2D Matrix Transformations Lecture Slides

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$Assignment 2: Math Review$

Basic <strong>Transformations</strong> Uniform/isotropic Non-uniform/anisotropic Rotations Cos( θ ) − Sin( θ ) p' = p Sin( θ ) Cos( θ ) Rotate Scale Rotate Translate Shear -- not really “basic” Rotations Rotations Rotations are counter-clockwise Right hand rule Could do the other way, but we don’t Preserve lengths/distance to origin Orthonormal matrices Det == 1, not –1 Note: rotate by zero degrees gives identity matrix Note: rotate +/- 360 degrees stays the same Scales Scale Uniform/isotropic Non-uniform/anisotropic sx 0 p' = p 0 s y Scales If Sx and Sy are the same scale is uniform Scaling by Sx or Sy < 0 causes problems If both are –1, it looks like 180 degree rotation 2
Shears Translation ( method 1) Shear 1 H yx p' = p H xy 1 Translate t p' = p + t x y True, but not the right way to think about shears…. Does not look like the others….. Arbitrary matrix transformation For everything but translation we have: p ' = Ap In a short while, we will make translation fit in too. What does an arbitrary A matrix mean Singular Value Decomposition (or polar decomposition) For any matrix A, we can write SVD: A = QSR where Q and R are orthonormal and S is diagonal Also write Polar Decomposition: A = QRSR (Different Q! Still orthonormal.) Decomposing matrices We can force Q and R to have Det=1 so they are rotations. Any matrix is now: – Rotation:Rotation:Scale:Rotation – That means shear is not primitive (in some sense) Multiple transformations What does something like the following mean p ' = BAp “Apply A to p and then apply B to the result.” p ' = B( Ap) = ( BA) p = Cp Lots of transformations all get composted into one. But translations are being left out… 3
Page 1: 2D Matrix Transformations Consider
Page 5 and 6: Example Example Step 1: Translate p

2D Matrix Transformations Lecture Slides

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