geometric transformations hearn & baker chapter 5 some slides are taken from robert thomsons...
TRANSCRIPT
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Geometric TransformationsHearn & Baker Chapter 5
Some slides are taken from Robert Thomsons notes.
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OVERVIEW
• Two dimensional transformations• Matrix representations• Inverse transformations• Three dimensional transformations
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Geometric transformation functions
• Translation• Rotation• Scaling• Reflection• Shearing
• We first discuss the 2D transformations, then we will continue with 3D.
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2D Translations
TPP
d
dT
y
xP
y
xP
dyydxx
yxP
yxPP
yx
Now
,,
torscolumn vec theDefine
axis.y toparallel d axis, x toparallel d distance a ),(Point totranslate
),,( as defined Point
y
x
yx
PP’
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2D Scaling from the origin.
y
x.
0
0
y
xor
Now
0
0
matrix theDefine
. ,.
axis.y thealong s and
axis, x thealong sfactor aby ),(Point to(stretch) scale a Perform
),,( as defined Point
y
x
y
x
y
x
yx
s
sPSP
s
sS
ysyxsx
yxP
yxPP
PP’
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2D Rotation about the origin.
y
x
r
r
P’(x’,y’)
P(x,y)
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2D Rotation about the origin.
y
x
r
r
P’(x’,y’)
P(x,y)
y
sin.
cos.
ry
rx
x
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2D Rotation about the origin.
y
x
r
r
P’(x’,y’)
P(x,y)
y
sin.
cos.
ry
rx
x
cos.sin.sin.cos.)sin(.
sin.sin.cos.cos.)cos(.
rrry
rrrx
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2D Rotation about the origin.
sin.
cos.
ry
rx
cos.sin.sin.cos.)sin(.
sin.sin.cos.cos.)cos(.
rrry
rrrx
Substituting for r :
Gives us :
cos.sin.
sin.cos.
yxy
yxx
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2D Rotation about the origin.
cos.sin.
sin.cos.
yxy
yxx
Rewriting in matrix form gives us :
y
x
y
x.
cossin
sincos
PRPR
,
cossin
sincosmatrix theDefine
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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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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.– (2,5,3) and (4,10,6) represent the same point.
• At least one component must be non-zero (0,0,0) is not defined.
• 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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Homogeneous coordinates
• If we represent (x,y,W) in 3-space, all triples representing the same point describe a line passing through the origin.
• If we homogenize the point, we get a point of form (x,y,1)– homogenised points form a plane at W=1.
P
X
Y
W
W=1 plane
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Translations in homogenised coordinates
• Transformation matrices for 2D translation are now 3x3.
1
.
100
10
01
1
y
x
d
d
y
x
y
x
11
y
x
dyy
dxx
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Concatenation.
• We perform 2 translations on the same point:
),(),(),(
:expect weSo
),(),(),(
),(
),(
21212211
21212211
22
11
yyxxyxyx
yyxxyxyx
yx
yx
ddddTddTddT
ddddTPddTddTPP
ddTPP
ddTPP
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Concatenation.
?
100
10
01
.
100
10
01
: is ),(),(product matrix The
2
2
1
1
2211
y
x
y
x
yxyx
d
d
d
d
ddTddT
Matrix product is variously referred to as compounding, concatenation, or composition
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Concatenation.
Matrix product is variously referred to as compounding, concatenation, or composition.This single matrix is called the Coordinate Transformation Matrix or CTM.
100
10
01
100
10
01
.
100
10
01
: is ),(),(product matrix The
21
21
2
2
1
1
2211
yy
xx
y
x
y
x
yxyx
dd
dd
d
d
d
d
ddTddT
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Properties of translations.
),(),(T 4.
),(),(),(),( 3.
),(),(),( 2.
)0,0( 1.
1-yxyx
yxyxyxyx
yyxxyxyx
ssTss
ssTttTttTssT
tstsTttTssT
IT
Note : 3. translation matrices are commutative.
