old rotation problem, for what we will check at least three solutions– Euler Theorem (Old school thing)
– Rodrigues Formula (The beautiful Construction)
– Quaternion (Latest hero in the town)
71
Euler Theorem
That old school thing ….
72
Euler Angles
Before considering our composite rotation around a rotation axis which ultimately will be called Euler Axis, we examine another way of rotating a body from one orientation to another orientation ….
These are the Euler Angles
73
Special note : Don’t confuse Euler theorem and Euler Angles. Euler angles are not concerned with our axis of composite rotation. Rather they specify a way to orient a co-ordinate system with another by means of defined rotationsEuler theorem is concerned with our rotation axis, which we are going to study once we are done with Euler angles.
Euler Angles
Euler angles a means of representing the
spatial orientation of any frame of the space as a
composition of rotations from a reference frame.
The fixed system is denoted in lower case (x,y,z)
and the rotated system in upper case letters
(X,Y,Z).
Line of nodes (N) The intersection of the xy
and the XY coordinate planes.
• α is the angle between the x-axis and the line of nodes.
• β is the angle between the z-axis and the Z-axis. • γ is the angle between the line of nodes and the X-axis.
Angle signs and ranges
α and γ range are defined modulo 2π radians. A valid range could be A valid range could be (-π, π]
β range covers π radians (but can't be said to be modulo π). For example could be [0, π] or [-π/2, π/2]
The angles α, β and γ are uniquely determined except for the singular case that – xy and the XY planes are identical,
– the z axis and the Z axis having the same or opposite directions. • if the z-axis and the Z-axis are the same, β = 0 and only (α+γ) is uniquely defined (not
the individual values),
• if the z-axis and the Z-axis are opposite, β = π and only (α-γ) is uniquely defined (not the individual values). These ambiguities are known as gimbal lock in applications.
75
Euler Rotations
Change one of the Euler angles while leaving the other two constant.
Neither expressed in terms of external or internal frame, but in mixture.
• first angle moves the line of nodes around the external axis z • second rotates around the line of nodes • third one is an intrinsic rotation around an axis fixed in the body
The three rotations are called Precession, Nutation, and intRinsic rotation
Euler rotations are commutative.Used in aerodynamics. We are not going to use them
76
Intrinsic or Extrinsic Rotations
Intrinsic rotation (in terms of moving body or local coordinate) or Extrinsic rotation (global coordinate) is
possible also possible with Euler angles.
77
78
Intrinsic and Extrinsic Rotations
Intrinsic Rotation
• Rotate the XYZ system about the z-axis by α.
The x-axis now lies on the line of nodes.• Rotate the XYZ system again about the new
rotated x-axis by β. • Rotate the XYZ system a third time about the
new z-axis by γ.
Extrinsic Rotation
• Rotate the XYZ-system about the z-axis by γ.
• Rotate the XYZ-system again about the x-axis
by β.• Rotate the XYZ-system a third time about the
z-axis by α.
Be careful, in this notation, rotations are around z-x-z. Here y is not involved. But there are other conventions also.
Order of rotations
3D rotations do not commuteEuler angles are not unique !!For a specific composite rotation, there are 12
possible orderings of three individual axis-aligned rotations, and each produce different Euler angles β1, β2, β3
One of them are called "proper" Euler angles and the other Tait–Bryan angles
79
Matrix of Euler Angles
– In z-x-z convention :
– Writing the matrices
– Denoting cos by c and sin by s
- Homework : Get Euler angles from the rotation matrix
80
)().().( zxz RRRR
Why Euler angles ?
When we want to go between one orientation to another, we may use the Euler anglesBut use of euler angles impose problems as
well like :
81
Gimbal Lock
Consider a case of a level sensing platform on an aircraft flying due North with its three gimbal axes mutually perpendicular (i.e., roll, pitch and yaw angles each zero). If the aircraft pitches up 90 degrees, the aircraft and platform's Yaw axis gimbal becomes parallel to the Roll axis gimbal, and changes about yaw can no longer be compensated for (see illustration, red ring is pivoted to green ring and green ring is pivoted to blue ring).
The word lock is misleading: no gimbal is restrained, all three gimbals can still rotate freely about their respective axis of suspension. Nevertheless, because of the parallel orientation of both the yaw and roll gimbal axes, there is no axis available to accommodate yaw rotation.
More later
82
Euler Theorem
We are now back to our previous problem, finding the rotation matrix of an arbitrary rotation around an axis.
