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BCET
Contents
n Translation
n Scaling
n Rotation
n Rotations about an arbitrary axis in space
n Other Transformations
n Coordinate Transformations
BCET
Transformation in 3D
n Transformation Matrix
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1131
1333
000 SLIFCKHEBJGDA
3´3 : Scaling, Reflection, Shearing, Rotation3´3 : Scaling, Reflection, Shearing, Rotation
3´1 : Translation3´1 : Translation
1´3 : Homogeneous representation1´3 : Homogeneous representation
1´1 : Uniform global Scaling1´1 : Uniform global Scaling
BCET
3D Translation
n Translation of a Point
zyx tzztyytxx +=+=+= ',','
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11000100010001
1'''
zyx
ttt
zyx
z
y
x
xz
y
BCET
3D Scaling
n Uniform Scaling
zyx szzsyysxx ×=×=×= ',','
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11000000000000
1'''
zyx
ss
s
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y
x
xz
y
BCET
Relative Scaling
n Scaling with a Selected Fixed Position
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---
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=---××
11000100010001
1000000000000
1000100010001
1'''
),,(),,(),,(zyx
zyx
ss
s
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zyxTsssSzyxTf
f
f
z
y
x
f
f
f
fffzyxfff
x x x xzzzz
y y y y
Original position Translate Scaling Inverse Translate
BCET
3D Rotation
n Coordinate-Axes Rotationsn X-axis rotationn Y-axis rotationn Z-axis rotation
n General 3D Rotationsn Rotation about an axis that is parallel to one of the
coordinate axesn Rotation about an arbitrary axis
BCET
Coordinate-Axes Rotations
n Z-Axis Rotation n X-Axis Rotation n Y-Axis Rotation
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11000010000cossin00sincos
1'''
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110000cossin00sincos00001
1'''
zyx
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-=
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110000cos0sin00100sin0cos
1'''
zyx
zyx
z
y
xz
y
x
BCET
Order of Rotations
n Order of Rotation Affects Final Positionn X-axisè Z-axis
n Z-axisè X-axis
BCET
General 3D Rotations
n Rotation about an Axis that is Parallel to One ofthe Coordinate Axesn Translate the object so that the rotation axis
coincides with the parallel coordinate axisn Perform the specified rotation about that axisn Translate the object so that the rotation axis is
moved back to its original position
BCET
General 3D Rotations
n Rotation about an Arbitrary Axis
Basic Idea1. Translate (x1, y1, z1) to the origin2. Rotate (x’2, y’2, z’2) on to the z
axis3. Rotate the object around the z-axis4. Rotate the axis to the original
orientation
5. Translate the rotation axis to the
original position
(x2,y2,z2)
(x1,y1,z1)
x
z
y
R-1
T-1
R
T
( ) ( ) ( ) ( ) ( ) ( )TRRRRRTR abqbaq xyzyx111 ---=
BCET
General 3D Rotations
n Step 1. Translation
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=
1000100010001
1
1
1
zyx
T
(x2,y2,z2)
(x1,y1,z1)
x
z
y
BCET
General 3D Rotations
n Step 2. Establish [ TR ]ax x axis
( )úúúú
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=
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=
10000//00//00001
10000cossin00sincos00001
dcdbdbdc
x aaaa
aR
(a,b,c)(0,b,c)
ProjectedPoint a a
RotatedPoint
dc
cbc
db
cbb
=+
=a
=+
=a
22
22
cos
sin
x
y
z
BCET
Arbitrary Axis Rotation
n Step 3. Rotate about y axis by
(a,b,c)
(a,0,d)b
l
d
22
222222
cos,sin
cbd
dacballd
la
+=
+=++=
== bb
( )úúúú
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=
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=
10000/0/00100/0/
10000cos0sin00100sin0cos
ldla
lald
y bb
bb
bRx
y
ProjectedPoint
zRotated
Point
b
BCET
Arbitrary Axis Rotation
n Step 4. Rotate about z axis by the desiredangle q
q
l
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qzR
y
x
z
BCET
Arbitrary Axis Rotation
n Step 5. Apply the reverse transformation toplace the axis back in its initial position
x
y
l
l
z
( ) ( )
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---
=---
10000cos0sin00100sin0cos
10000cossin00sincos00001
1000100010001
1
1
1
111
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BCET
Find the new coordinates of a unit cube 90º-rotatedabout an axis defined by its endpoints A(2,1,0) and
B(3,3,1).
