Math Lover
11-06-2008, 12:10 PM
Euler Angleshttp://mathworld.wolfram.com/images/entries/underline.gifhttp://mathworld.wolfram.com/images/entries/underline.gifhttp://mathworld.wolfram.com/images/spacer.gifhttp://mathworld.wolfram.com/images/entries/dnld-nb.gif (http://mathworld.wolfram.com/notebooks/DifferentialGeometry/EulerAngles.nb)
http://mathworld.wolfram.com/images/eps-gif/EulerAngles_600.gif
According to Euler's rotation theorem (http://mathworld.wolfram.com/EulersRotationTheorem.html), any rotation (http://mathworld.wolfram.com/Rotation.html) may be described using three angles (http://mathworld.wolfram.com/Angle.html). If the rotations (http://mathworld.wolfram.com/Rotation.html) are written in terms of rotation matrices (http://mathworld.wolfram.com/RotationMatrix.html) http://mathworld.wolfram.com/images/equations/EulerAngles/Inline1.gif, http://mathworld.wolfram.com/images/equations/EulerAngles/Inline2.gif, and http://mathworld.wolfram.com/images/equations/EulerAngles/Inline3.gif, then a general rotation (http://mathworld.wolfram.com/Rotation.html) http://mathworld.wolfram.com/images/equations/EulerAngles/Inline4.gif can be written as
http://mathworld.wolfram.com/images/equations/EulerAngles/NumberedEquation1.gif(1)
The three angles giving the three rotation matrices are called Euler angles. There are several conventions for Euler angles, depending on the axes about which the rotations are carried out. Write the matrix (http://mathworld.wolfram.com/Matrix.html) http://mathworld.wolfram.com/images/equations/EulerAngles/Inline5.gif as
http://mathworld.wolfram.com/images/equations/EulerAngles/NumberedEquation2.gif(2)
The so-called "http://mathworld.wolfram.com/images/equations/EulerAngles/Inline6.gif-convention," illustrated above, is the most common definition. In this convention, the rotation given by Euler angles http://mathworld.wolfram.com/images/equations/EulerAngles/Inline7.gif, where the first rotation is by an angle http://mathworld.wolfram.com/images/equations/EulerAngles/Inline8.gif about the z-axis (http://mathworld.wolfram.com/z-Axis.html), the second is by an angle http://mathworld.wolfram.com/images/equations/EulerAngles/Inline9.gif about the x-axis (http://mathworld.wolfram.com/x-Axis.html), and the third is by an angle http://mathworld.wolfram.com/images/equations/EulerAngles/Inline10.gif about the z-axis (http://mathworld.wolfram.com/z-Axis.html) (again). Note, however, that several notational conventions for the angles are in common use. Goldstein (1980, pp. 145-148) and Landau and Lifschitz (1976) use http://mathworld.wolfram.com/images/equations/EulerAngles/Inline11.gif, Tuma (1974) says http://mathworld.wolfram.com/images/equations/EulerAngles/Inline12.gif is used in aeronautical engineering in the analysis of space vehicles (but claims that http://mathworld.wolfram.com/images/equations/EulerAngles/Inline13.gif is used in the analysis of gyroscopic motion), while Bate et al. (1971) use http://mathworld.wolfram.com/images/equations/EulerAngles/Inline14.gif. Goldstein remarks that continental authors usually use http://mathworld.wolfram.com/images/equations/EulerAngles/Inline15.gif, and warns that left-handed coordinate systems are also in occasional use (Osgood 1937, Margenau and Murphy 1956-64). Varshalovich (1988, pp. 21-23) uses the notation http://mathworld.wolfram.com/images/equations/EulerAngles/Inline16.gif or http://mathworld.wolfram.com/images/equations/EulerAngles/Inline17.gif to denote the Euler angles, and gives three different angle conventions, none of which corresponds to the http://mathworld.wolfram.com/images/equations/EulerAngles/Inline18.gif-convention.
