Lecture 2 Transformations · Lecture 2 1 Lecture 2 Transformations. Transformations. What is a...
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Computer Graphics Lecture 2 1 Lecture 2 Transformations
Transcript of Lecture 2 Transformations · Lecture 2 1 Lecture 2 Transformations. Transformations. What is a...
Computer Graphics
Lecture 2 1
Lecture 2
Transformations
Transformations.
What is a transformation?
• P′=T(P) What does it do?
Transform the coordinates / normal vectors of objects
Why use them?
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• Modelling
-Moving the objects to the desired location in the environment -Multiple instances of a prototype shape
-Kinematics of linkages/skeletons – character animation
• Viewing– Virtual camera: parallel and perspective projections
Types of Transformations
– Geometric Transformations• Translation• Rotation• scaling
•• Linear Linear (preserves parallel lines) • Non-uniform scales, shears or skews
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• Non-uniform scales, shears or skews– Projection (preserves lines)
• Perspective projection• Parallel projection
– Non-linear (lines become curves) • Twists, bends, warps, morphs,
Geometric Transformation
– Once the models are prepared, we need to place them in the environment
– Objects are defined in their own local coordinate system
– We need to translate, rotate and scale them to put them into the world coordinate system
22/09/2011 Lecture 1 4
2D Translations.
dyydxx
yxP
yxPP
yx +=′+=′
′′′
torscolumn vec theDefine
axis.y toparallel d axis, x toparallel d distance a ),(Point totranslate
),,( as defined Point
yx
5
TPP
d
dT
y
xP
y
xP
+=′
=
′′
=′
=
Now
,,
torscolumn vec theDefine
y
x
PP’
2D Scaling from the origin.
=′=′
′′′
. ,.
axis.y thealong s and
axis, x thealong sfactor aby ),(Point to(stretch) scale a Perform
),,( as defined Point
y
x
yx ysyxsx
yxP
yxPP
6
=
′′
⋅=′
=
y
x.
0
0
y
xor
Now
0
0
matrix theDefine
y
x
y
x
s
sPSP
s
sS
PP’
2D Rotation about the origin.
y
P’(x’,y’) θ
7
x
r
r
P’(x’,y’)
P(x,y)
2D Rotation about the origin.
y
P’(x’,y’)
8
x
r
r
P’(x’,y’)
P(x,y)
θφ
y
φφ
sin.
cos.
ry
rx
==
x
2D Rotation about the origin.
y
P’(x’,y’)
θφθφφθθφθφφθ
cos.sin.sin.cos.)sin(.
sin.sin.cos.cos.)cos(.
rrry
rrrx
+=+=′−=+=′
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x
r
r
P’(x’,y’)
P(x,y)
θφ
yφφ
sin.
cos.
ry
rx
==
x
2D Rotation about the origin.
θφθφφθθφθφφθ
cos.sin.sin.cos.)sin(.
sin.sin.cos.cos.)cos(.
rrry
rrrx
+=+=′−=+=′
Substituting for r :
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φφ
sin.
cos.
ry
rx
==
Gives us :
θθθθ
cos.sin.
sin.cos.
yxy
yxx
+=′−=′
2D Rotation about the origin.
θθθθ
cos.sin.
sin.cos.
yxy
yxx
+=′−=′
Rewritingin matrix form gives us :
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Rewritingin matrix form gives us :
−=
′′
y
x
y
x.
cossin
sincos
θθθθ
PRPR ⋅=′
−= ,
cossin
sincosmatrix theDefine
θθθθ
Transformations.• Translation.
– P′=T + P
• Scale
– P′=S ⋅ P
• Rotation
– P′=R ⋅ P
• We would like all transformations to be multiplications so we can concatenate them
• ⇒ express points in homogenous coordinates.
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−=
′′
y
x
y
x.
cossin
sincos
θθθθ
Homogeneous coordinates
• Add an extra coordinate, W, to a point.
– P(x,y,W).
• Two sets of homogeneous coordinates represent the
same point if they are a multiple of each other.same point if they are a multiple of each other.
– (2,5,3) and (4,10,6) represent the same point.
