Transcript Transformations

Matrices, Rotations,
Translations,
Transformations
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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
2 3
a
6
=
7 4
b
8
Translations
• Translations slide an object a fixed
distance in a given direction
2D Translations
• 2D translations can be represented by
a vector with 2 numbers
[x, y] = [6, 0]
Rotations
• Rotations turn an object about a fixed
point (center of rotation) a fixed angle.
Y
X
2D Rotations
• 2D rotations can be represented by one
number (in degrees or radians)
[90]
3D Translations
• 3D translations can be represented by
a vector with 3 numbers
[x, y, z] = [6, 0, 3]’
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]
cos(pi/2) = 0
sin(pi/2) =1
3D Rotations
• Order of axes in which you rotate is
important!
• Changing the order of rotations
changes the final location of the object!
Transformation
• A transformation is a combination of a
rotation and a translation
• Whether you rotate or translate first is
also important!
Uses
• Transformations used extensively in
computer vision and robotics.