1. 3D Geometric Transformation
고려대학교 그래픽스 연구실
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2. Contents Translation Scaling Rotation Other Transformations
Coordinate Transformations
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3. Transformation in 3D Transformation Matrix
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
Translation of a Point y x z
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5. 3D Scaling
Uniform Scaling y x z
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6. Relative Scaling Scaling with a Selected Fixed Position y y y y x x xz z z z
Original position
Translate
Scaling
Inverse Translate
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7. 3D Rotation Coordinate-Axes Rotations General 3D Rotations
X-axis rotation
Y-axis rotation
Z-axis rotation
General 3D Rotations
Rotation about an axis that is parallel to one of the coordinate axes
Rotation about an arbitrary axis
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8. Coordinate-Axes Rotations
Z-Axis Rotation
X-Axis Rotation
Y-Axis Rotation y y y x x x z z z
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9. Order of Rotations Order of Rotation Affects Final Position
X-axis  Z-axis
Z-axis  X-axis
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10. General 3D Rotations
Rotation about an Axis that is Parallel to One of the Coordinate Axes
Translate the object so that the rotation axis coincides with the parallel coordinate axis
Perform the specified rotation about that axis
Translate the object so that the rotation axis is moved back to its original position
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11. General 3D Rotations Rotation about an Arbitrary Axis Basic Idea T R
Translate (x1, y1, z1) to the origin
Rotate (x’2, y’2, z’2) on to the z axis
Rotate the object around the z-axis
Rotate the axis to the original orientation
Translate the rotation axis to the original position y T
(x2,y2,z2) R
(x1,y1,z1)
R-1 x z
T-1
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12. General 3D Rotations Step 1. Translation y (x2,y2,z2) (x1,y1,z1) x z
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13. General 3D Rotations Step 2. Establish [ TR ]x x axis y (0,b,c)
(a,b,c)
Projected Point   x z
Rotated Point
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14. Arbitrary Axis Rotation
Step 3. Rotate about y axis by  y
(a,b,c)
Projected Point l d x 
(a,0,d)
Rotated Point z
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15. Arbitrary Axis Rotation
Step 4. Rotate about z axis by the desired angle  y l x  z
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16. Arbitrary Axis Rotation
Step 5. Apply the reverse transformation to place the axis back in its initial position y l l x z
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17. Example
Ex) 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. Step1. Translate point A to the origin
Example
Step1. Translate point A to the origin x z y
B’(1,2,1)
A’(0,0,0)
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19. Example
Step 2. Rotate axis A’B’ about the x axis by and angle , until it lies on the xz plane. y
Projected point
(0,2,1)
B’(1,2,1) l  x z
B”(1,0,5)
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20. Example
Step 3. Rotate axis A’B’’ about the y axis by and angle , until it coincides with the z axis. y l  x
(0,0,6)
B”(1,0,  5) z
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21. Step 4. Rotate the cube 90° about the z axis
Example
Step 4. Rotate the cube 90° about the z axis
Finally, the concatenated rotation matrix about the arbitrary
axis AB becomes,
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22. Example
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23. Example Multiplying [TR]AB by the point matrix of the original cube
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24. Other Transformations
Reflection Relative to the xy Plane
Z-axis Shear y y z z x x
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25. Coordinate Transformations
Multiple Coordinate System
Hierarchical Modeling
As tractor moves, tractor coordinate system and front-wheel coordinate system move in world coordinate system
front wheels rotate in wheel coordinate system
When tractor turns, wheel coordinate system rotates in tractor system
World Coordinate System
Tractor Coordinate System
Front-Wheel Coordinate System
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26. Coordinate Transformations
Transformation of an Object Description from One Coordinate System to Another
Set up a translation that brings the new coordinate origin to the position of the other coordinate origin
Rotations that corresponding coordinate axes
Scaling transformation, if different scales are used in the two coordinates systems
Example y’ z’ x’
u’y
u’z
u’x
(x0,y0,z0) x y z
(0,0,0)
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