• If W≠ 0 , divide by it to get Cartesian coordinates of point
(x/W,y/W,1).
• If W=0, point is said to be at infinity.
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Translations in homogenised
coordinates
• Transformation matrices for 2D translation
are now 3x3.are now 3x3.
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=
′′
1
.
100
10
01
1
y
x
d
d
y
x
y
x
11
=
+=′+=′
y
x
dyy
dxx
Concatenation.
• When we perform 2 translations on the same point
),( 11 yx PddTP ⋅=′
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),(),(),(
:expect weSo
),(),(),(
),(
21212211
21212211
22
yyxxyxyx
yyxxyxyx
yx
ddddTddTddT
PddddTPddTddTP
PddTP
++=⋅
⋅++=⋅⋅=′′
′⋅=′′
Concatenation.
0101
: is ),(),(product matrix The
21
2211
⋅
xx
yxyx
dd
ddTddT
16
?
100
10.
100
10 2
2
1
1
=
y
x
y
x
dd
Matrix product is variously referred to as compounding, concatenation, or composition
Concatenation.
++
=
⋅
010101
: is ),(),(product matrix The
2121
2211
xxxx
yxyx
dddd
ddTddT
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Matrix product is variously referred to as compounding, concatenation, or composition.
+=
100
10
100
10.
100
10 2121 yyyy dddd
Properties of translations.
),(),(),(),( 3.
),(),(),( 2.
)0,0( 1.
yyxxyxyx
ssTttTttTssT
tstsTttTssT
IT
⋅=⋅
++=⋅=
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),(),(T 4.
),(),(),(),( 3.1-
yxyx
yxyxyxyx
ssTss
ssTttTttTssT
−−=
⋅=⋅
Note : 3. translation matrices are commutative.
Homogeneous form of scale.
=
y
x
yx s
sssS
0
0),(
Recall the (x,y) form of Scale :
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=100
00
00
),( y
x
yx s
s
ssS
In homogeneous coordinates :
Concatenation of scales.
00ss00s00s
: is ),(),(product matrix The
x2x1x2x1
2211
⋅
⋅ yxyx ssSssS
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!multiply easy to -matrix in the elements diagonalOnly
100
0ss0
00ss
100
0s0
00s
.
100
0s0
00s
y2y1
x2x1
y2
x2
y1
x1
⋅⋅
=
Homogeneous form of rotation.
−=
′′
1
.
100
0cossin
0sincos
1
y
x
y
x
θθθθ
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)()(
: i.e ,orthogonal are matricesRotation
).()(
matrices,rotation For
1
1
θθ
θθ
TRR
RR
=
−=
−
−
Orthogonality of rotation
matrices.
−=
−
= 0cossin
0sincos
)( ,0cossin
0sincos
)( θθθθ
θθθθθ
θ TRR
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−=
=100
0cossin)( ,
100
0cossin)( θθθθθθ TRR
−=
−−−−−
=−100
0cossin
0sincos
100
0cossin
0sincos
)( θθθθ
θθθθ
θR
Other properties of rotation.
and
)()()(
)0(
φθφθ RRR
IR
+=⋅=
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careful more be toneed ,rotations 3DFor
same theis
rotation of axis thebecauseonly is But this
)()()()(
and
θφφθ RRRR ⋅=⋅
How are transforms combined?
(0,0)
(1,1) (2,2)
(0,0)
(5,3)
(3,1) Scale(2,2) Translate(3,1)
Scale then Translate
Use matrix multiplication: p' = T ( S p ) = TS p
02/10/09 Lecture 4 24
TS =2
0
0
2
0
0
1
0
0
1
3
1
2
0
0
2
3
1=
Use matrix multiplication: p' = T ( S p ) = TS p
Caution: matrix multiplication is NOT commutative!
