C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 1/29
GeometricTransformations
Readings: Chapters 5-6
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 2/29
Announcement
• Proj 1 posted on 2/2, due two weeks from today.
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 3/29
• Translate [1,3] by [7,9]
• Scale [2,3] by 5 in the X direction and 10 in the Y direction
• Rotate [2,2] by 90° (π/2)
• What is the result for each of the following two cases applied to the house in the figure.
Exercises
Case 1: Translate by
x=6, y=0 then rotate by 45º
0 1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
Case 2: Rotate by 45 and then
translate by x=6, y=0
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 4/29
• Translate [1,3] by [7,9]
• Scale [2,3] by 5 in the X direction and 10 in the Y direction
• Rotate [2,2] by 90° (π/2)
Examples
1
12
8
1
3
1
100
910
701
1
30
10
1
3
2
100
0100
005
1
2
2
1
2
2
100
001
010
1
2
2
100
0)2/cos()2/sin(
0)2/sin()2/cos(
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 5/29
• Scene graph (DAG) construction from primitives
• Object motion (this lecture)• Camera motion (next lecture on viewing)
ROBOT transformation
upper body lower body
head trunk arm
base
Scenegraph(see Sceneview assignment)
stanchion
“is composed of hierarchy”
Why do we need geometric Transformations (T,R,S) in Graphics?
More on scenegraphs on Slide 18
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 6/29
2D Transformations
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 7/29
• Component-wise addition of vectors
v’ = v + t where
and x’ = x + dx
y’ = y + dy
To move polygons: translate vertices (vectors) and redraw lines between them
• Preserves lengths (isometric)
• Preserves angles (conformal)
• Translation is thus "rigid-body"
dx = 2dy = 3
Y
X 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
dy
dxt
y
xv
y
xv ,
'
'',
2D Translation
3
3
6
5
(Points designate origin of object's local
coordinate system)
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 8/29
• Component-wise scalar multiplication of vectors
v’ = Sv where
and
• Does not preserve lengths• Does not preserve angles (except when scaling is
uniform)
'
'',
y
xv
y
xv
y
x
s
sS
0
0
ysy
xsx
y
x
'
'
2
3
y
x
s
s
2D Scaling
Side effect: House shifts position relative to origin
1
2
1
3
2
6
2
9
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 9/29
v’ = Rθ v where
and x’ = x cos Ө – y sin Ө y’ = x sin Ө + y cos Ө
A rotation by 0 angle, i.e. no rotation at all, gives us the identity matrix
• Preserves lengths in objects, and angles between parts of objects
• Rotation is rigid-body
cossin
sincosR
'
'',
y
xv
y
xv
6
2D Rotation
Side effect: House shifts position relative to origin
Rotate by about the
origin
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 10/29
• Suppose object is not centered at origin and we want to scale and rotate it.
• Solution: move to the origin, scale and/or rotate in its local coordinate system, then move it back.
• This sequence suggests the need to compose successive transformations…
2D Rotation and Scale are Relative to Origin
Object’s local coordinate system origin
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 11/29
• Translation, scaling and rotation are expressed as:
• Composition is difficult to express
– translation is not expressed as a matrix multiplication
• Homogeneous coordinates allows expression of all three transformations as 3x3 matrices for easy composition
• w is 1 for affine transformations in graphics
• Note:
This conversion does not transform p. It is only changing notation to show it can be viewed as a point on w = 1 hyperplane
translation:
scale:
rotation:
v’ = v + t
v’ = Sv
v’ = Rv
w
y
w
xPyxP
wwyxP
wwwywxPyxP
dd
h
hd
',
'),(
0),,','(
0),,,(),(
22
2
Homogenous Coordinates
y x, p y,1 x, p becomes
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 12/29
• For points written in homogeneous coordinates,
translation, scaling and rotation relative to the origin are expressed homogeneously as:
1
y
x
2D Homogeneous CoordinateTransformations
100
00
00
, y
x
ss s
s
Syx vss yx
Sv ,'
vdd yxTv ,'
100
10
01
, y
x
dd d
d
Tyx
100
0cossin
0sincos
R vRv '
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 13/29
• Avoiding unwanted translation when scaling or rotating an object not centered at origin:
– translate object to origin, perform scale or rotate, translate back.
• How would you scale the house by 2 in “its” y and rotate it through 90° ?
