3D transformations Dr Nicolas Holzschuch University of Cape Town Modified by Longin Jan Latecki

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Presentation transcript:

3D transformations Dr Nicolas Holzschuch University of Cape Town Modified by Longin Jan Latecki

Map of the lecture Homogeneous coordinates in 3D Geometric transformations in 3D –translations, rotations, scaling,…

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

Translations in 3D

Scaling 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 Sanity check: a rotation of  /2 should change y in z, and z in - y x- axis is unmodified x- axis is unmodified

Rotation around y -axis Sanity check: a rotation of  /2 should change z in x, and x in - z y- axis is unmodified y- axis is unmodified

Rotation about z -axis Sanity check: a rotation of  /2 should change x in y, and y in - x z- axis is unmodified z- axis is unmodified

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