Find the new coordinates of a unit cube 90º-rotatedabout an axis defined by its endpoints A(2,1,0) and
B(3,3,1).
A Unit Cube
Example
BCET
Example
n Step1. Translate point A to the origin
A’(0,0,0)x
z
y
B’(1,2,1)
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=
1000010010102001
T
BCET
xz
y
l
( )
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-=
1000
055
5520
05
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0001
axR
6121
55
51cos
552
52
122sin
222
22
=++=
==a
==+
=a
lB’(1,2,1)
a
Projected point(0,2,1)
B”(1,0,Ö5)
Example
n Step 2. Rotate axis A’B’ about the x axis by andangle a, until it lies on the xz plane.
BCET
x
z
y
l
( )
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=
1000
06300
66
0010
0660
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byR
630
65cos
66
61sin
==
==
b
b
b
B”(1,0, Ö 5)(0,0,Ö6)
Example
n Step 3. Rotate axis A’B’’ about the y axis by andangle , until it coincides with the z axis.b
BCET
Example
n Step 4. Rotate the cube 90° about the z axis
n Finally, the concatenated rotation matrix about thearbitrary axis AB becomes,
( ) ( ) ( ) ( ) ( ) ( )TRRRRRTR abbaq xyzyx °= --- 90111
( )úúúú
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=°
1000010000010010
90zR
BCET
( )
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-
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=
1000560.0167.0741.0650.0151.1075.0667.0742.0
742.1983.0075.0166.0
1000010010102001
1000
055
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1000010000010010
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Example
BCET
[ ] ( ) [ ]PRP ×=¢ q
[ ]
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=
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=¢
11111111076.0091.0560.0726.0817.0650.0301.1467.1483.0409.0151.1225.1184.0258.0484.0558.0
891.2909.1742.1725.2816.2834.1667.1650.2
11111111100110010000111111001100
1000560.0167.0741.0650.0151.1075.0667.0742.0
742.1983.0075.0166.0
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Example
n Multiplying R(θ) by the point matrix of the original cube
BCET
Other Transformations
n Reflection Relative to the xy Plane
n Z-axis Shearxz
y
x
z
y
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11000010000100001
1zyx
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'''
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BCET
Other Transformations
n Z-axis ShearThe effect of this transformation matrix is to alter x and y-coordinatevalues by an amount that is proportional to the z value, while leaving thez coordinate unchanged.
q Boundaries of planes that are perpendicular to the z axis are thus shiftedby an amount proportional to z.
q An example of the effect of this shearing matrix on a unit cube isshown in Fig., for shearing values a = b = 1. Shearing matrices forthe x axis and y axis are defined similarly.
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110000100010001
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BCET
Coordinate Transformations
n Multiple Coordinate Systemn Local (modeling) coordinate systemn World coordinate scene
Local Coordinate System
BCET
Coordinate Transformations
n Multiple Coordinate Systemn Local (modeling) coordinate systemn World coordinate scene
Word Coordinate System
BCET
Coordinate Transformations
n Example – Simulation of Tractor movementn As tractor moves, tractor coordinate system and
front-wheel coordinate system move in worldcoordinate system
n Front wheels rotate in wheel coordinate system
BCET
Coordinate Transformations
n Transformation of an Object Description fromOne Coordinate System to Another
n Transformation Matrixn Bring the two coordinates systems into alignment1. Translation
x
y
z (0,0,0)
y’
z’
x’u'y
u'z
u'x(x0,y0,z0)x
y
z (0,0,0)
u'y
u'z
u'x
( )000 ,, zyx ---T
BCET
Coordinate Transformations
2. Rotation & Scaling
x
y
z (0,0,0)
y’
z’
x’u'y
u'z
u'x x
y
z (0,0,0) x
y
z (0,0,0)
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¢¢¢¢¢¢¢¢¢
=
1000000
321
321
321
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R