• Remember: matrix multiplication is not commutative! Hence order matters! (Refer to class website: resources/SIGGRAPH2001_courseNotes_source/transformation for camera transformation)
HdydxTRdydxTHdydxTRHdydxTHHouse ),()(),(),()(),()(
HSRH,SHHouse )2,1()2/()21()(
Matrix Compositions: Using Translation
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 14/29
Matrix Multiplication is NOT Commutative
Translate by x=6, y=0 then rotate by 45º
Rotate by 45º then translate by
x=6, y=0
0 1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
Y
X 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 15/29
3D Transformations
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CMSC 435 / 634 Transformations 16/29
• Translation
• Scaling
(right-handed coordinate system)
1000
100
010
001
dz
dy
dx
1000
000
000
000
z
y
x
s
s
s
x
y
z
3D Basic Transformations (1/2)
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 17/29
• Rotation about X-axis
(right-handed coordinate system)
1000
0cossin0
0sincos0
0001
1000
0cos0sin
0010
0sin0cos
1000
0100
00cossin
00sincos
• Rotation about Y-axis
• Rotation about Z-axis
3D Basic Transformations (2/2)
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 18/29
More transformation matrices
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 19/29
• “Skew” a scene to the side:
• Squares become parallelograms - x coordinates skew to right, y coordinates stay same
• Degree between axes becomes Ө
• Like pushing top of deck of cards to the side – each card shifts relative to the one below
• Notice that the base of the house (at y=1) remains horizontal, but shifts to the right.
Y
X 0
1
1
2
2
3 4 5 6 7 8 9 10
3
4
5
6
4
NB: A skew of 0 angle, i.e. no skew at all, gives us the identity matrix, as it should
Skew/Shear/Translate (1/2)
10tan
11
Skew
2D non-homogeneous
100
010
0tan
11
Skew
2D homogeneous
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 20/29
Scene graph manipulation
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 21/29
• 3D scenes are often stored in a scene graph: – Open Scene Graph– Sun’s Java3D™– X3D ™ (VRML ™ was a precursor to X3D)
• Typical scene graph format:– objects (cubes, sphere, cone, polyhedra etc.)
– stored as nodes (default: unit size at origin)– attributes (color, texture map, etc.) and
transformations are also nodes in scene graph (labeled edges on slide 2 are an abstraction)
Transformations in Scene Graphs (1/3)
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 22/29
ROBOT
upper body lower body
head trunk arm
stanchion base
1. Leaves of tree are standard size object primitives
2. We transform them
3. To make sub-groups
4. Transform subgroups
5. To get final scene
Closer look at Scenegraph from slide 2 …
Transformations in Scene Graphs (2/3)
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 23/29
• Transformations affect all child nodes
• Sub-trees can be reused, called group nodes– instances of a group can have different transformations
applied to them (e.g. group3 is used twice– once under t1 and once under t4)
– must be defined before use
object nodes (geometry)
transformation nodes
group nodes
group3
obj3 obj4
t5 t6
t4
root
t0
group1
t1 t2
obj1 group3
t3
group2
group3obj2
Transformations in Scene Graphs (3/3)
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 24/29
• Transformation nodes contain at least a matrix that handles the transformation;
– may also contain individual transformation parameters
• To determine final composite transformation matrix (CTM) for object node:
– compose all parent transformations during prefix graph traversal
– exact detail of how this is done varies from package to package, so be careful
Composing Transformations in a Scene Graph (1/2)
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 25/29
• Example:
- for o1, CTM = m1
- for o2, CTM = m2* m3
- for o3, CTM = m2* m4* m5
- for a vertex v in o3, position in the world (root) coordinate system is:
CTM v = (m2*m4*m5)v
g: group nodes
m: matrices of transform nodes
o: object nodes
m5
m1 m2
m3 m4
o1
o2
o3
g1
g2
g3
Composing Transformations in a Scene Graph (2/2)
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 26/29
Exercises
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 27/29
• Create a transform matrix that takes points in the rectangle [xl, xh] x [yl, yh] to the rectangle [xl’, xh’], [yl’, yh’] (Shirley Figure 6.18)
E1: Windowing transforms
=
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 28/29
• Create a transform matrix that takes a point in the (X,Y) coordinate system to the point in the (U, V) coordinate system; and vice versa (Shirley Figure 6.20)
E2: Transformation between two coordinate systems
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 29/29
Take-home exercise
E3: Rotation about arbitrary axis
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 30/29
(right-handed coordinate system)
Rotation axis specification
C O M P U T E R G R A P H I C S
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CMSC 435 / 634 Transformations 31/29
Rotation about arbitrary axis