Lecture_Chapter4_a

Directions and scaling factors are given by

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This document was uploaded on 03/12/2014 for the course ECS 175 at UC Davis.

Ask a homework question - tutors are online