EulerAngles

EulerAngles - Euler Angle Formulas David Eberly Geometric...

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Euler Angle Formulas David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c ± 1998-2008. All Rights Reserved. Created: December 1, 1999 Last Modified: March 1, 2008 Contents 1 Introduction 3 2 Factor as a Product of Three Rotation Matrices 3 2.1 Factor as R x R y R z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Factor as R x R z R y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Factor as R y R x R z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Factor as R y R z R x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Factor as R z R x R y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.6 Factor as R z R y R x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.7 Factor as R x 0 R y R x 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.8 Factor as R x 0 R z R x 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.9 Factor as R y 0 R x R y 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.10 Factor as R y 0 R z R y 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.11 Factor as R z 0 R x R z 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.12 Factor as R z 0 R y R z 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Factor as a Product of Two Rotation Matrices 19 3.1 Factor P x P y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Factor P y P x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Factor P x P z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 Factor P z P x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.5 Factor P y P z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1
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3.6 Factor P z P y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2
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1 Introduction Rotations about the coordinate axes are easy to define and work with. Rotation about the x -axis by angle θ is R x ( θ ) = 1 0 0 0 cos θ - sin θ 0 sin θ cos θ where θ > 0 indicates a counterclockwise rotation in the plane x = 0. The observer is assumed to be positioned on the side of the plane with x > 0 and looking at the origin. Rotation about the y -axis by angle θ is R y ( θ ) = cos θ 0 sin θ 0 1 0 - sin θ 0 cos θ where θ > 0 indicates a counterclockwise rotation in the plane y = 0. The observer is assumed to be positioned on the side of the plane with
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EulerAngles - Euler Angle Formulas David Eberly Geometric...

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