two-dimensional geometric transformations a two dimensional transformation is any operation on a...
TRANSCRIPT
Two-Dimensional Geometric Transformations
A two dimensional transformation is any operation on a point in space (x y) that maps
that points coordinates into a new set of coordinates (xrsquo yrsquo)
Instead of applying a transformation to every point in every line that makes up an object the transformation is applied only to the vertices of
the object and then new lines are drawn between the resulting endpoints
Two-Dimensional Geometric Transformations
bull Basic Transformations bull Translation bull Rotation bull Scaling
bull Composite Transformations
bull Other transformations bull Reflection bull Shear
Transformation include change in size shape amp orientation
Translation
bull Translation transformation
bull Translation vector or shift vector T = (tx ty)
bull Rigid-body transformationbull Moves objects without deformation
xtxx ytyy
y
xp
PrsquoT
y
x
T
Matrix Representation
Translation matrix
Row vector representation
P = (x y)T = (tx ty)P = (x y)
Prsquo= (xrsquo yrsquo) Prsquo = P + T
x lsquo = x + txyrsquo = y + ty
Rotation
Repositioning along a circular path in the xy plane y
xr
yr ө
P
Prsquo
(xr yr) is the rotation point ө is the rotation angle
Rotation
Rotation transformation
xrsquo=rcos(φ+θ)= rcos φ cos θ -rsin φ sin θ
yrsquo=rsin(φ + θ)= rcos φ sin θ +rsin φ cos θ
x=rcos φ y=rsin φ
Prsquo= R P
cossin
sincosR
xrsquo = x cos ө - y sin ө
yrsquo = x sin ө + y Cos ө rr
Φ
ө
Prsquo
P
(xrsquo yrsquo)
(x y)
(xrsquoyrsquo)
(xryr)
ө Φ
Rotation about arbitrary pivot point position
xrsquo = xr + (x - xr) cosө - (y ndash yr) sin ө
yrsquo = yr + (x - xr) sin ө - (y ndash yr) cos ө
General Pivot-Point Rotation
bull Rotations about any selected pivot point (xryr)
bull Translate-rotate-translate
General Pivot Point Rotation
To rotate about P1 (x1y1) the following sequence of the fundamental transformations are needed
1Translate the object by (-x1 -y1) 2Rotate (Oslash)3Translate the object by (x1 y1)
General Pivot-Point Rotation
100
10
01
100
0cossin
0sincos
100
10
01
r
r
r
r
y
x
y
x
100
sin)cos1(cossin
sin)cos1(sincos
rr
rr
xy
yx
)()()()( rrrrrr yxRyxTRyxT
Scaling
Alters the size of an object Scaling an object is implemented by
scaling the X and Y coordinates of each vertex in the object
Scaling
Scaling transformation
Scaling factors sx and sy
Uniform scaling if sx = sy
xsxx
ysyy
y
x
s
s
y
x
y
x
0
0
PSP
x
y
x
y2xs1ys
Matrix Representations and Homogeneous Coordinates- allow all transformations as matrix
multiplications
bull Homogeneous Coordinates
bull Matrix representations bull Translation
)()( hyxyx hh h
xx h
h
yy h
1100
10
01
1
y
x
t
t
y
x
y
x
h =1 for 2 D transformations
Matrix Representations
Matrix representations Scaling
Rotation
1100
00
00
1
y
x
s
s
y
x
y
x
1100
0cossin
0sincos
1
y
x
y
x
Composite Transformations-sequence of transformations
Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
)()( 1122 PttTttTP yxyx
PttTttT yxyx )()( 1122
)()()( 21211122 yyxxyxyx ttttTttTttT
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
Two-Dimensional Geometric Transformations
bull Basic Transformations bull Translation bull Rotation bull Scaling
bull Composite Transformations
bull Other transformations bull Reflection bull Shear
Transformation include change in size shape amp orientation
Translation
bull Translation transformation
bull Translation vector or shift vector T = (tx ty)
bull Rigid-body transformationbull Moves objects without deformation
xtxx ytyy
y
xp
PrsquoT
y
x
T
Matrix Representation
