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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
 Linear Algebra, Matrices, Cartesian Coordinate System, θ

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