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旋转矩阵

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Euler Angle Formulas

David Eberly, Geometric Tools, Redmond WA 98052

https://www.geometrictools.com/

This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy

of this license, visit http://creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons,

PO Box 1866, Mountain View, CA 94042, USA.

Created: December 1, 1999

Last Modiﬁed: April 28, 2020

Contents

1 Introduction 3

2 Factor as a Product of Three Rotation Matrices 3

2.1 Factor as R

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Factor as R

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Factor as R

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Factor as R

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 Factor as R

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.6 Factor as R

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.7 Factor as R

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.8 Factor as R

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.9 Factor as R

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.10 Factor as R

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.11 Factor as R

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.12 Factor as R

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Factor as a Product of Two Rotation Matrices 17

3.1 Factor P

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Factor P

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Factor P

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Factor P

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.5 Factor P

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1 Introduction

Rotations about the coordinate axes are easy to deﬁne and work with. Rotation about the x-axis by angle

θ is

(θ) =







1 0 0

0 cos θ − sin θ

0 sin θ cos θ







(1)

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

(θ) =







cos θ 0 sin θ

0 1 0

− sin θ 0 cos θ







(2)

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 y > 0 and looking at the origin. Rotation about the z-axis by angle

θ is

(θ) =







cos θ − sin θ 0

sin θ cos θ 0

0 0 1







(3)

where θ > 0 indicates a counterclockwise rotation in the plane z = 0. The observer is assumed to be

positioned on the side of the plane with z > 0 and looking at the origin. Rotation by an angle θ about an

arbitrary axis containing the origin and having unit length direction U = (U

, U

) is given by

(θ) = I + (sin θ)S + (1 − cos θ)S

(4)

where I is the identity matrix,

S =







0 −U

−U







(5)

and θ > 0 indicates a counterclockwise rotation in the plane U · (x, y, z) = 0. The observer is assumed to be

positioned on the side of the plane to which U points and is looking at the origin.

2 Factor as a Product of Three Rotation Matrices

A common problem is to factor a rotation matrix as a product of rotations about the coordinate axes.

The form of the factorization depends on the needs of the application and what ordering is speciﬁed. For

example, one might want to factor a rotation as R = R

(θ

) for some angles θ

, θ

, and θ

The ordering is xyz. Five other possibilities are xzy, yxz, yzx, zxy, and zyx. It is also possible to factor

as R = R

(θ

), the ordering referred to as xyx. Five other possibilites are xzx, yxy, yzy,

zxz, and zyz. These are also discussed here. The following discussion uses the notation c

= cos(θ

) and

= sin(θ

) for a = x, y, z.

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