Lecture_Chapter4_b

# 175 chapter 4 object transformation 45 rotation

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Unformatted text preview: ) 0 0 Chapter 4: Object Transformation 0 p1 − p1 cos(α) + p2 sin(α) 0 p2 − p1 sin(α) − p2 cos(α) · vlocal 1 0 0 1 44 Arbitrary Rotation •  Arbitrary rotations can be expressed as successive rotations around coordinate axes R = Rx ( α ) Ry ( β ) Rz ( γ ) •  Decomposition is not unique Rz (90◦ )Ry (90◦ ) = Ry (90◦ )Rx (−90◦ ) ECS 175 Chapter 4: Object Transformation 45 Rotation Around Arbitrary Axis •  Rotation around arbitrary axis v by α : •  Move fixed point (e.g., center of object) to origin: T (−p) •  Rotate axis of rotation, such that it aligns with z-axis: •  Rotate around z-axis by •  Undo alignment rotation: •  Undo translation: Ry ( β 2 ) R x ( β 1 ) α : Rz ( α ) Rx ( − β 1 ) Ry ( − β 2 ) T ( p) A = T ( p) R x ( − β 1 ) R y ( − β 2 ) R z ( α ) R y ( β 2 ) R x ( β 1 ) T ( − p) ‘Only’ need to find ECS 175 β 1 , β2 Chapter 4: Object Transformation 46 Rotation Around Arbitrary Axis A sequence of two rotations can align any vector with the z-axis ECS 175 Chapter 4: Object Transformation 47 Rotation Around Arbitrary Axis •  β1 , β2 do not need to be computed explicitly ￿ 2 2 d = vy + vz 1 0 0 0 vz /d −vy /d Rx ( β 1 ) = 0 vy /d vz /d 0 0 0 d 0 −vx 0 0 1 0 0 Ry ( β 2 ) = vx 0 d 0 00 0 1 ECS 175 Chapter 4: Object Transformation 0 0 0 1 48 Homogeneous Coordinates •  How to interpret a transformation matrix vworld 0. 5 0 = 0 0 0 0.5 0 0 0 0 0.5 0 p1 p2 · vlocal p3 1 Local coordinate frame (source) expressed in global coordinates (target). Inverse matrix reverses transformation. ECS 175 Chapter 4: Object Transformation 49 Review of Tuesday •  Transformation matrix in homogeneous coordinates vwo...
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## This document was uploaded on 03/12/2014 for the course ECS 175 at UC Davis.

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