Matrices, Rotations, Translations, Transformations

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Matrices, Rotations, Translations, Transformations

• 600.161
• Exploring Vision in the Real World

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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

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Translations

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

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2D Translations

• 2D translations can be represented by a vector
  with 2 numbers

x, y 6, 0
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Rotations

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

Y
X
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2D Rotations

• 2D rotations can be represented by one number (in
  degrees or radians)

90
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3D Translations

• 3D translations can be represented by a vector
  with 3 numbers

x, y, z 6, 0, 3
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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
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cos(pi/2) 0 sin(pi/2) 1
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3D Rotations

• Order of axes in which you rotate is important!
• Changing the order of rotations changes the final
  location of the object!

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Transformation

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

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Uses

• Transformations used extensively in computer
  vision and robotics.