3D transformations Dr Nicolas Holzschuch University of Cape Town

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Presentation transcript:

3D transformations Dr Nicolas Holzschuch University of Cape Town e-mail: holzschu@cs.uct.ac.za Modified by Longin Jan Latecki latecki@temple.edu Sep. 11, 2002

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

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

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