Translation matrix
Row vector representation
P = (x y)T = (tx ty)P = (x y)
Prsquo= (xrsquo yrsquo) Prsquo = P + T
x lsquo = x + txyrsquo = y + ty
Rotation
Repositioning along a circular path in the xy plane y
xr
yr ө
P
Prsquo
(xr yr) is the rotation point ө is the rotation angle
Rotation
Rotation transformation
xrsquo=rcos(φ+θ)= rcos φ cos θ -rsin φ sin θ
yrsquo=rsin(φ + θ)= rcos φ sin θ +rsin φ cos θ
x=rcos φ y=rsin φ
Prsquo= R P
cossin
sincosR
xrsquo = x cos ө - y sin ө
yrsquo = x sin ө + y Cos ө rr
Φ
ө
Prsquo
P
(xrsquo yrsquo)
(x y)
(xrsquoyrsquo)
(xryr)
ө Φ
Rotation about arbitrary pivot point position
xrsquo = xr + (x - xr) cosө - (y ndash yr) sin ө
yrsquo = yr + (x - xr) sin ө - (y ndash yr) cos ө
General Pivot-Point Rotation
bull Rotations about any selected pivot point (xryr)
bull Translate-rotate-translate
General Pivot Point Rotation
To rotate about P1 (x1y1) the following sequence of the fundamental transformations are needed
1Translate the object by (-x1 -y1) 2Rotate (Oslash)3Translate the object by (x1 y1)
General Pivot-Point Rotation
100
10
01
100
0cossin
0sincos
100
10
01
r
r
r
r
y
x
y
x
100
sin)cos1(cossin
sin)cos1(sincos
rr
rr
xy
yx
)()()()( rrrrrr yxRyxTRyxT
Scaling
Alters the size of an object Scaling an object is implemented by
scaling the X and Y coordinates of each vertex in the object
Scaling
Scaling transformation
Scaling factors sx and sy
Uniform scaling if sx = sy
xsxx
ysyy
y
x
s
s
y
x
y
x
0
0
PSP
x
y
x
y2xs1ys
Matrix Representations and Homogeneous Coordinates- allow all transformations as matrix
multiplications
bull Homogeneous Coordinates
bull Matrix representations bull Translation
)()( hyxyx hh h
xx h
h
yy h
1100
10
01
1
y
x
t
t
y
x
y
x
h =1 for 2 D transformations
Matrix Representations
Matrix representations Scaling
Rotation
1100
00
00
1
y
x
s
s
y
x
y
x
1100
0cossin
0sincos
1
y
x
y
x
Composite Transformations-sequence of transformations
Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
)()( 1122 PttTttTP yxyx
PttTttT yxyx )()( 1122
)()()( 21211122 yyxxyxyx ttttTttTttT
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
Translation
bull Translation transformation
bull Translation vector or shift vector T = (tx ty)
bull Rigid-body transformationbull Moves objects without deformation
xtxx ytyy
y
xp
PrsquoT
y
x
T
Matrix Representation
Translation matrix
Row vector representation
P = (x y)T = (tx ty)P = (x y)
Prsquo= (xrsquo yrsquo) Prsquo = P + T
x lsquo = x + txyrsquo = y + ty
Rotation
Repositioning along a circular path in the xy plane y
xr
yr ө
P
Prsquo
(xr yr) is the rotation point ө is the rotation angle
Rotation
Rotation transformation
xrsquo=rcos(φ+θ)= rcos φ cos θ -rsin φ sin θ
yrsquo=rsin(φ + θ)= rcos φ sin θ +rsin φ cos θ
x=rcos φ y=rsin φ
Prsquo= R P
cossin
sincosR
xrsquo = x cos ө - y sin ө
yrsquo = x sin ө + y Cos ө rr
Φ
ө
Prsquo
P
(xrsquo yrsquo)
(x y)
(xrsquoyrsquo)
(xryr)
ө Φ
Rotation about arbitrary pivot point position
xrsquo = xr + (x - xr) cosө - (y ndash yr) sin ө
yrsquo = yr + (x - xr) sin ө - (y ndash yr) cos ө
General Pivot-Point Rotation
bull Rotations about any selected pivot point (xryr)
bull Translate-rotate-translate
General Pivot Point Rotation
To rotate about P1 (x1y1) the following sequence of the fundamental transformations are needed
1Translate the object by (-x1 -y1) 2Rotate (Oslash)3Translate the object by (x1 y1)
General Pivot-Point Rotation
100
10
01
100
0cossin
0sincos
100
10
01
r
r
r
r
y
x
y
x
100
sin)cos1(cossin
sin)cos1(sincos
rr
rr
xy
yx
)()()()( rrrrrr yxRyxTRyxT
Scaling
Alters the size of an object Scaling an object is implemented by
