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5.2 Three-Dimensional Geometric and Modeling Transformations
2D
3DConsideration for the z coordinate
5.2.1 Translation From position P=(x, y, z) to P’ (x’, y’,
z’)
11000
100
010
001
1
'
'
'
z
y
x
t
t
t
z
y
x
z
y
x
Or P’=T· P
An equivalent representation:
x’=x +tx y’=y + ty z’=z + tz
5.2.2 Rotation Designate an axis of rotation and the
amount of angular rotation
Coordinate-Axes Rotations
11000
0100
00cossin
00sincos
1
z
y
x
z
y
x
x' = x cosθ - y sinθ
y' = x sinθ + y cosθ
z' = z
Z-axis rotation equation:
Homogeneous coordinate form
Or P’= Rz(θ)· P
a cyclic permutation of the coordinate parameters x, y
x → y → z → x
x' = x cosθ - y sinθ
y' = x sinθ + y cosθ
z' = z
Z-axis rotation equation:
y' = y cosθ - z sinθ
z' = y sinθ + z cosθ
x' = x
X-axis rotation equation:
11000
0100
00cossin
00sincos
1
z
y
x
z
y
x
11000
0cossin0
0sincos0
0001
1
z
y
x
z
y
x
Or P’= Rz(θ)· P Or P’= Rx(θ)· P
y' = y cosθ - z sinθ
z' = y sinθ + z cosθ
x' = x
X-axis rotation equation:
11000
0cossin0
0sincos0
0001
1
z
y
x
z
y
x
z' = z cosθ – x sinθ
x' = z sinθ + x cosθ
y' = y
Y-axis rotation equation:
11000
0cos0sin
0010
0sin0cos
1
z
y
x
z
y
x
Or P’= Ry(θ)· POr P’= Rx(θ)· P
General Three-Dimensional Rotations
an object is to be rotated about an axis that is parallel to one of the coordinate axes
Step 1: Translate the object so that the rotation axis coincides with the parallel coordinate axis.
General Three-Dimensional Rotations
Step 2: Perform the specified rotation about that axis.
General Three-Dimensional Rotations
Step 3: Translate the object so that the rotation axis is moved back to its original position.
General Three-Dimensional Rotations
rotation about an arbitrary axis (five steps)Step 1: Translate the object so that the
rotation axis passes through the coordinate origin.
rotation about an arbitrary axis (five steps)Step 2:Rotate the object so that the axis of rotation coincides with one of the coordinate axes.Step 3:Perform the specified rotation about that coordinate axis.
rotation about an arbitrary axis (five steps)Step 4: Apply inverse rotations to bring the rotation
axis back to its original orientation.Step 5: Apply the inverse translation to bring the
rotation axis back to its original position.
5.2.3 Scaling From position P=(x, y, z) to P’ (x’, y’,
z’)
11000
000
000
000
1
'
'
'
z
y
x
s
s
s
z
y
x
z
y
x
Or P’=S· P
An equivalent representation:
x' = x' · sx, y' = y · sy, z' = z · sz
Sx=Sy=Sz=2
Scaling with respect to a fixed position (xf, yf, zf,)
Step 1: Translate the fixed point to the origin.Step 2: Scale the object relative to the coordinate origin .Step 3: Translate the fixed point back to its original
position.
5.2.4 Other Transformation ---- reflection
1000
0100
0010
0001
zRF
The matrix representation for this reflection of points relative to the xy plane is
5.2.4 Other Transformation
1000
0100
0010
0001
xRF
The matrix representation for this reflection of points relative to the yz plane is
1000
0100
0010
0001
RFy
The matrix representation for this reflection of points relative to the zx plane is
5.2.4 Other Transformation ---- shear
As an example of three-dimensional shearing, the following transformation produces a z-axis shear:
1000
0100
010
001
b
a
SH z
11000
0100
010
001
1
z
y
x
b
a
z
y
x
a = b = 1