1) The document discusses various 3D geometric transformations including translation, scaling, rotation, and coordinate transformations. 2) It explains transformations like uniform scaling, translation of a point, and rotation about the X, Y, and Z axes. 3) Rotation about an arbitrary axis is described as translating the axis to the origin, rotating it onto an axis, rotating the object, and translating the axis back.

1. Graphics
3D Geometric
Transformation
Sudipta Mondal
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2. Contents
BCET
n Translation
n Scaling
n Rotation
n Rotations about an arbitrary axis in space
n Other Transformations
n Coordinate Transformations
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3. Transformation in 3D
BCET
n Transformation Matrix
éA D G Jù é ù
êB E H Kú ê 3´ 3 3 ´ 1ú
ê ú Þ ê ú
êC F I Lú ê ú
ê ú ê ú
ë0 0 0 Sû ë 1´ 3 1 ´ 1û
3´3 : Scaling, Reflection, Shearing, Rotation
3´1 : Translation
1´1 : Uniform global Scaling
1´3 : Homogeneous representation
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4. 3D Translation
BCET
n Translation of a Point
x' = x + t x , y' = y + t y , z' = z + t z
y
é x 'ù é1 0 0 tx ùé xù
ê y 'ú ê 0 1 0 ty úê yú
ê ú=ê úê ú
ê z ' ú ê0 0 1 tz úê z ú
ê ú ê úê ú
x ë 1 û ë0 0 0 1 ûë1 û
z
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5. 3D Scaling
BCET
n Uniform Scaling
x' = x × s x , y' = y × s y , z' = z × sz
y
é x 'ù é s x 0 0 0ù é x ù
ê y 'ú ê 0 sy 0 0ú ê y ú
ê ú=ê úê ú
ê z 'ú ê 0 0 sz 0ú ê z ú
ê ú ê úê ú
x
ë1û ë 0 0 0 1û ë1 û
z
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6. Relative Scaling
BCET
n Scaling with a Selected Fixed Position
y y y y
z x z x z x z x
Original position Translate Scaling Inverse Translate
éx'ù é1 0 0 x f ùésx 0 0 0ùé1 0 0 - x f ùéxù
ê y'ú ê0 1 0 y f úê 0 sy 0 0úê0 1 0 - y f úê yú
T (x f , y f , z f ) × S(sx , sy , sz ) ×T(-x f ,-y f ,-z f ) = ê ú = ê úê úê úê ú
ê z'ú ê0 0 1 z f úê 0 0 sz 0úê0 0 1 - z f úê z ú
ê ú ê úê úê úê ú
ë 1 û ë0 0 0 1 ûë 0 0 0 1ûë0 0 0 1 ûë1û
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7. 3D Rotation
BCET
n Coordinate-Axes Rotations
n X-axis rotation
n Y-axis rotation
n Z-axis rotation
n General 3D Rotations
n Rotation about an axis that is parallel to one of the
coordinate axes
n Rotation about an arbitrary axis
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8. Coordinate-Axes Rotations
BCET
n Z-Axis Rotation n X-Axis Rotation n Y-Axis Rotation
é x'ù écos q - sin q 0 0ù é x ù é x'ù é1 0 0 0ù é x ù é x'ù é cos q 0 sin q 0ù é x ù
ê y'ú ê sin q cos q 0 0ú ê y ú ê y'ú ê0 cos q - sin q 0ú ê y ú ê y'ú ê 0 1 0 0ú ê y ú
ê ú=ê úê ú ê ú=ê úê ú ê ú=ê úê ú
ê z 'ú ê 0 0 1 0ú ê z ú ê z 'ú ê0 sin q cos q 0ú ê z ú ê z 'ú ê- sin q 0 cos q 0ú ê z ú
ê ú ê úê ú ê ú ê úê ú ê ú ê úê ú
ë1û ë 0 0 0 1û ë 1 û ë 1 û ë0 0 0 1û ë 1 û ë1û ë 0 0 0 1û ë 1 û
y y y
x x x
z z z
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9. Order of Rotations
BCET
n Order of Rotation Affects Final Position
n X-axis è Z-axis
n Z-axis è X-axis
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10. General 3D Rotations
BCET
n Rotation about an Axis that is Parallel to One of
the Coordinate Axes
n Translate the object so that the rotation axis