The result is due to Euler
83
Special note : Don’t confuse Euler theorem and Euler Angles. Euler theorem is concerned with our rotation axis, we are going to study it now.Euler angles are not concerned with our axis of composite rotation. Rather they specify a way to orient a co-ordinate system with another by means of defined rotations.
Euler’s Theorem
Euler’s Theorem states that– Any rotation (or sequence of rotations) about a
point is equivalent to a single rotation about some axis through that
84
Ref : Hill’s Book, page 239
(Original Statement) Quomodocunque sphaera circa centrum suum conuertatur, semper assignari potest diameter, cuius directio in situ translato conueniat cum situ initiali.
Translation: When a sphere is moved around its centre it is always possible to find a diameter whose direction in the displaced position is the same as in the initial position.
Ref : Wikipedia
This axis is called Euler
Axis
Euler’s Theorem
Implication of Euler’s Theorem – The θ angle rotation around vector a can be built
from composing axis-aligned rotations– We are now going to find that …
85
Finding the Rotation Matrix
Our previous method Step 1,2 Perform two rotations so that a becomes aligned
with the z-axis (two rotations are necessary) Step 3 Do the required θ rotation around z-axis Step 4,5 Undo the alignment rotations to restore a to its
original direction
86
Alignment
Rotation
Trace Back
Trivial but you should be careful when doing in hands
Step 1,2
We now study a composite transformationAV,N = aligning a vector V with a vector N
To find the rotation matrix, we need to find Av,k
87
88
AV : aligning vector V with k
Av = R,i
V = aI + bJ + cK
x
y
z
b
a
c
k
22λ
λcos
λsin
by axis-about x Rotate :1 Step
cbc
b
b
|V|
x
y
z
b
a
k
( 0, b,c )b
|V|
x
y
z
a
k
|V|
( a, 0, )
( 0, 0, )
( 0, b,c )
89
AV : aligning vector V with k
Av = R,i R-,j *
22λ
λcos
λsin
by axis-about x Rotate :1 Step
cbc
b
222
|V|
|V|
λ)cos(
|V|)sin(
-by axis-yabout V Rotate :2 Step
cba
a
P( a, b, c)
b
x
y
z
b
a
c
k
|V|
( a, 0, )( 0, b,c )b
x
y
z
b
a
c
|V|
( 0, 0, |V|)
( 0, b,c )
a
90
AV : aligning vector V with k
AV-1 = AV
T
AV,N = AN-1 * AV
1000
0
00
0
λλ
λ-
λ-λ
Vc
Vb
Va
bc
Vac
Vab
V
VA
Finding the rotation matrix
Now we have done step 1,2 : AV
For step 3, we have to rotate theta angle around z axisThen take the inverse of step 1,2,
AV-1 = AV
T
There are actually 5 rotations here. None of which are Euler angles α or β or γ
91
vzTv ARAR )(
Axis Angle Notation
92
sin)cos1(
100
010
001
cos),(2
2
2
zzyzx
zyyyx
zxyxx
aaaaa
aaaaa
aaaaa
aR
0
0
0
xy
xz
yz
aa
aa
aa
Rod
rigu
es F
orm
ula
The Geometry of a Rotation
93
The Geometry of a Rotation
94
The Geometry of Rotation
95
The Geometry of Rotation
So we find, the rotation vector is :
This is a vector equation, which is goodBut, we need a matrix form also, to work
easily.
96
Rodrigues Formula
Vecto
r For
m
Matrix Form
sin)cos1(
100
010
001
cos),(2
2
2
zzyzx
zyyyx
zxyxx
aaaaa
aaaaa
aaaaa
kR
0
0
0
xy
xz
yz
aa
aa
aa
Rodrigues Formula
Vector and Matrix forms
How we find these equivalent matrix and vector forms ?Check next slide
97
sin)cos1(
100
010
001
cos),(
),()(
2
2
2
zzyzx
zyyyx
zxyxx
T
aaaaa
aaaaa
aaaaa
kR
XkRxR
zyxX
0
0
0
xy
xz
yz
aa
aa
aa
Vector and Matrix algebra
98
Cross product in Matrix Form
99
Finding rotations from a rotation matrixGiven, R is a pure rotation matrixFind axis of rotation K(kx,ky,kz)
and angle theta from RRecall rodrigues formula, it gives R Check that :Trace(R ) = Sum of diagonal elements of the 3x3
rotation matrix
100
2
1)(cos
RTrace
R
Finding rotations from a rotation matrixCheck that :
101
T
xy
xz
yz
RR
kk
kk
kk
)sin(2
1
0
0
0
)()sin(2
1
)()sin(2
1
)sin(2
1
2,11,2
1,33,1
3,22,3
RRk
RRk
RRk
z
y
x
R
Modeling Transformation - Class 5
102
Quaternion
Have you ever thought about the operations of vector algebra ? We can add vectors, subtract vectors … But multiplication ?