scaling the X and Y coordinates of each vertex in the object
Scaling
Scaling transformation
Scaling factors sx and sy
Uniform scaling if sx = sy
xsxx
ysyy
y
x
s
s
y
x
y
x
0
0
PSP
x
y
x
y2xs1ys
Matrix Representations and Homogeneous Coordinates- allow all transformations as matrix
multiplications
bull Homogeneous Coordinates
bull Matrix representations bull Translation
)()( hyxyx hh h
xx h
h
yy h
1100
10
01
1
y
x
t
t
y
x
y
x
h =1 for 2 D transformations
Matrix Representations
Matrix representations Scaling
Rotation
1100
00
00
1
y
x
s
s
y
x
y
x
1100
0cossin
0sincos
1
y
x
y
x
Composite Transformations-sequence of transformations
Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
)()( 1122 PttTttTP yxyx
PttTttT yxyx )()( 1122
)()()( 21211122 yyxxyxyx ttttTttTttT
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
Matrix Representation
Translation matrix
Row vector representation
P = (x y)T = (tx ty)P = (x y)
Prsquo= (xrsquo yrsquo) Prsquo = P + T
x lsquo = x + txyrsquo = y + ty
Rotation
Repositioning along a circular path in the xy plane y
xr
yr ө
P
Prsquo
(xr yr) is the rotation point ө is the rotation angle
Rotation
Rotation transformation
xrsquo=rcos(φ+θ)= rcos φ cos θ -rsin φ sin θ
yrsquo=rsin(φ + θ)= rcos φ sin θ +rsin φ cos θ
x=rcos φ y=rsin φ
Prsquo= R P
cossin
sincosR
xrsquo = x cos ө - y sin ө
yrsquo = x sin ө + y Cos ө rr
Φ
ө
Prsquo
P
(xrsquo yrsquo)
(x y)
(xrsquoyrsquo)
(xryr)
ө Φ
Rotation about arbitrary pivot point position
xrsquo = xr + (x - xr) cosө - (y ndash yr) sin ө
yrsquo = yr + (x - xr) sin ө - (y ndash yr) cos ө
General Pivot-Point Rotation
bull Rotations about any selected pivot point (xryr)
bull Translate-rotate-translate
General Pivot Point Rotation
To rotate about P1 (x1y1) the following sequence of the fundamental transformations are needed
1Translate the object by (-x1 -y1) 2Rotate (Oslash)3Translate the object by (x1 y1)
General Pivot-Point Rotation
100
10
01
100
0cossin
0sincos
100
10
01
r
r
r
r
y
x
y
x
100
sin)cos1(cossin
sin)cos1(sincos
rr
rr
xy
yx
)()()()( rrrrrr yxRyxTRyxT
Scaling
Alters the size of an object Scaling an object is implemented by
scaling the X and Y coordinates of each vertex in the object
Scaling
Scaling transformation
Scaling factors sx and sy
Uniform scaling if sx = sy
xsxx
ysyy
y
x
s
s
y
x
y
x
0
0
PSP
x
y
x
y2xs1ys
Matrix Representations and Homogeneous Coordinates- allow all transformations as matrix
multiplications
bull Homogeneous Coordinates
bull Matrix representations bull Translation
)()( hyxyx hh h
xx h
h
yy h
1100
10
01
1
y
x
t
t
y
x
y
x
h =1 for 2 D transformations
Matrix Representations
Matrix representations Scaling
Rotation
1100
00
00
1
y
x
s
s
y
x
y
x
1100
0cossin
0sincos
1
y
x
y
x
Composite Transformations-sequence of transformations
Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
)()( 1122 PttTttTP yxyx
PttTttT yxyx )()( 1122
)()()( 21211122 yyxxyxyx ttttTttTttT
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
Rotation
Repositioning along a circular path in the xy plane y
xr
yr ө
P
Prsquo
(xr yr) is the rotation point ө is the rotation angle
Rotation
Rotation transformation
xrsquo=rcos(φ+θ)= rcos φ cos θ -rsin φ sin θ
yrsquo=rsin(φ + θ)= rcos φ sin θ +rsin φ cos θ
x=rcos φ y=rsin φ
Prsquo= R P
cossin
sincosR
xrsquo = x cos ө - y sin ө
yrsquo = x sin ө + y Cos ө rr
Φ
ө
Prsquo
P
(xrsquo yrsquo)
(x y)
(xrsquoyrsquo)
(xryr)
ө Φ
Rotation about arbitrary pivot point position
xrsquo = xr + (x - xr) cosө - (y ndash yr) sin ө
yrsquo = yr + (x - xr) sin ө - (y ndash yr) cos ө
General Pivot-Point Rotation
bull Rotations about any selected pivot point (xryr)
