Cayley-Klein Parameters

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博客介绍了Cayley - Klein参数，其与欧拉角类似，可唯一表征固体的方向，满足特定恒等式。还给出了基于该参数的变换矩阵，此矩阵可表征具有复轴的线性空间的变换，且满足一定条件，同时提及相关文献资料。

The parameters , , , and which, like the three Euler angles, provide a way to uniquely characterize the orientation of a solid body. These parameters satisfy the identities

			(1)
			(2)
			(3)
			(4)
			(5)

and

			(6)
			(7)

where denotes the complex conjugate. In terms of the Euler angles , , and , the Cayley-Klein parameters are given by

			(8)
			(9)
			(10)
			(11)

(Goldstein 1960, p. 155).

The transformation matrix is given in terms of the Cayley-Klein parameters by

A==[1/2(alpha^2-gamma^2+delta^2-beta^2) 1/2i(gamma^2-alpha^2+delta^2-beta^2) gammadelta-alphabeta; 1/2i(alpha^2+gamma^2-beta^2-delta^2) 1/2(alpha^2+gamma^2+beta^2+delta^2) -i(alphabeta+gammadelta); betadelta-alphagamma i(alphagamma+betadelta) alphadelta+betagamma]

(12)

(Goldstein 1960, p. 153).

The Cayley-Klein parameters may be viewed as parameters of a matrix (denoted for its close relationship with quaternions)

(13)

which characterizes the transformations

			(14)
			(15)

of a linear space having complex axes. This matrix satisfies

(16)

where is the identity matrix and the conjugate transpose, as well as

(17)

In terms of the Euler parameters and the Pauli matrices , the -matrix can be written as

(18)

(Goldstein 1980, p. 156).

REFERENCES:

Goldstein, H. "The Cayley-Klein Parameters and Related Quantities." §4-5 in Classical Mechanics, 2nd ed. Reading, MA: Addison-Wesley, pp. 148-158, 1980.

Varshalovich, D. A.; Moskalev, A. N.; and Khersonskii, V. K. "Description of Rotations in Terms of Unitary Matrices. Cayley-Klein Parameters." §1.4.3 in Quantum Theory of Angular Momentum. Singapore: World Scientific, pp. 24-27, 1988.

CITE THIS AS:

Eric W. Weisstein. "Cayley-Klein Parameters." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Cayley-KleinParameters.html