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Unformatted text preview: CS 455 Computer Graphics TwoDimensional Transformations The Point Transforms can be reduced to matrix operations. The reduction is an equivalence. Properties of matrix operations are properties of transformations. Properties of transformations are properties of matrix operations. The reduction is very useful. Two languages or frameworks in which to think about the problem. Linear Algebra Review: Matrices Matrices: An m n matrix A has m rows and n columns A matrix is square if m = n Matrix element a ij is at row i and column j The transpose of an m n matrix A is an n m matrix B = A T where b ij = a ji The zero matrix has a ij = 0 for all 1 i m and 1 j n The identity matrix I is a square matrix ( m = n ) such that:a ii = 1 for 1 i ma ij = 0 for i j = = 32 22 12 31 21 11 T a a a a a a A B = 32 31 22 21 12 11 a a a a a a A A is a 3 2 matrix = = 1 1 1 I Linear Algebra Review: Matrix Addition Matrix addition: Two m n matrices, A and B , can be added if they have the same number of rows and columns Matrix addition is defined as: C = A + B where c ij = a ij + b ij for 1 i m and 1 j n Matrix addition is associative, commutative, and distributive:Associative: ( A + B ) + C = A + ( B + C )Commutative: A + B = B + ADistributive: s ( A + B ) = s A + s B + + + + + + = + = + = 32 32 31 31 22 22 21 21 12 12 11 11 32 31 22 21 12 11 32 31 22 21 12 11 b a b a b a b a b a b a b b b b b b a a a a a a B A C Linear Algebra Review Matrix multiplication: An m l matrix A can be multiplied with an l n matrix B if the number of columns of A equals the number of rows of B Matrix multiplication is defined as: C = AB where for 1 i m and 1 j n Matrix multiplication is associative and distributive (but not commutative)Associative: ( AB ) C = A ( BC )Distributive: ( A + B ) C = AC + BC + + + + + + = = = 22 32 12 31 21 32 11 31 22 22 12 21 21 22 11 21 22 12 12 11 21 12 11 11 22 21 12 11 32 31 22 21 12 11 b a b a b a b a b a b a b a b a b a b a b a b a b b b b a a a a a a AB C = = l k kj ik ij b a c 1 Linear Algebra Review Vectors: A vector is an n 1 column matrix Dot (or inner) product: Dot Product v1 dot v2 = = = 3 2 1 2 2 2 1 2 1 1 v v v v v v T [ ] 7 3 2 1 2 3 2 1 2 2 1 = = = T v v [ ] 4 2 1 2 3 1 = = v v ORTHOGONAL General Transformations Want to be able to manipulate graphical objects:...
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 Winter '08
 Jones,M

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