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Matrices, Rotations, Translations, Transformations

Matrices, Rotations, Translations, Transformations. 600.161 Exploring Vision in the Real World. Matrix representation. A linear system can be represented in a matrix multiplication format: 2a + 3b = 6 7a + 4b = 8. a b. =. 6 8. 3 7 4. Translations.

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Matrices, Rotations, Translations, Transformations

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  1. Matrices, Rotations, Translations, Transformations 600.161 Exploring Vision in the Real World

  2. Matrix representation A linear system can be represented in a matrix multiplication format: 2a + 3b = 6 7a + 4b = 8 a b = 6 8 3 7 4

  3. Translations • Translations slide an object a fixed distance in a given direction

  4. 2D Translations • 2D translations can be represented by a vector with 2 numbers [x, y] = [6, 0]

  5. Rotations • Rotations turn an object about a fixed point (center of rotation) a fixed angle. Y X

  6. 2D Rotations • 2D rotations can be represented by one number (in degrees or radians) [90]

  7. 3D Translations • 3D translations can be represented by a vector with 3 numbers [x, y, z] = [6, 0, 3]’

  8. 3D Rotations • 3D rotations can be represented by a vector with 3 numbers. The angle to rotate about each axis. about [x, y, z] = [90, 45, 180]

  9. cos(pi/2) = 0 sin(pi/2) =1

  10. 3D Rotations • Order of axes in which you rotate is important! • Changing the order of rotations changes the final location of the object!

  11. Transformation • A transformation is a combination of a rotation and a translation • Whether you rotate or translate first is also important!

  12. Uses • Transformations used extensively in computer vision and robotics.

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