0 0 1 0 0 1 0 0 1
Non-commutative Composition
Scale then Translate: p' = T ( S p ) = TS p
(0,0)
(1,1) (2,2)
(0,0)
(5,3)
(3,1) Scale(2,2) Translate(3,1)
02/10/09 Lecture 4 25
Translate then Scale: p' = S ( T p ) = ST p
(0,0)
(1,1) (4,2)
(3,1)
(8,4)
(6,2) Translate(3,1) Scale(2,2)
Non-commutative Composition
TS =2
0
0
0
2
0
0
0
1
1
0
0
0
1
0
3
1
1
Scale then Translate: p' = T ( S p ) = TS p
2
0
0
0
2
0
3
1
1
=
02/10/09 Lecture 4 26
ST =2
0
0
2
0
0
1
0
0
1
3
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
How are transforms combined?
(0,0)
(1,1) (0,sqrt(2))
(0,0)
(3,sqrt(2))
(3,0)
Rotate 45 deg Translate(3,0)
Rotate then Translate
Translate then Rotate
02/10/09 Lecture 4 27
Caution: matrix multiplication is NOT commutative!
Translate then Rotate
(0,0)
(1,1)
(0,0) (3,0)
Rotate 45 deg Translate(3,0) (3/sqrt(2),3/sqrt(2))
Non-commutative Composition
TR =0
1
0
-1
0
0
0
0
1
1
0
0
0
1
0
3
1
1
Rotate then Translate: p' = T ( R p ) = TR p
0
0
0
-1
0
0
3
1
1=
0
1
0
-1
0
0
0
0
1
0
1
0
-1
0
0
0
0
1
02/10/09 Lecture 4 28
RT =1
0
0
1
3
1
0
1
0
-1
0
0
-1
3
1=
=Translate then Rotate: p' = R ( T p ) = RT p
0 0 1
0
1
0
-1
0
0
0
0
1
0
1
0
-1
0
0
0
0
1
0
1
0
-1
0
0
0
0
1
Types of transformations.
• Rotation and translation
– Angles and distances are preserved
– Unit cube is always unit cube
– Rigid-Body transformations.
• Rotation, translation and scale.
– Angles & distances not preserved.
– But parallel lines are.
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Transformations of coordinate
systems.• Have been discussing transformations as
transforming points.
• Always need to think the transformation in the • Always need to think the transformation in the world coordinate system
• Sometimes this might not be so convenient
– i.e. rotating objects at its location
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Transformations of coordinate
systems - Example • Concatenate local
transformation matricesfrom left to right
• Can obtain the local –
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• Can obtain the local –world transformationmatrix
• p’,p’’,p’’’ are the world coordinates of p after each transformation
Transformations of coordinate
systems - example
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• is the world coordinate of point p after n transformations
Quiz
• I sat in the car, and find the side mirror is 0.4m on my right and 0.3m in my front
• I started my car and drove 5m forward, turned 30 degrees to right, moved 5m forward again, and
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degrees to right, moved 5m forward again, and turned 45 degrees to the right, and stopped
• What is the position of the side mirror now, relative to where I was sitting in the beginning?
Solution
• The side mirror position is locally
(0,4,0.3)
• The matrix of first driving
forward 5m isforward 5m is
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=100
510
001
1T
Solution
• The matrix to turn to the right
30 and 45 degrees (rotating -30
and -45 degrees around the
origin) are origin) are
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lyrespective ,
100
02
1
2
1
02
1
2
1
,
100
02
3
2
1
02
1
2
3
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−=
−= RR
Solution
The local-to-global transformation matrix at the last configuration of the car is
−
−
= 11
02
1
2
1
00131
02
1
2
3
001
The final position of the side mirror can be computed by TR1TR2 p which is around (2.89331, 9.0214)
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−
−
=
100
02
1
2
1
100
510
100
02
3
2
1
100
51021TRTR
This is convenient for character
animation / robotics
• In robotics / animation, we often want to know what is the current 3D location of the end effectors (like the
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the end effectors (like the hand)
• Can concatenate matricesfrom the origin of the body towards the end effecter
Transformations of coordinate
systems.
)()(
ji
)(
i systemin point a toj systemin point a converts that transform theas M Define
system coordinatein point a as Define
←
←
⋅= jji
i
i
PMP
iP
Lecture 4 38
1
)()(
:shown that be alsocan It
:onsubstitutiby obtain we
and
−←←
←←←
←
←
=
⋅=
⋅=
jiij
kjjiki
kkj
j
ji
MM
MMM
PMP
3D Transformations.