coincides with the parallel coordinate axis
n Perform the specified rotation about that axis
n Translate the object so that the rotation axis is
moved back to its original position
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11. General 3D Rotations
BCET
n Rotation about an Arbitrary Axis
y Basic Idea
T 1. Translate (x1, y1, z1) to the origin
(x2,y2,z2) 2. Rotate (x’2, y’2, z’2) on to the z
R
axis
(x1,y1,z1) 3. Rotate the object around the z-axis
4. Rotate the axis to the original
x R-1
orientation
z 5. Translate the rotation axis to the
T-1
original position
R (q ) = T -1R -1 (a )R -1 (b )R z (q )R y (b )R x (a )T
x y
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12. General 3D Rotations
BCET
n Step 1. Translation
y
(x2,y2,z2) é1 0 0 - x1 ù
ê0 1 0 - y1 ú
T=ê ú
(x1,y1,z1) ê0 0 1 - z1 ú
ê ú
x
ë0 0 0 1 û
z
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13. General 3D Rotations
BCET
n Step 2. Establish [ TR ]ax x axis
y b b
sin a = =
b +c
2 2 d
(0,b,c) c c
(a,b,c) cos a = =
b +c
2 2 d
Projected
Point a a é1 0 0 0ù é1 0 0 0ù
ê0 cos a - sin a 0 ú ê0 c / d -b/d 0ú
x R x (a ) = ê ú=ê ú
ê0 sin a cos a 0 ú ê0 b / d c/d 0ú
z ê ú ê ú
ë0 0 0 1 û ë0 0 0 1û
Rotated
Point
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14. Arbitrary Axis Rotation
BCET
n Step 3. Rotate about y axis by b
y a d
sin b = , cos b =
l l
l 2 = a 2 + b2 + c2 = a2 + d 2
(a,b,c)
d = b2 + c2
Projected l
Point écos b 0 - sin b 0ù éd / l 0 - a / l 0ù
d ê 0 1 0 0ú ê 0 1 0 0ú
b x R y (b ) = ê ú=ê ú
ê sin b 0 cos b 0ú ê a / l 0 d / l 0ú
(a,0,d) ê ú ê ú
ë 0 0 0 1û ë 0 0 0 1û
Rotated
z Point
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15. Arbitrary Axis Rotation
BCET
n Step 4. Rotate about z axis by the desired
angle q
y
écosq - sin q 0 0ù
ê sin q cosq 0 0ú
R z (q ) = ê ú
ê 0 0 1 0ú
l
x ê ú
q ë 0 0 0 1û
z
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16. Arbitrary Axis Rotation
BCET
n Step 5. Apply the reverse transformation to
place the axis back in its initial position
y
é1 0 0 - x1 ù é1 0 0 0ù
ê0 1 0 - y1 ú ê0 cos a sin a 0ú
T -1R -1 (a )R -1 ( )= ê
y b
úê ú
l
x
ê0 0 1 - z1 ú ê0 - sin a cos a 0ú
ê úê ú
ë0 0 0 1 û ë0 0 0 1û
é cos b 0 sin b 0ù
l ê 0 1 0 0ú
x ê ú
ê- sin b 0 cos b 0ú
ê ú
ë 0 0 0 1û
z
R (q ) = T -1R -1 (a )R -1 (b )R z (q )R y (b )R x (a )T
x y
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17. Example
BCET
Find the new coordinates of a unit cube 90º-rotated
about an axis defined by its endpoints A(2,1,0) and
B(3,3,1).
A Unit Cube
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18. Example
BCET
n Step1. Translate point A to the origin
y
é1 0 0 - 2ù
B’(1,2,1) ê0 1 0 - 1ú
T=ê ú
ê0 0 1 0ú
A’(0,0,0)
x ê ú
ë0 0 0 1û
z
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19. Example
BCET
n Step 2. Rotate axis A’B’ about the x axis by and
angle a, until it lies on the xz plane.
2 2 2 5
sin a = = =
2 2 + 12 5 5
1 5
cos a = =
y 5 5
Projected point
(0,2,1) B’(1,2,1)
l = 12 + 2 2 + 12 = 6
a l
é1 0 0 0ù
x ê 5 2 5 ú
z ê0 - 0ú
B”(1,0,Ö5) R x (a ) = ê 5 5 ú
ê0 2 5 5
0ú
ê 5 5 ú
ê0
ë 0 0 1ú
û
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20. Example
BCET
n Step 3. Rotate axis A’B’’ about the y axis by and
angle b , until it coincides with the z axis.
