3D Geometric Transformations

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Presentation transcript:

3D Geometric Transformations Computer Graphics 3D Geometric Transformations

Contents transformations continued transitioning from 2D to 3D y y x z Computer Graphics – IIa 2007/2008

Geometric Transformations in 3D same approach as in 2D also use homogeneous coordinates (for the same reasons) vectors/points from 3D to 4D transformation matrices are now 4×4 (instead of 3×3) Computer Graphics – IIa 2007/2008

Transformation Matrices in 3D translation translation vector (dx, dy, dz)T scaling for uniform scaling sx = sy = sz otherwise individual factors may differ mirroring using factors of -1 and 1 depending on the mirror plane Computer Graphics – IIa 2007/2008

Transformation Matrices in 3D rotation rotation around 3 axes possible now each has individual rotation matrix rotation around positive angles in right-handed coordinate system rotation axis stays unit vector in matrix Computer Graphics – IIa 2007/2008

Transformation Matrices in 3D rotation why are signs different from 2D case for Ry? y x contrast to 2D case: mirroring on x-axis  (x, -y) x z right-handed coordinate system z Computer Graphics – IIa 2007/2008

Concatenating Transformations in 3D matrix multiplication just as in 3D general transformation matrix in 3D scaling rotation translation Computer Graphics – IIa 2007/2008

Transformations: Summary geometric transformations: linear mapping from n to n we are interested in 2  2 and 3  3 transformations most relevant for CG: translation rotation scaling mirroring shearing Computer Graphics – IIa 2007/2008

Transformations: Summary unified representation of geometric transformations as matrices in homogeneous coordinates concatenation of transformation by multiplying the respective matrices order matters: with row vectors, the first transformation comes last in the sequence concatenated transformations can be pre-computed (saving run-time) Computer Graphics – IIa 2007/2008