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3D transformations. Dr Nicolas Holzschuch University of Cape Town e-mail: [email protected] Modified by Longin Jan Latecki [email protected] Sep. 11, 2002. Map of the lecture. Homogeneous coordinates in 3D Geometric transformations in 3D translations, rotations, scaling,….

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3d transformations

3D transformations

Dr Nicolas Holzschuch

University of Cape Town

e-mail: [email protected]

Modified by Longin Jan Latecki

[email protected]

Sep. 11, 2002

map of the lecture
Map of the lecture
  • Homogeneous coordinates in 3D
  • Geometric transformations in 3D
    • translations, rotations, scaling,…
homogeneous coordinates in 3d
Homogeneous coordinates in 3D
  • In order to model all transformations as matrices:
    • introduce a fourth coordinate, w
    • two vectors are equal if: x/w = x’/w’, y/w = y’/w’ and z/w=z’/w’
  • All transformations are 4x4 matrices
rotations in 3d
Rotations in 3D
  • One rotation: one axis and one angle
  • Matrix depends on both axis and angle
    • direct expression possible, from axis and angle, using cross-products
  • Rotations about axis have simple expression
    • other rotations express as composition of these rotations
rotation around x axis
Rotation around x-axis

x-axis is

unmodified

Sanity check: a rotation of p/2

should change y in z, and z in -y

rotation around y axis
Rotation around y-axis

y-axis is

unmodified

Sanity check: a rotation of p/2

should change z in x, and x in -z

rotation about z axis
Rotation about z-axis

z-axis is

unmodified

Sanity check: a rotation of p/2

should change x in y, and y in -x

any transformation in 3d
Any transformation in 3D
  • All transformations in 3D can be expressed as combinations of translations, rotations, scaling
    • expressed using matrix multiplication
  • Transformations can be expressed as 4x4 matrices
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