– We can multiply vectors, But, two ways !!– Dot product outputs scalar– Cross product outputs vector
Hey, what about division – Man….r u insane ? vector division !!
103
Quaternion
Vector algebra contains subtraction as the inverse operation for addition, Null vector being the identity element. But there is no inverse vector for
multiplication operation !! That’s why we can’t divide vectors.Why no inverse ?
– A . B is scalar, inverse is out of question– A × B is vector, can we have an inverse ?
104
In search of an vector inverse…
So, we want an inverse A-1 for a vector A, so that A × A-1 = Identity Or may be, we want that
– If B × A = C, then• Given C and B, find A.
• A= B-1 × C
– Is it actually possible ??
105
NONO
Why ?
If A × B = C then– Given C and A, find B– If A, B, C are scalars, 5 × 3 = 15. Hence
15 / 5 = 3, uniquely defined.– But if A, B, C are vectors. B is not uniquely
defined now !!• We just know, B lies on the plane normal to C.
• B can be any vector lying on the plane normal to C, as long as
106
||
||sin||
A
CB
ϕ
Why we are craving for inverse
So, what is the reason behind our wild goose search for an inverse of cross ?Because, inverse of cross product could
capture rotation around an axisThink other wayIf C and B is given,
– A is ϕ rotation away– On the plane
normal to C107
Axis of rotation
Unique
How to uniquely determine B ??– The problem is not complete actually.– Only, C and A alone are not sufficient for
rotation– We need either |B| or ϕ (or sinϕ or cosϕ) to get
A from B and C– Which means, Vector B and C alone aren’t
sufficient, a scalar is needed also.– Here comes quaternion : scalar + 3D vector
108
ϕ
Is it really new ?
No, actuallyWe already know something … Think about complex numbers A point (x,y) on a euclidean space can be
thought as a vector, where inverse is not possible.But if we think it as a complex number,
then inverse is possible !!!
109
Complex Numbers
So, a 2D vector, ai+bj is not invertibleBut a complex number a+ib has inverse…
– Why ???
– Because there is a defined Identity element
– More importantly, there is a necessary relationship for a multiplicative space
i2 = -1And this lets us to have the idea of
conjugation110
Complex Number
Actually, a complex number is not really a 2D vector.Complex Number = Scalar + 1D VectorI said Quaternion = Scalar + 3D VectorSo, what necessary properties do we need
now ?– We need something to operate with
multiplication, something like i2 = -1
111
Great minds in History
Hamilton knew that the complex numbers could be viewed as points in a plane, and he was looking for a way to do the same for points in space. Points in space can be represented by their coordinates, which are triples of numbers, and for many years Hamilton had known how to add and multiply triples of numbers. But he had been stuck on the problem of division: He did not know how to take the quotient of two points in space
The breakthrough finally came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy where he was going to preside at a council meeting. While walking along the towpath of the Royal Canal with his wife, the concept behind quaternions was taking shape in his mind. Hamilton could not resist the impulse to carve the formulae for the quaternions into the stone of Brougham Bridge as he passed by it.