bull Translate-rotate-translate
General Pivot Point Rotation
To rotate about P1 (x1y1) the following sequence of the fundamental transformations are needed
1Translate the object by (-x1 -y1) 2Rotate (Oslash)3Translate the object by (x1 y1)
General Pivot-Point Rotation
100
10
01
100
0cossin
0sincos
100
10
01
r
r
r
r
y
x
y
x
100
sin)cos1(cossin
sin)cos1(sincos
rr
rr
xy
yx
)()()()( rrrrrr yxRyxTRyxT
Scaling
Alters the size of an object Scaling an object is implemented by
scaling the X and Y coordinates of each vertex in the object
Scaling
Scaling transformation
Scaling factors sx and sy
Uniform scaling if sx = sy
xsxx
ysyy
y
x
s
s
y
x
y
x
0
0
PSP
x
y
x
y2xs1ys
Matrix Representations and Homogeneous Coordinates- allow all transformations as matrix
multiplications
bull Homogeneous Coordinates
bull Matrix representations bull Translation
)()( hyxyx hh h
xx h
h
yy h
1100
10
01
1
y
x
t
t
y
x
y
x
h =1 for 2 D transformations
Matrix Representations
Matrix representations Scaling
Rotation
1100
00
00
1
y
x
s
s
y
x
y
x
1100
0cossin
0sincos
1
y
x
y
x
Composite Transformations-sequence of transformations
Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
)()( 1122 PttTttTP yxyx
PttTttT yxyx )()( 1122
)()()( 21211122 yyxxyxyx ttttTttTttT
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
Rotation
Rotation transformation
xrsquo=rcos(φ+θ)= rcos φ cos θ -rsin φ sin θ
yrsquo=rsin(φ + θ)= rcos φ sin θ +rsin φ cos θ
x=rcos φ y=rsin φ
Prsquo= R P
cossin
sincosR
xrsquo = x cos ө - y sin ө
yrsquo = x sin ө + y Cos ө rr
Φ
ө
Prsquo
P
(xrsquo yrsquo)
(x y)
(xrsquoyrsquo)
(xryr)
ө Φ
Rotation about arbitrary pivot point position
xrsquo = xr + (x - xr) cosө - (y ndash yr) sin ө
yrsquo = yr + (x - xr) sin ө - (y ndash yr) cos ө
General Pivot-Point Rotation
bull Rotations about any selected pivot point (xryr)
bull Translate-rotate-translate
General Pivot Point Rotation
To rotate about P1 (x1y1) the following sequence of the fundamental transformations are needed
1Translate the object by (-x1 -y1) 2Rotate (Oslash)3Translate the object by (x1 y1)
General Pivot-Point Rotation
100
10
01
100
0cossin
0sincos
100
10
01
r
r
r
r
y
x
y
x
100
sin)cos1(cossin
sin)cos1(sincos
rr
rr
xy
yx
)()()()( rrrrrr yxRyxTRyxT
Scaling
Alters the size of an object Scaling an object is implemented by
scaling the X and Y coordinates of each vertex in the object
Scaling
Scaling transformation
Scaling factors sx and sy
Uniform scaling if sx = sy
xsxx
ysyy
y
x
s
s
y
x
y
x
0
0
PSP
x
y
x
y2xs1ys
Matrix Representations and Homogeneous Coordinates- allow all transformations as matrix
multiplications
bull Homogeneous Coordinates
bull Matrix representations bull Translation
)()( hyxyx hh h
xx h
h
yy h
1100
10
01
1
y
x
t
t
y
x
y
x
h =1 for 2 D transformations
Matrix Representations
Matrix representations Scaling
Rotation
1100
00
00
1
y
x
s
s
y
x
y
x
1100
0cossin
0sincos
1
y
x
y
x
Composite Transformations-sequence of transformations
Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
)()( 1122 PttTttTP yxyx
PttTttT yxyx )()( 1122
)()()( 21211122 yyxxyxyx ttttTttTttT
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
(xrsquoyrsquo)
(xryr)
ө Φ
Rotation about arbitrary pivot point position
xrsquo = xr + (x - xr) cosө - (y ndash yr) sin ө
yrsquo = yr + (x - xr) sin ө - (y ndash yr) cos ө
General Pivot-Point Rotation
bull Rotations about any selected pivot point (xryr)
bull Translate-rotate-translate
General Pivot Point Rotation
To rotate about P1 (x1y1) the following sequence of the fundamental transformations are needed
1Translate the object by (-x1 -y1) 2Rotate (Oslash)3Translate the object by (x1 y1)