1 6
sin b = =
y 6 6
5 30
cos b = =
6 6
l é 30 6 ù
b x ê 0 - 0ú
(0,0,Ö6)
ê 6 6 ú
B”(1,0, Ö 5)
R y (b ) = ê 0 1 0 0ú
ê 6 30
z 0 0ú
ê 6 6 ú
ê
ë 0 0 0 1ú
û
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21. Example
BCET
n Step 4. Rotate the cube 90° about the z axis
é0 - 1 0 0 ù
ê1 0 0 0ú
R z (90°) = ê ú
ê0 0 1 0 ú
ê ú
ë0 0 0 1 û
n Finally, the concatenated rotation matrix about the
arbitrary axis AB becomes,
R (q ) = T -1R -1 (a )R -1 (b )R z (90°)R y (b )R x (a )T
x y
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22. Example
BCET
é1 0 0 0ù é 30 6 ù
é1 0 0 2ù ê 5 2 5 úê 0 0ú é 0 - 1 0 0ù
ê0 ú ê0 0ú ê 6 6 úê
1 0 1ú 5 5 0ú ê 1 0 0 0ú
R (q ) = ê ê úê 0 1 0 ú
ê0 0 1 0ú ê 2 5 5 6 30 ê0 0 1 0ú
ê ú 0 - 0ú ê - 0 0ú ê ú
ë0 0 0 1û ê 5 5 úê 6 6 ú 0 0
ë 0 1û
ê0
ë 0 0 1ú ê 0
ûë 0 0 1ú
û
é 30 6 ù é1 0 0 0ù
ê 0 - 0ú ê 5 2 5 ú é1 0 0 - 2ù
6 6 - 0 ú ê0
ê
0 1 0
ú ê0
0ú ê 5 5 1 0 - 1ú
ê úê ú
ê 6 30 2 5 5 ê0 0 1 0ú
0 0 ú ê0 0ú ê ú
ê 6 6 úê 5 5 ú 0 0 0 1û
ê ë
ë 0 0 0 1 ú ê0
ûë 0 0 1úû
é 0.166 - 0.075 0.983 1.742 ù
ê 0.742 0.667 0.075 - 1.151ú
=ê ú
ê - 0.650 0.741 0.167 0.560 ú
ê ú
ë 0 0 0 1 û
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23. Example
BCET
n Multiplying R(θ) by the point matrix of the original cube
[P¢] =R(q)×[P]
é 0.166 - 0.075 0.983 1.742 ù é0 0 1 1 0 0 1 1ù
ê 0.742 0.667 0.075 - 1.151ú ê1 1 1 1 0 0 0 0ú
[P¢] = ê úê ú
ê- 0.650 0.741 0.167 0.560 ú ê1 0 0 1 1 0 0 1ú
ê úê ú
ë 0 0 0 1 û ë1 1 1 1 1 1 1 1û
é 2.650 1.667 1.834 2.816 2.725 1.742 1.909 2.891 ù
ê- 0.558 - 0.484 0.258 0.184 - 1.225 - 1.151 - 0.409 - 0.483ú
=ê ú
ê 1.467 1.301 0.650 0.817 0.726 0.560 - 0.091 0.076 ú
ê ú
ë 1 1 1 1 1 1 1 1 û
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24. Other Transformations
BCET
n Reflection Relative to the xy Plane
é x' ù é1 0 0 0ù é x ù
y y ê y 'ú ê 0 1 0 0ú ê y ú
ê ú=ê úê ú
z ê z' ú ê0 0 - 1 0ú ê z ú
ê ú ê úê ú
z x x ë 1 û ë0 0 0 1û ë1 û
n Z-axis Shear
é x 'ù é1 0 a 0ù é x ù
ê y 'ú ê 0 1 b 0ú ê y ú
ê ú=ê úê ú
ê z ' ú ê0 0 1 0ú ê z ú
ê ú ê úê ú
ë 1 û ë0 0 0 1ûë1 û
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25. Other Transformations
BCET
n Z-axis Shear
The effect of this transformation matrix is to alter x and y-coordinate
values by an amount that is proportional to the z value, while leaving the
z coordinate unchanged.
q Boundaries of planes that are perpendicular to the z axis are thus shifted
by an amount proportional to z.
q An example of the effect of this shearing matrix on a unit cube is
shown in Fig., for shearing values a = b = 1. Shearing matrices for
the x axis and y axis are defined similarly.
é x 'ù é1 0 a 0ù é x ù
ê y 'ú ê 0 1 b 0ú ê y ú
ê ú=ê úê ú
ê z ' ú ê0 0 1 0ú ê z ú
ê ú ê úê ú
ë 1 û ë0 0 0 1ûë1 û
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26. Coordinate Transformations
BCET
n Multiple Coordinate System
n Local (modeling) coordinate system
n World coordinate scene
Local Coordinate System
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27. Coordinate Transformations
BCET
n Multiple Coordinate System
n Local (modeling) coordinate system
n World coordinate scene
Word Coordinate System
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28. Coordinate Transformations
BCET
n Example – Simulation of Tractor movement
n As tractor moves, tractor coordinate system and
front-wheel coordinate system move in world
coordinate system
n Front wheels rotate in wheel coordinate system
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29. Coordinate Transformations
BCET
n Transformation of an Object Description from
One Coordinate System to Another
n Transformation Matrix
n Bring the two coordinates systems into alignment
1. Translation
y’
y
u'y x’
y u'x u'y
(x0,y0,z0) u'x
x
z’ u'z z (0,0,0)
u'z
(0,0,0) x
z
T(- x0 , - y0 , - z0 )
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30. Coordinate Transformations
BCET
2. Rotation & Scaling
y’ y y y
u'y x’
u'x
x x (0,0,0) x
z (0,0,0) z (0,0,0)
z
z’ u'z
é u ¢1
x u¢2x u¢3x 0ù
êu ¢ u ¢y 2 u ¢y 3 0ú
R = ê y1 ú
ê u ¢1
z u ¢2
z u ¢3
z 0ú
ê ú
ë 0 0 0 1û
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