Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13CSC 480 Computer GraphicsK. KirbySpring 2006Characteristics of Linear TransformationsA qualitative reviewLinear algebra reviewMaps R2 R2 and R3 R3 Linear maps R2 R2 and R3 R3 It's all about what happens to the basis.Definition of linearity.Some Kinds of 2D Linear OperatorsIdentityUniform scaleNon-uniform scaleSimple reflectionRotationShear Singular operatorsSymmetric operatorsGeneral operatorsPropertiesDeterminantInverseRankImageKernelSingular values & vectorsCondition numberEigenvalues & eigenvectorsDeterminants| det M | = the factor by which M changes measuresign det M = the change in orientation caused by Mdet M = 1det M = -1/2det M = 0Q: Why does det AB = det A det B ?RankThe rank of M is the dimension of its image.Four different 3D operators M with different ranks.rank 2rank 1rank 0“singular”det = 0rank 3“invertible”det  0Singular ValuesThe singular values of M are the principle radii of the imageof the unit sphere under M.This image is an ellipse in 2D, an ellipsoid in 3D, etc. 1 = 42 = 2/3MThe condition  of M is the ratio of the largestto smallest nonzero singular value.It measures the “squash” of M.   1.  = 6If  is large, then Mx=b is hard to solve numerically for x.Why?Eigenvalues and EigenvectorsIf 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. xMx = 2xyMydifferent direction-y not a real eigenvectorsame direction-x is a real eigenvectorwith eigenvalue 2MOrthogonal OperatorsAn operator is orthogonal if it maps the standardbasis (e1, e2, e3) to an orthonormal set (one-to-one). e1e2e3m1m2m3This means m1•m1 = 1, m1• m2 = 0, etc.In short: MTM = ISo for an orthogonal matrix, the inverse is merely the transpose. A rotation is an orthogonal operator with det = 1. m3m2m1M =Affine TransformationsAn affine transformation is a linear transformation followedby a translation: A(x) = Mx + d.dTranslations in N dimensions can be represented byshears in N+1 dimensions. z=1 planeM = 1 0 d1 0 1 d2 0 0 1dxyAffine Transformations: PracticeGive 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.1. A translation that takes the point (6,7,8) to the point (2,2,2). 2. A rotation by 45 degrees about the line from (1,1,1) to (2,1,1). 3. A reflection through the plane y= 2. 4. 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.) 5. The inverse of the transformation in #4.Representation of Spatial RotationsAngle-axis representation),ˆ(uuˆMatrix representation333231232221131211rrrrrrrrrRxRx 321ˆˆˆrrrRepresentation of Spatial Rotations : Example60oMatrix representation22112221231R[ 1 1 1 ]TAngle-axis representationwe will showthis in class 0.667 -0.333 0.667 0 0.667 0.667 -0.333 0-0.333 0.667 0.667 0 0 0 0 1double m[16] ;glMatrixMode( GL_MODELVIEW ) ;glLoadIdentity() ;glRotated( 60.0, 1,1,1 ) ;glGetDoublev( GL_MODELVIEW_MATRIX, m );print(m) ;You can confirm this in