General Pivot-Point Rotation
100
10
01
100
0cossin
0sincos
100
10
01
r
r
r
r
y
x
y
x
100
sin)cos1(cossin
sin)cos1(sincos
rr
rr
xy
yx
)()()()( rrrrrr yxRyxTRyxT
Scaling
Alters the size of an object Scaling an object is implemented by
scaling the X and Y coordinates of each vertex in the object
Scaling
Scaling transformation
Scaling factors sx and sy
Uniform scaling if sx = sy
xsxx
ysyy
y
x
s
s
y
x
y
x
0
0
PSP
x
y
x
y2xs1ys
Matrix Representations and Homogeneous Coordinates- allow all transformations as matrix
multiplications
bull Homogeneous Coordinates
bull Matrix representations bull Translation
)()( hyxyx hh h
xx h
h
yy h
1100
10
01
1
y
x
t
t
y
x
y
x
h =1 for 2 D transformations
Matrix Representations
Matrix representations Scaling
Rotation
1100
00
00
1
y
x
s
s
y
x
y
x
1100
0cossin
0sincos
1
y
x
y
x
Composite Transformations-sequence of transformations
Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
)()( 1122 PttTttTP yxyx
PttTttT yxyx )()( 1122
)()()( 21211122 yyxxyxyx ttttTttTttT
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
General Pivot-Point Rotation
bull Rotations about any selected pivot point (xryr)
bull Translate-rotate-translate
General Pivot Point Rotation
To rotate about P1 (x1y1) the following sequence of the fundamental transformations are needed
1Translate the object by (-x1 -y1) 2Rotate (Oslash)3Translate the object by (x1 y1)
General Pivot-Point Rotation
100
10
01
100
0cossin
0sincos
100
10
01
r
r
r
r
y
x
y
x
100
sin)cos1(cossin
sin)cos1(sincos
rr
rr
xy
yx
)()()()( rrrrrr yxRyxTRyxT
Scaling
Alters the size of an object Scaling an object is implemented by
scaling the X and Y coordinates of each vertex in the object
Scaling
Scaling transformation
Scaling factors sx and sy
Uniform scaling if sx = sy
xsxx
ysyy
y
x
s
s
y
x
y
x
0
0
PSP
x
y
x
y2xs1ys
Matrix Representations and Homogeneous Coordinates- allow all transformations as matrix
multiplications
bull Homogeneous Coordinates
bull Matrix representations bull Translation
)()( hyxyx hh h
xx h
h
yy h
1100
10
01
1
y
x
t
t
y
x
y
x
h =1 for 2 D transformations
Matrix Representations
Matrix representations Scaling
Rotation
1100
00
00
1
y
x
s
s
y
x
y
x
1100
0cossin
0sincos
1
y
x
y
x
Composite Transformations-sequence of transformations
Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
)()( 1122 PttTttTP yxyx
PttTttT yxyx )()( 1122
)()()( 21211122 yyxxyxyx ttttTttTttT
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
General Pivot Point Rotation
To rotate about P1 (x1y1) the following sequence of the fundamental transformations are needed
1Translate the object by (-x1 -y1) 2Rotate (Oslash)3Translate the object by (x1 y1)
General Pivot-Point Rotation
100
10
01
100
0cossin
0sincos
100
10
01
r
r
r
r
y
x
y
x
100
sin)cos1(cossin
sin)cos1(sincos
rr
rr
xy
yx
)()()()( rrrrrr yxRyxTRyxT
Scaling
Alters the size of an object Scaling an object is implemented by
scaling the X and Y coordinates of each vertex in the object
Scaling
Scaling transformation
Scaling factors sx and sy
Uniform scaling if sx = sy
xsxx
ysyy
y
x
s
s
y
x
y
x
0
0
PSP
x
y
x
y2xs1ys
Matrix Representations and Homogeneous Coordinates- allow all transformations as matrix
multiplications
bull Homogeneous Coordinates
bull Matrix representations bull Translation
)()( hyxyx hh h
xx h
h
yy h
1100
10
01
1
y
x
t
t
y
x
y
x
h =1 for 2 D transformations
Matrix Representations
Matrix representations Scaling
Rotation
1100
00
00
1
y
x
s
s
y
x
y
x
1100
0cossin
0sincos
1
y
x
y
x
Composite Transformations-sequence of transformations
Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
)()( 1122 PttTttTP yxyx
PttTttT yxyx )()( 1122
)()()( 21211122 yyxxyxyx ttttTttTttT
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
To rotate about P1 (x1y1) the following sequence of the fundamental transformations are needed
1Translate the object by (-x1 -y1) 2Rotate (Oslash)3Translate the object by (x1 y1)
General Pivot-Point Rotation
100
10
01
100
0cossin
0sincos
100
10
01
r
r
r
r
y
x
y
x
100
sin)cos1(cossin
sin)cos1(sincos
rr
rr
xy
yx
)()()()( rrrrrr yxRyxTRyxT
Scaling
Alters the size of an object Scaling an object is implemented by
scaling the X and Y coordinates of each vertex in the object
Scaling
Scaling transformation
Scaling factors sx and sy
Uniform scaling if sx = sy
xsxx
ysyy
y
x
s
s
y
x
y
x
0
0
PSP
x
y
x
y2xs1ys
Matrix Representations and Homogeneous Coordinates- allow all transformations as matrix
multiplications
bull Homogeneous Coordinates
bull Matrix representations bull Translation
)()( hyxyx hh h
xx h
h
yy h
1100
10
01
1
y
x
t
t
y
x
y
x
h =1 for 2 D transformations
Matrix Representations
Matrix representations Scaling
Rotation
1100
00
00
1
y
x
s
s
y
x
y
x
1100
0cossin
0sincos
1
y
x
y
x
Composite Transformations-sequence of transformations
Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
)()( 1122 PttTttTP yxyx
PttTttT yxyx )()( 1122
)()()( 21211122 yyxxyxyx ttttTttTttT
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
General Pivot-Point Rotation
100
10
01
100
0cossin
0sincos
100
10
01
r
r
r
r
y
x
y
x
100
sin)cos1(cossin
sin)cos1(sincos
rr
rr
xy
yx
)()()()( rrrrrr yxRyxTRyxT
Scaling
Alters the size of an object Scaling an object is implemented by
scaling the X and Y coordinates of each vertex in the object
Scaling
Scaling transformation
Scaling factors sx and sy
Uniform scaling if sx = sy
xsxx
ysyy
y
x
s
s
y
x
y
x
0
0
PSP
x
y
x
y2xs1ys
Matrix Representations and Homogeneous Coordinates- allow all transformations as matrix
multiplications
bull Homogeneous Coordinates
bull Matrix representations bull Translation
)()( hyxyx hh h
xx h
h
yy h
1100
10
01
1
y
x
t
t
y
x
y
x
h =1 for 2 D transformations
Matrix Representations
Matrix representations Scaling
Rotation
1100
00
00
1
y
x
s
s
y
x
y
x
1100
0cossin
0sincos
1
y
x
y
x
Composite Transformations-sequence of transformations
Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
)()( 1122 PttTttTP yxyx
PttTttT yxyx )()( 1122
)()()( 21211122 yyxxyxyx ttttTttTttT
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
Scaling
Alters the size of an object Scaling an object is implemented by
scaling the X and Y coordinates of each vertex in the object
Scaling
Scaling transformation
Scaling factors sx and sy
Uniform scaling if sx = sy
xsxx
ysyy
y
x
s
s
y
x
y
x
0
0
PSP
x
y
x
y2xs1ys
Matrix Representations and Homogeneous Coordinates- allow all transformations as matrix
multiplications
bull Homogeneous Coordinates
bull Matrix representations bull Translation
)()( hyxyx hh h
xx h
h
yy h
1100
10
01
1
y
x
t
t
y
x
y
x
h =1 for 2 D transformations
Matrix Representations
Matrix representations Scaling
Rotation
1100
00
00
1
y
x
s
s
y
x
y
x
1100
0cossin
0sincos
1
y
x
y
x
Composite Transformations-sequence of transformations
Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
)()( 1122 PttTttTP yxyx
PttTttT yxyx )()( 1122
)()()( 21211122 yyxxyxyx ttttTttTttT
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
Scaling
Scaling transformation
Scaling factors sx and sy
Uniform scaling if sx = sy
xsxx
ysyy
y
x
s
s
y
x
y
x
0
0
PSP
x
y
x
y2xs1ys
Matrix Representations and Homogeneous Coordinates- allow all transformations as matrix
