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3-D Geometric Transformations

3-D Geometric Transformations. Some of the material in these slides may have been adapted from the lecture notes of Graphics Lab @ Korea University. Contents. Translation Scaling Rotation Other Transformations Coordinate Transformations. Transformation in 3D. Transformation Matrix.

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3-D Geometric Transformations

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  1. 3-D GeometricTransformations Some of the material in these slides may have been adapted from the lecture notes of Graphics Lab @ Korea University

  2. Contents • Translation • Scaling • Rotation • Other Transformations • Coordinate Transformations

  3. Transformation in 3D • Transformation Matrix 33 : Scaling, Reflection, Shearing, Rotation 31 : Translation 11 : Uniform global Scaling 13 : Homogeneous representation

  4. 3D Translation • Translation of a Point y x z

  5. 3D Scaling • Uniform Scaling y x z

  6. Relative Scaling • Scaling with a Selected Fixed Position y y y y x x x x z z z z Original position Translate Scaling Inverse Translate

  7. 3D Rotation • Coordinate-Axes 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

  8. Rotation is counterclockwise

  9. Coordinate-Axes Rotations • Z-Axis Rotation • X-Axis Rotation • Y-Axis Rotation y y y x x x z z z

  10. 3D line equation • Given A (x0, y0, z0) and B (x1, y1, z1) forming a line. • The equation of this line can be formulated as follows:

  11. Order of Rotations • Order of Rotation Affects Final Position • X-axis  Z-axis • Z-axis  X-axis

  12. General 3D Rotations • Rotation about an Axis that is Parallel to One of the Coordinate Axes • Translatethe 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

  13. Rotation about an arbitrary axis

  14. General 3D Rotations • Rotation about an Arbitrary Axis • Basic Idea • 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

  15. y (x2,y2,z2) (x1,y1,z1) x z General 3D Rotations • Step 1. Translation

  16. General 3D Rotations • Step 2. Establish [ TR ]xx axis y (0,b,c) (a,b,c) Projected Point   x z Rotated Point

  17. General 3D Rotations • Step 2. Establish [ TR ]xx axis y (0,b,c) (a,b,c) Projected Point   x a (0,b,c) (0,b,c) (a,b,c) z Rotated Point b  (0,0,c) (0,0,0) c (0,0,0)

  18. Arbitrary Axis Rotation • Step 3. Rotate about y axis by  y (a,b,c) l Projected Point d x  (a,0,d) Rotated Point (0,0,0) z  l d (a,0,d) a (0,0,d)

  19. Arbitrary Axis Rotation • Step 4. Rotate about z axis by the desired angle  y l x  z

  20. Arbitrary Axis Rotation • Step 5. Apply the reverse transformation to place the axis back in its initial position y l l x z

  21. Example 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

  22. y B’(1,2,1) x z Example • Step1. Translate point A to the origin A’(0,0,0)

  23. Example • Step 2. Rotate axis A’B’ about the x axis by an 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)

  24. 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

  25. Example • Step 4. Rotate the cube 90° about the z axis • Finally, the concatenated rotation matrix about the arbitrary axis AB becomes,

  26. Example

  27. Example • Multiplying R(θ) by the point matrix of the original cube

  28. Other Transformations • Reflection Relative to the xy Plane • Z-axis Shear y y z x z x

  29. Coordinate Transformations • Multiple Coordinate System • Local (modeling) coordinate system • World coordinate scene Local Coordinate System

  30. Coordinate Transformations • Multiple Coordinate System • Local (modeling) coordinate system • World coordinate scene Word Coordinate System

  31. Coordinate Transformations • Example – Simulation of Tractor movement • As tractor moves, tractor coordinate system and front-wheel coordinate systemmove in world coordinate system • Front wheelsrotate in wheel coordinate system

  32. y’ y y x’ u'y u'x (x0,y0,z0) x x (0,0,0) (0,0,0) z z u'z z’ Coordinate Transformations • Transformation of an Object Description from One Coordinate System to Another • Transformation Matrix • Bring the two coordinates systems into alignment 1. Translation u'y u'x u'z

  33. y y y x x x (0,0,0) (0,0,0) (0,0,0) z z z Coordinate Transformations 2. Rotation & Scaling y’ x’ u'y u'y u'y u'x u'x u'x u'z u'z u'z z’

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