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Homogeneous form of scale.
y
xyx s
sssS
0
0),(
Recall the (x,y) form of Scale :
100
00
00
),( y
x
yx s
s
ssS
In homogeneous coordinates :
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Concatenation of scales.
!multiply easy to -matrix in the elements diagonalOnly
100
0ss0
00ss
100
0s0
00s
.
100
0s0
00s
: is ),(),(product matrix The
y2y1
x2x1
y2
x2
y1
x1
2211
yxyx ssSssS
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Reflection
corresponds to negative scale factors
originalsx = -1 sy = 1
sx = -1 sy = -1 sx = 1 sy = -1
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Homogeneous form of rotation.
1
.
100
0cossin
0sincos
1
y
x
y
x
)()(
: i.e ,orthogonal are matricesRotation
).()(
matrices,rotation For
1
1
TRR
RR
i.e. the inverse is the transpose
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Orthogonality of rotation matrices.
100
0cossin
0sincos
)( ,
100
0cossin
0sincos
)(
TRR
100
0cossin
0sincos
100
0cossin
0sincos
)(
R
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Other properties of rotation.
matters.order Otherwise
same, thearerotation of centres theifonly
)()()()(
and
)()()(
)0(
RRRR
RRR
IR
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2D Composite Transformations
• Combine transformations of different type– translate, rotate, translate– translate, scale, translate– translate, reflect, translate
• Use to rotate or scale an object w.r.t. a point that is not the origin
• Implement by multiplication of the corresponding homogeneous matrices
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reflection in x and y axes reflection in origin
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1 0
1
x x
y xb b y
1
0 1
x x y
y y
a a
Shx gives the slope of a vertical line after shear, as dx/dy
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Example 1
y
yx
x
y
xyxT
4.0
14.0
01),(
)1,0(
)0,1(
)1,1(
)0,0(
S
x
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Example 1
x
y
yx
x
yxT
4.0
),(
S
)1,0(
)0,1(
)1,1(
)0,0(
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Example 1
y
)1,0(
)4.0,1(
)0,0(
T(S)
yx
x
yxT
4.0
),()4.1,1(
x
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Example 2
0.6
0.6
1( , )
0 1
xT x y
y
x y
y
y
Sx
)1,0(
)0,1(
)1,1(
)0,0(
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Example 2
0.6( , )
x yT x y
y
x
y
S
)1,0(
)0,1(
)1,1(
)0,0(
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Example 2
( , ) 0.6x y
T x yy
x
y
)1,0(
)0,1()0,0(
T(S)
(0.6,1) (1.6,1)
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Summary
Shear in x:
Shear in y:
y
yshx
y
xshSh xx
x 10
1
yxsh
x
y
x
shSh
yyy .1
01
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Double Shear: not commutative!
ab)(10
1
1
01
1
1
b
aa
b
1
1 ab)(
1
01
10
1
b
a
b
a
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Shear Matrix defined by angle, not slope
e.g. simple shear along x axis
x’ = x + y cot y’ = yz’ = z
1000
0100
0010
00cot 1
H() =
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3D Transformations.
• Use homogeneous coordinates, just as in 2D case.• Transformations are now 4x4 matrices.• We will use a right-handed (world) coordinate system -
( z out of page ).
z (out of page)
y
x
Note:Convenient to think of display asBeing left-handed !!( z into the screen )
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Simple extension to the 3D case:
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Scale in 3D.
1000
000
000
000
),,(z
y
x
zyx s
s
s
sssS
Simple extension to the 3D case:
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Rotation in 3D
• Need to specify which axis the rotation is about.• z-axis rotation is the same as the 2D case.
1000
0100
00cossin
00sincos
)(
zR
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Rotation around an axis parallel to x-axis
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H&B p271
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Rows of upper-left 3x3 submatrix, when rotated by R lie on the x,y and z axes
1
1
0
0
1
1
0
1
0
1
1
0
0
1
1
33
32
31
23
22
21
13
12
11
r
r
r
Rr
r
r
Rr
r
r
R
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Scaling an object with this transformation will also move its position relative to the origin - so move it to the origin, scale it, then move it back...
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