i2 = j2 = k2 = ijk = − 1
112
Quaternion Arsenal
– A quaternion H = [v,w]• v is vector, w is scalar
• v is 3D vector, hence
• H = [(x,y,z),w]
• Real numbers [0,s]
• Pure Vectors [v,0]
113
Quaternion Arsenal
114
Quaternion Arsenal
115
Quaternion Arsenal
116
Quaternions as Rotation
A unit quaternion aids for vector rotationIf q is a unit quaternion, p is a point stored in homogenous
coordinate [x,y,z,w] as a quaternionqpq-1 rotates p around
117
Rotating a vector
118
Representing objects in 3D
119
120
Orientation
Thumb points to +ve Z-axisFingers show +ve rotation from X to Y
axis
Y
X
Z (out of page)
Y
X
Z (larger z areaway from viewer)
Right-handed orentation Left-handed orentation
121
Vectors in 3D
Have length and direction
V = [xv , yv , zv]Length is given by the Euclidean Norm
||V|| = ( xv2 + yv
2 + zv2 )
Dot Product
V • U = [xv, yv, zv]•[xu, yu, zu]
= xv*xu + yv*yu + zv*zu = ||V|| || U|| cos ß
Cross Product V U
= [yv*zu - zv yu , -xv*zu + zv*xu , xv*yu – yv*xu ]= ||V|| || U|| sin ßV U = - ( U x V)
(xv,yv,zv)V=aI+bJ+cK
x
y
z
122
3D Equation of Curve & Line
Parametric equations of Curve & LineCurve
Line
bta
thz
tgy
tfx
C
:
VtPPPtPL
t
tzzzz
tyyyy
txxxx
L
PPPPV
0010
010
010
010
0110
)(
10:
P0(x0,y0,z0)
P1(x1,y1,z1)
t > 1
Vt < 0
t =1
t = 00 < t < 1
x
y
z
C
123
3D Equation of Surface & Plane
Parametric equations of Surface & PlaneSurface
Plane : with Normal, N
dtc
bsa
tshz
tsgy
tsfx
S
,
,
,
:
P0
N
kCjBiAN
DCzByAx
ˆˆˆ
0
124
3D Plane
Ways of defining a plane
1. 3 points P0, P1, P2 on the plane
2. Plane Normal N & P0 on plane
3. Plane Normal N & a vector V on the plane
Plane Passing through P0, P1, P2
P0
P1
P2
N
V
)(
0
0ˆ)(ˆ)(ˆ)()ˆˆˆ(
0
ˆˆˆ
000
000
0
2010
CzByAxD
DCzByAx
kzzjyyixxkCjBiA
PPN
kCjBiAPPPPN
where
plane the on is z) y,P(x, if
125
Revisit : Affine Transformation
Transformation – is a function that takes a point (or vector) and maps that point (or vector) into another point (or vector).
A coordinate transformation of the form:
x’ = axx x + axy y + axz z + bx ,
y’ = ayx x + ayy y + ayz z + by ,
z’ = azx x + azy y + azz z + bz ,
is called a 3D affine transformation.
11000
'
'
'
z
y
x
baaa
baaa
baaa
w
z
y
x
zzzzyzx
yyzyyyx
xxzxyxx
The 4th row for affine transformation is always [0 0 0 1]. Properties of affine transformation:
– translation, scaling, shearing, rotation (or any combination of them) are examples affine transformations.
– Lines and planes are preserved.– parallelism of lines and planes are also preserved, but not angles and
length.
Composite 3D Transformations
126
127
Object Transformation
Line: Can be transformed by transforming the end pointsPlane:(described by 3-points) Can be
transformed by transforming the 3-pointsPlane:(described by a point and Normal)
Point is transformed as usual. Special treatment is needed for transforming Normal
128
Composite Transformations – 3D
Some of the composite transformations to be studied are:AV,N = aligning a vector V with a vector N
– We already studied it
R,L = rotation about an axis L( V, P )
MN,P = Mirror Reflection w.r.t a plane
Aligning a plane P with a plane Q
129
R,L : rotation about an axis L
Let the axis L be represented by vector V and passing through point P
1.Translate P to the origin
2. Align V with vector k
3. Rotate about k
4. Reverse step 2
5. Reverse step 1
R,L = T-PAV *R,k *AV-1 *T-P
-1 *
V
P
Q
Q'
L
z
x
y
k
130
MN,P : Mirror reflection
Let the plane be represented by plane normal N and a point P in that plane
x
y
z N
P
131
MN,P : Mirror reflection
Let the plane be represented by plane normal N and a point P in that plane
1.Translate P to the origin
MN,P = T-P
x
y
z
N
P
132
MN,P : Mirror reflection
Let the plane be represented by plane normal N and a point P in that plane
1.Translate P to the origin
2. Align N with vector k
MN,P = T-PAN *
N
P
x
y
z
133
MN,P : Mirror reflection
Let the plane be represented by plane normal N and a point P in that plane
1.Translate P to the origin
2. Align N with vector k
3. Reflect w.r.t xy-plane
MN,P = T-PAN *S1,1,-1 *
N
P
x
y
z
134
x
y
z
MN,P : Mirror reflection
Let the plane be represented by plane normal N and a point P in that plane
1.Translate P to the origin
2. Align N with vector k
3. Reflect w.r.t xy-plane
4. Reverse step 2
MN,P = T-PAN *S1,1,-1 *AN-1 *
135
MN,P : Mirror reflection
Let the plane be represented by plane normal N and a point P in that plane
1.Translate P to the origin
2. Align N with vector k
3. Reflect w.r.t xy-plane
4. Reverse step 2
5. Reverse step 1
MN,P = T-PAN *S1,1,-1 *AN-1 *T-P
-1 *
x
y
z N
P
136
Further Composition
The Composite Transform must have
– Translation of P1 to Origin T
zx
y3P
1P2PT
– Some Combination of Rotations R
Rx
y
z 2P
3P 1P
z
x
y3P
1P2P
Fig. 1 Fig. 2
Translate points in fig. 1 into points in fig 2 such that:– P3 is moved to yz plane
– P2 is on z-axis
– P1 is at Origin
137
Finding R
xx
zyx
zyx
zzz
yyy
xxx
Rx.x R
RRR
RRR
zRyRxR
zRyRxR
zRyRxR
rrr
rrr
rrr
R
R
vextor ofcomponent :Note
other each to
lar perpendicu are ii)
vectorsunit are i)
Transform body-Rigid isR
be Let
,,
,,
...