multiplications
bull Homogeneous Coordinates
bull Matrix representations bull Translation
)()( hyxyx hh h
xx h
h
yy h
1100
10
01
1
y
x
t
t
y
x
y
x
h =1 for 2 D transformations
Matrix Representations
Matrix representations Scaling
Rotation
1100
00
00
1
y
x
s
s
y
x
y
x
1100
0cossin
0sincos
1
y
x
y
x
Composite Transformations-sequence of transformations
Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
)()( 1122 PttTttTP yxyx
PttTttT yxyx )()( 1122
)()()( 21211122 yyxxyxyx ttttTttTttT
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
Matrix Representations and Homogeneous Coordinates- allow all transformations as matrix
multiplications
bull Homogeneous Coordinates
bull Matrix representations bull Translation
)()( hyxyx hh h
xx h
h
yy h
1100
10
01
1
y
x
t
t
y
x
y
x
h =1 for 2 D transformations
Matrix Representations
Matrix representations Scaling
Rotation
1100
00
00
1
y
x
s
s
y
x
y
x
1100
0cossin
0sincos
1
y
x
y
x
Composite Transformations-sequence of transformations
Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
)()( 1122 PttTttTP yxyx
PttTttT yxyx )()( 1122
)()()( 21211122 yyxxyxyx ttttTttTttT
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
Matrix Representations
Matrix representations Scaling
Rotation
1100
00
00
1
y
x
s
s
y
x
y
x
1100
0cossin
0sincos
1
y
x
y
x
Composite Transformations-sequence of transformations
Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
)()( 1122 PttTttTP yxyx
PttTttT yxyx )()( 1122
)()()( 21211122 yyxxyxyx ttttTttTttT
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
Composite Transformations-sequence of transformations
Translations
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
)()( 1122 PttTttTP yxyx
PttTttT yxyx )()( 1122
)()()( 21211122 yyxxyxyx ttttTttTttT
100
10
01
100
10
01
100
10
01
21
21
1
1
2
2
yy
xx
y
x
y
x
tt
tt
t
t
t
t
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
Composite Transformations
Scaling
100
00
00
100
00
00
100
00
00
21
21
1
1
2
2
yy
xx
y
x
y
x
ss
ss
s
s
s
s
)()()( 21211122 yyxxyxyx ssssSssSssS
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
Composite Transformations
Rotations )()( 12 PRRP
PRR )()( 12
)()()( 2112 RRR
100
0)cos()sin(
0)sin()cos(
100
0cossin
0sincos
100
0cossin
0sincos
2121
2121
11
11
22
22
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
General Scaling Directions
Scaling factors sx and sy scale objects along the x and y directions
We scale an object in other directions with scaling factors s1 and s2
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
General Scaling Directions
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
Concatenation Properties
Matrix multiplication is associative
AmiddotB middotC = (AmiddotB )middotC = Amiddot(B middotC) Transformation products may not be commutative
Be careful about the order in which the composite matrix is evaluated
Except for some special cases Two successive rotations Two successive translations Two successive scaling rotation and uniform scaling
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
Concatenation Properties
Reversing the order A sequence of transformations is performed may affect
the transformed position of an object
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane
Reflection
A transformation produces a mirror image of an object Axis of reflection
A line in the xy plane A line perpendicular to the xy plane The mirror image is obtained by rotating the object
1800 about the reflection axis Rotation path
Axis in xy plane in a plane perpendicular to the xy plane
Axis perpendicular to xy plane in the xy plane