january 19, 20161. y x z translations objects are usually defined relative to their own coordinate...
DESCRIPTION
Translations Objects are usually defined relative to their own coordinate system. We can translate points in space to new positions by adding offsets to their coordinates. The following figure shows the effect of translating a teapot. January 19, 20163TRANSCRIPT
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y
X
Z
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Translations
Objects are usually defined relative to their own coordinate system. We can translate points in space to new positions by adding offsets to their coordinates.
The following figure shows the effect of translating a teapot.
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Properties of Translation
v=v)0,0,0(T=v),,(),,( zyxzyx tttTsssT
=
=v),,(1zyx tttT
v),,(),,( zyxzyx tttTsssT v),,(),,( zyxzyx sssTtttT
v),,( zzyyxx tststsT
v),,( zyx tttT
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Translation Revisited
=
1
'
'
'
zyx
1000100010001
ttt
z
y
x
1zyx
=T(tx, ty, tz )
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Scaling
zsysxs
zyx
z
y
x
'''
z
y
x
zyx
ss
ssssS
000000
),,(
Uniform scaling iff zyx sss
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3D Rotations
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An Alternative View
• We can view the rotation around an arbitrary axis as a set of simpler steps
• We know how to rotate and translate around the world coordinate system
• Can we use this knowledge to perform the rotation?
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Rotations about an arbitrary axis
Rotate by around a unit axis r
r
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Rotation about an arbitrary axisA rotation matrix for any axis that does not coincide with a coordinate axis can be set up as a composite transformation involving combination of translations and the coordinate-axes rotations.
1.Translate the object so that the rotation axis passes through the coordinate origin
2. Rotate the object so that the axis rotation coincides with one of the coordinate axes
3. Perform the specified rotation about that coordinate axis
4. Apply inverse rotation axis back to its original orientation
5. Apply the inverse translation to bring the rotation axis back to its original position
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1010000100001
zyx
T=
1010000100001
zyx
T-1=
•Translate origin to rotation axis
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A, B, C
V
)0 ,B, C(
θ
CBV 22
Sin θ = B / V
Cos θ = C / V
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10000cossin00sincos00001
Rx=
10000//00//00001
VCVBVBVC
Rx=
Sin θ = B / V
Cos θ = C / V
10000//00//00001
VCVBVBVC Rx
-1 =
• Rotation about X-axis
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Lß
A
V
CBAL 222
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• Rotation about Y-axis
10000cos0sin00100sin0cos
Ry=
sin = V / L
cos = A / L
10000/0/00100/0/
LVLA
LALV
Ry=
Ry-1=
10000/0/00100/0/
LVLA
LALV
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• Finally Rotate about Z-axis with angle
1000010000cossin00sincos
Rz =
Rotation about an arbitrary axis with a angle
R = T Rx Ry Rz Ry-1 Rx
-1 T-1
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Shear (Kxy, Kxz, Kyz)=
1000010001001
KKK
yz
xzxy
Shear
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