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CSC 480 Computer Graphics K. Kirby Spring 2006

CSC 480 Computer Graphics K. Kirby Spring 2006. Characteristics of Linear Transformations A qualitative review. Linear maps R 2  R 2 and R 3  R 3. It's all about what happens to the basis. Definition of linearity. Linear algebra review. Maps R 2  R 2 and R 3  R 3.

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CSC 480 Computer Graphics K. Kirby Spring 2006

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  1. CSC 480 Computer Graphics K. Kirby Spring 2006 Characteristics of Linear Transformations A qualitative review

  2. Linear maps R2R2 and R3R3 It's all about what happens to the basis. Definition of linearity. Linear algebra review Maps R2R2 and R3R3

  3. Properties Determinant Inverse Rank Image Kernel Singular values & vectors Condition number Eigenvalues & eigenvectors Some Kinds of 2D Linear Operators Identity Uniform scale Non-uniform scale Simple reflection Rotation Shear Singular operators Symmetric operators General operators

  4. det M = 1 det M = -1/2 det M = 0 Determinants | det M | = the factor by which M changes measure sign det M = the change in orientation caused by M Q: Why does det AB = det A det B ?

  5. “invertible” det  0 rank 3 rank 2 “singular” det = 0 rank 1 rank 0 Rank The rank of M is the dimension of its image. Four different 3D operators M with different ranks.

  6. The conditionof M is the ratio of the largest to smallest nonzero singular value. It measures the “squash” of M.  1.  = 6 Singular Values The singular values of M are the principle radii of the image of the unit sphere under M. This image is an ellipse in 2D, an ellipsoid in 3D, etc. 1 = 4 2 = 2/3 M If  is large, then Mx=b is hard to solve numerically for x. Why?

  7. Eigenvalues and Eigenvectors If a M leaves a the direction of a vector x unchanged, x is called a real eigenvector of M. The factor by which the length of x changes is called the eigenvalue of M for x. different direction- y not a real eigenvector y My M x Mx = 2x same direction- x is a real eigenvector with eigenvalue 2

  8. m1 m2 m3 m2 m1 m3 Orthogonal Operators An operator is orthogonal if it maps the standard basis (e1, e2, e3) to an orthonormal set (one-to-one). e2 M = e1 e3 This means m1•m1 = 1, m1• m2 = 0, etc. In short: MTM = I So for an orthogonal matrix, the inverse is merely the transpose. A rotation is an orthogonal operator with det = 1.

  9. Translations in N dimensions can be represented by shears in N+1 dimensions. d z=1 plane M = 1 0 d1 0 1 d2 0 0 1 y x Affine Transformations An affine transformation is a linear transformation followed by a translation: A(x) = Mx + d. d

  10. Affine Transformations: Practice • Give 4x4 matrices for the following affine transformations of three dimensional space. • You may leave the matrices in factored form if you like, but each entry must be in rational radical form. • A translation that takes the point (6,7,8) to the point (2,2,2). • A rotation by 45 degrees about the line from (1,1,1) to (2,1,1). • A reflection through the plane y= 2. • A transformation that takes the shape “A>” on the xy-plane centered at (1,1,0) to the shape “>” centered at the origin, still lying in the xy plane, scaled uniformly to half its size. (The z dimension is not affected.) • The inverse of the transformation in #4.

  11. Representation of Spatial Rotations Angle-axis representation x Matrix representation Rx

  12. Representation of Spatial Rotations : Example Angle-axis representation [ 1 1 1 ]T 60o You can confirm this in OpenGL: double m[16] ; glMatrixMode( GL_MODELVIEW ) ; glLoadIdentity() ; glRotated( 60.0, 1,1,1 ) ; glGetDoublev( GL_MODELVIEW_MATRIX, m ); print(m) ; we will show this in class Matrix representation 0.667 -0.333 0.667 0 0.667 0.667 -0.333 0 -0.333 0.667 0.667 0 0 0 0 1

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