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CSI4130
Review Questions – 2D Transformations University of Ottawa – Universit ́e d’Ottawa
Jochen Lang
These questions are meant as a review of lecture material. The style of these questions is not necessarily a good indication of the style of the midterm (see the midterm examples instead).
1 2D Transformation 1.1 Scaling
Give the 2 × 2 matrix S for a scaling by 3 along the x-axis and by 2 along the y-axis. 1.2 Shear
Givethe2×2matrixHx forashearby2inxand2×2matrixHy forashearby3iny. Can you combine the two?
1.3 Rotation
Give the 2 × 2 matrix for a rotation by −90◦. 2 Homogenous Transform
For the matrices in question 1, add to each a translation by x = −1 and y = 2 which should take place after the scaling, shear, and rotation, respectively. Give the 3 × 3 homogeneous transform.
3 Matrix Decomposition
Use Matlab (or similar) to calculate the following decomposition of the matrix: M =
√2 3√2 √√
−232
3.1 SVD Decomposition
Give the matrices U, S and V for the SVD-Decomposition of M. Verify that M = USVT .
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3.2 Eigen Decomposition
Find the symmetric matrix A = MMT and calculate its Eigendecomposition A = RDRT
4 Point and Normal Transform in 3D
Apply a scaling in x = 2, followed by a rotation around x by 90◦ and a translation by y = 1
T √2 √2 T with the tangent vectors t0 = 2 − 2 0
2 −1 −5
t1=0 0 −1T anditsnormaln=√2 √2 0T.Verifythatthetransformednormalis
to the 3D vertex p =
still orthogonal to the transformed tangent vectors.
and
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