...
...
333231
232221
131211
138
Finding Rz
z
z
z
z
zyx
zyx
zyx
TT
R
zR
yR
xR
zRzRzR
yRyRyR
xRxRxR
RRkR
21
21
21
21
21
21
PP
PP
PP
PP
PP
PP
.
.
.
1
0
0
...
...
...
ˆ 1
R
z
x
y
3P
1P2P
x
y
z 2P
3P 1P
Rz
k
kR
R
ˆ
21
21
21
PP
PP
axis-z along PP aligns
139
R
x
y
z 2P
3P 1P
z
x
y
3P
1P 2P
Finding Rx
x
x
x
x
zyx
zyx
zyx
R
zR
yR
xR
zRzRzR
yRyRyR
xRxRxR
iR
2131
2131
2131
2131
2131
2131
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
.
.
.
0
0
1
...
...
...
ˆ1 Rx
i
iR
R
ˆ
2131
2131
2131
PPPP
PPPP
axis-x along PPPP aligns
Rz
k
140
Finding Ry
y
y
y
y
zyx
zyx
zyx
R
zR
yR
xR
zRzRzR
yRyRyR
xRxRxR
jR
xz
xz
xz
RR
RR
RR
.
.
.
0
1
0
...
...
...
ˆ1
jR
R
ˆ
xz
xz
RR
axis- yalong RR aligns
R
x
y
z 2P
3P 1P
z
x
y
3P
1P 2P
Rx
i
Rz
k
Ry
j
141
Problems to be solved:
Schaum’s outline series:
Problems: 6.1
6.2, 6.5, 6.9, 6.10, 6.11, 6.12 Av
6.3, 6.4 R,L 6.6, 6.7, 6.8 MN,P
Transforming Normal Correctly
142
What is the problem
You have known of Normal vectorsNormal vector to a plane is used for
lighting and shading purposeWhen a Plane is transformed, the normal
vector may not be transformed correctly.The programmer needs to specify correct
normal after transformation
143
144
Normal
Surface Normal: unit vector that is locally perpendicular to the surface
145
Why is the Normal important?
It's used for shading — makes things look 3D!
object color only Diffuse Shading
146
Visualization of Surface Normal
± x = Red± y = Green± z = Blue
147
How do we transform normals?
Object SpaceWorld Space
nOS
nWS
6.837 Fall 06 – Durand 148
Transform Normal like Object?
translation? rotation? isotropic scale? scale? reflection? shear?perspective?
149
Transform Normal like Object? translation? rotation? isotropic scale?scale? reflection?shear?perspective?
150
Similitudes
What class of transforms?
TranslationRotation
Rigid / Euclidean
Linear
Affine
Projective
Similitudes
Isotropic Scaling
Scaling
Shear
Reflection
Perspective
IdentityTranslation
RotationIsotropic Scaling
IdentityReflection
a.k.a. Orthogonal Transforms
151
Transformation for shear and scale
IncorrectNormal
Transformation
CorrectNormal
Transformation
152
More Normal Visualizations
Incorrect Normal Transformation Correct Normal Transformation
153
Think about transforming the tangent plane to the normal, not the normal vector
So how do we do it right?