• Use homogeneous coordinates, just as in 2D case.
• Transformations are now 4x4 matrices.
• We will use a right-handed (world) coordinate • We will use a right-handed (world) coordinate
system - ( z out of page ).
02/10/09 Lecture 4 39
Translation in 3D.
Simple extension to the 3D case:
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=
1000
100
010
001
),,(z
y
x
zyx d
d
d
dddT
Scale in 3D.
Simple extension to the 3D case:
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=
1000
000
000
000
),,(z
y
x
zyx s
s
s
sssS
Rotation in 3D
• Need to specify which axis the rotation is about.
• z-axis rotation is the same as the 2D case.
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−
=
1000
0100
00cossin
00sincos
)(θθθθ
θzR
Rotating About the x-axis Rx(θ)
0001 xx
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⋅
−=
′′
11000
0θcosθsin0
0θsinθcos0
0001
1
z
y
x
z
y
x
Rotating About the y-axis
Ry(θ)
′ 0θsin0θcos xx
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⋅
−=
′
′
11000
0θcos0θsin
0010
0θsin0θcos
1
z
y
x
z
y
x
Rotation About the z-axis
Rz(θ)
− ′ 00θsinθcos xx
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⋅
−
=
′′
11000
0100
00θcosθsin
00θsinθcos
1
z
y
x
z
y
x
Rotation about an
arbitrary axis• About (ux, uy, uz), a unit
vector on an arbitrary axis
x' xuxux(1-c)+c 0uzux(1-c)-uzs uxuz(1-c)+uys
Rotate(k, θ)
x
y
z
θ
u
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x'
y'
z'
1
=
x
y
z
1
uxux(1-c)+c
uyux(1-c)+uzs
uzux(1-c)-uys
0
0
0
0
1
uzux(1-c)-uzs
uzux(1-c)+c
uyuz(1-c)+uxs
0
uxuz(1-c)+uys
uyuz(1-c)-uxs
uzuz(1-c)+c
0
where c = cos θ & s = sin θ
Rotation
• Not commutative if the axis of rotation are
not parallel
)()()()( αββα xyyx RRRR ≠
02/10/09
)()()()( αββα xyyx RRRR ≠
Shearing
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1000
0100
001
0001
a
1000
0100
0010
001 a
Calculating the world coordinates of all vertices
• For each object, there is a local-to-global transformation matrix
• So we apply the transformations to all the vertices of each object
• We now know the world coordinates of all • We now know the world coordinates of all the points in the scene
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Normal Vectors
• We also need to know the direction of the normal vectors in the world coordinate system
• This is going to be used at the shading operation • We only want to rotate the normal vector• Do not want to translate it
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Normal Vectors - (2)
• We need to set elements of the translation part to zero
0111111111111 rrrtrrr x
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⇒
1000
0
0
0
1000111111
111111
111111
111111
111111
111111
rrr
rrr
rrr
trrr
trrr
trrr
z
y
x
Viewing
• Now we have the world coordinates of all the vertices
• Now we want to convert the scene so that it appears in front of the camera
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View Transformation
�We want to know the positions in the camera coordinate system
�We can compute the camera-to-world transformation matrix using the orientation and translation of the camera from the origin of the world coordinate system
Mw←c
Lecture 4 53
Mw←c
View Transformation
�We want to know the positions in the camera coordinate system
vw = Mw←c vc
Point in theworld coordinate
Point in thecamera coordinate
Camera-to-world transformation
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vc = Mw ← c vw
= Mc←w vw
-1
world coordinate
Summary.
• Transformations: translation, rotation and scaling
• Using homogeneous transformation, 2D (3D) transformations can be represented by multiplication of a 3x3 (4x4) matrix
• Multiplication from left-to-right can be • Multiplication from left-to-right can be considered as the transformation of the coordinate system
• Need to multiply the camera matrix from the left at the end
• Reading: Foley et al. Chapter 5, Appendix 2 sections A1 to A5 for revision and further background (Chapter 5)
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