Here, the notation http://mathworld.wolfram.com/images/equations/EulerAngles/Inline19.gif is used, a convention that could be used in versions of Mathematica (http://www.wolfram.com/products/mathematica/) prior to 6 as RotationMatrix3D[phi, theta, psi] (which could be run after loading Geometry`Rotations`) and RotateShape[g, phi, theta, psi] (which could be run after loading Geometry`Shapes`). In the http://mathworld.wolfram.com/images/equations/EulerAngles/Inline20.gif-convention, the component rotations are then given by
http://mathworld.wolfram.com/images/eps-gif/EulerAngles_600.gif
According to Euler's rotation theorem (http://mathworld.wolfram.com/EulersRotationTheorem.html), any rotation (http://mathworld.wolfram.com/Rotation.html) may be described using three angles (http://mathworld.wolfram.com/Angle.html). If the rotations (http://mathworld.wolfram.com/Rotation.html) are written in terms of rotation matrices (http://mathworld.wolfram.com/RotationMatrix.html) http://mathworld.wolfram.com/images/equations/EulerAngles/Inline1.gif, http://mathworld.wolfram.com/images/equations/EulerAngles/Inline2.gif, and http://mathworld.wolfram.com/images/equations/EulerAngles/Inline3.gif, then a general rotation (http://mathworld.wolfram.com/Rotation.html) http://mathworld.wolfram.com/images/equations/EulerAngles/Inline4.gif can be written as
http://mathworld.wolfram.com/images/equations/EulerAngles/NumberedEquation1.gif(1)
The three angles giving the three rotation matrices are called Euler angles. There are several conventions for Euler angles, depending on the axes about which the rotations are carried out. Write the matrix (http://mathworld.wolfram.com/Matrix.html) http://mathworld.wolfram.com/images/equations/EulerAngles/Inline5.gif as
http://mathworld.wolfram.com/images/equations/EulerAngles/NumberedEquation2.gif(2)
The so-called "http://mathworld.wolfram.com/images/equations/EulerAngles/Inline6.gif-convention," illustrated above, is the most common definition. In this convention, the rotation given by Euler angles http://mathworld.wolfram.com/images/equations/EulerAngles/Inline7.gif, where the first rotation is by an angle http://mathworld.wolfram.com/images/equations/EulerAngles/Inline8.gif about the z-axis (http://mathworld.wolfram.com/z-Axis.html), the second is by an angle http://mathworld.wolfram.com/images/equations/EulerAngles/Inline9.gif about the x-axis (http://mathworld.wolfram.com/x-Axis.html), and the third is by an angle http://mathworld.wolfram.com/images/equations/EulerAngles/Inline10.gif about the z-axis (http://mathworld.wolfram.com/z-Axis.html) (again). Note, however, that several notational conventions for the angles are in common use. Goldstein (1980, pp. 145-148) and Landau and Lifschitz (1976) use http://mathworld.wolfram.com/images/equations/EulerAngles/Inline11.gif, Tuma (1974) says http://mathworld.wolfram.com/images/equations/EulerAngles/Inline12.gif is used in aeronautical engineering in the analysis of space vehicles (but claims that http://mathworld.wolfram.com/images/equations/EulerAngles/Inline13.gif is used in the analysis of gyroscopic motion), while Bate et al. (1971) use http://mathworld.wolfram.com/images/equations/EulerAngles/Inline14.gif. Goldstein remarks that continental authors usually use http://mathworld.wolfram.com/images/equations/EulerAngles/Inline15.gif, and warns that left-handed coordinate systems are also in occasional use (Osgood 1937, Margenau and Murphy 1956-64). Varshalovich (1988, pp. 21-23) uses the notation http://mathworld.wolfram.com/images/equations/EulerAngles/Inline16.gif or http://mathworld.wolfram.com/images/equations/EulerAngles/Inline17.gif to denote the Euler angles, and gives three different angle conventions, none of which corresponds to the http://mathworld.wolfram.com/images/equations/EulerAngles/Inline18.gif-convention.
Here, the notation http://mathworld.wolfram.com/images/equations/EulerAngles/Inline19.gif is used, a convention that could be used in versions of Mathematica (http://www.wolfram.com/products/mathematica/) prior to 6 as RotationMatrix3D[phi, theta, psi] (which could be run after loading Geometry`Rotations`) and RotateShape[g, phi, theta, psi] (which could be run after loading Geometry`Shapes`). In the http://mathworld.wolfram.com/images/equations/EulerAngles/Inline20.gif-convention, the component rotations are then given by