Original Incorrect Correct
nOS
Pick any vector vOS in the tangent plane,how is it transformed by matrix M?
vOSvWS
nWS
vWS = M vOS
154
Transform tangent vector vv is perpendicular to normal n:
nOST vOS = 0
nOST (M-1 M) vOS = 0
nWST = nOS
T (M-1)
(nOST M-1) (M vOS) = 0
(nOST M-1) vWS = 0
nWST vWS = 0
vWS is perpendicular to normal nWS:
nWS = (M-1)T nOS
nOS
vWS
nWS
vOS
Dot product
155
Comment
So the correct way to transform normals is:
But why did nWS = M nOS work for similitudes?Because for similitude / similarity
transforms,
(M-1)T = Me.g. for orthonormal basis:
nWS = (M-1)T nOS
xu
yu
zu
xv
yv
zv
xn
yn
zn
ux
vx
nx
uy
vy
ny
uz
vz
nz
M-1 =M =
Sometimes noted M-T
Rotational Interpolation
156
Camera Interpolation
How to smoothly interpolate a camera between two views ?
157
More about Euler Angles
Durand slideTait Bryan Angles
158
Flight Dynamics
Roll Pitch YawTrai
159
Roll, Pitch, Yaw
160
Interpolation by Euler AngleGimbal LockHow gimbal lock prevented (4th axis)Interpolation by Quaternion
161
162
Hierarchical Modeling
Many graphical objects are structured Exploit structure for
– Efficient rendering– Concise specification of model parameters– Physical realism
Often we need several instances of an object– Wheels of a car– Arms or legs of a figure– Chess pieces
Encapsulate basic object in a function Object instances are created in “standard” form Apply transformations to different instances Typical order: scaling, rotation, translation
163
OpenGL & Hierarchical Model
ABCC
glPushMatrix
– void glPushMatrix(void);
AB
glPushMatrix
– void glPoipMatrix(void);
ABC C
m
glGetFloatv
– void glGetFloatv(GL_MODELVIEW_MATRIX, *m);
ABC
Some of the OpenGL functions helpful for hierarchical modeling are:
164
Scene Graph
A scene graph is a hierarchical representation of a scene We will use trees for representing hierarchical objects
such that:– Nodes represent parts of an object
– Topology is maintained using parent-child relationship
– Edges represent transformations that applies to a part and all the subparts connected to that part
typedef struct treenode {GLfloat m[16]; // Transformationvoid (*f) ( ); // Draw functionstruct treenode *sibling;struct treenode *child;
} treenode;
Scene
Sun Star X
Earth Venus Saturn
Moon Ring
165
Example - Torso
Initializing transformation matrix for node
treenode torso, head, ...;
/* in init function */
glLoadIdentity();
glRotatef(...);
glGetFloatv(GL_MODELVIEW_MATRIX, torso.m);
Initializing pointers
torso.f = drawTorso;
torso.sibling = NULL;
torso.child = &head;
166
Generic Traversal
To render the hierarchy:– Traverse the scene graph depth-first– Going down an edge:
• push the top matrix onto the stack
• apply the edge's transformation(s)
– At each node, render with the top matrix– Going up an edge:
• pop the top matrix off the stack
167
Generic Traversal : Torso
Recursive definitionvoid traverse (treenode *root) {
if (root == NULL) return;
glPushMatrix();
glMultMatrixf(root->m);
root->f();
if (root->child != NULL) traverse(root->child);
glPopMatrix();
if (root->sibling != NULL) traverse(root->sibling);
}
C is really not the right language for this !!
168
Viewing Transformation
169
Viewing Pipeline Revisited
xw
yw
zw
pw
Modeling Transform
WorldCoordinates
Pw
yo
xo
zo
po
GraphicsPrimitives Po
ObjectCoordinates
ye
xe
-ze
pe
Viewing Transform Pe
EyeCoordinates
170
Viewing Transformation in OpenGL To setup the modelview matrix, OpenGL provides the
following function:
y
x
z
eye(eyex, eyey, eyez)
center(centerx, centery, centerz)
up(upx, upy, upz)
gluLookAtgluLookAt(( eyex, eyey, eyezeyex, eyey, eyez, , centerx, centery, centerzcenterx, centery, centerz, , upx, upy, upzupx, upy, upz ))
171
Implementation
We want to construct an Orthogonal FrameOrthogonal Frame such that,
xzy
x
z
eCeCeC
puvnormalizeeC
veC
eyecenternormalizev
eyeOC
...
.
.
.
C.e z
C.O(eye)
center
up(upx, upy, upz)
v
C.e x
C.e y
(1)(1) its origin is the point eyepoint eye
(2)(2) its -z basis vector points towards the point point centercenter(3)(3) the up vectorup vector projects to the up direction (+ve y-axis)
Let CC (for camera) denote this frame. Clearly,
172
Thank You