Lecture_Chapter4_a

# Directions and scaling factors are given by

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Unformatted text preview: t Transformation 22 A Brief Review of Linear Algebra •  Scaling along coordinate axes ￿ sx 0 ￿￿ ￿ ￿ ￿ 0 a sx a = sy b sy b Uniform (isotropic) scaling if sx = sy Anisotropic scaling if sx ￿= sy Area (volume) change: sx · sy ECS 175 Chapter 4: Object Transformation 23 A Brief Review of Linear Algebra •  Scaling ￿ sx sxy ￿￿ ￿ ￿ ￿ sxy a s a + sxy b =x sy b sxy a + sy b Symmetric matrices scale objects along “arbitrary” directions. Directions and scaling factors are given by eigenvectors and eigenvalues. M e 1 = λ1 e 1 M e 2 = λ2 e 2 Area (volume) change: Absolute value of determinant (product of eigenvalues) ECS 175 Chapter 4: Object Transformation 24 A Brief Review of Linear Algebra •  Determinant ￿ ￿ ￿a b c ￿ ￿ ￿ ￿d e f ￿ = aei + bf g + cdh − ceg − bdi − af h det(M ) = ￿ ￿ ￿g h i ￿ •  Eigenvalues det(M − λI ) = 0 solve for λ (eigenvalues are solutions) det(M − λI ) = det ￿￿ 2 0 ￿ ￿ 0 1 −λ 3 0 ￿￿ ￿ 0 2−λ = det 1 0 0 3−λ ￿ •  Eigenvectors M e 1 = λ1 e 1 ECS 175 M e 2 = λ2 e 2 Chapter 4: Object Transformation solve for x,y components of e1,e2 25 A Brief Review of Linear Algebra •  Examples: Scaling ￿ ￿ ECS 175 2 0.5 2 0 0 1 ￿ ￿ 0.5 1 Chapter 4: Object Transformation 26 A Brief Review of Linear Algebra •  Rotation ￿ cos α sin α − sin α cos α ￿ counterclockwise Determinant: cos α · cos α − sin α · (− sin α) = 1 Inverse M −1 = M T ECS 175 Chapter 4: Object Transformation 27 A Brief Review of Linear Algebra   Constructing a rotation matrix M (1, 0)T = (cos α, sin α)T M (0, 1)T = (− sin α, cos α)T (a, b)T = a(1, 0)T + b(0, 1)T M (a, b)T = aM (1, 0)T + bM (0, 1)T M (a, b)T = (a cos α − b sin α, a sin α + b cos α)T ￿ ￿ cos α − sin α sin α cos α columns correspond to axes of new coordinate frame ECS 175 Chapter 4: Object Transformation 28...
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## This document was uploaded on 03/12/2014 for the course ECS 175 at UC Davis.

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