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Versors are an algebraic parametrisation of rotations. In classical quaternion theory a versor is a quaternion of norm one (a ''unit quaternion''). Each versor has the form : where the r2 = −1 condition means that r is a 3-dimensional unit vector. In case , the versor is termed a right versor. The corresponding 3-dimensional rotation has the angle ''a'' about the axis r in axis–angle representation. The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by William Rowan Hamilton in the context of his quaternion theory. For historical reasons, it sometimes is used synonymously with a "unit quaternion" without a reference to rotations. ==Versors, rotations, and Lie groups== In the quaternion algebra a versor will rotate any quaternion ''v'' through the sandwiching product map such that the scalar part of ''v'' conserves. If this scalar part (the fourth dimension of the quaternion space) is zero, i.e. ''v'' is a Euclidean vector in three dimensions, then the formula above defines the rotation through the angle 2''a'' around the unit vector r. For this case, this formula expresses the adjoint representation of the Spin(3) Lie group in its respective Lie algebra of 3-dimensional Euclidean vectors, and the factor "2" is due to the double covering of Spin(3) over the rotation group SO(3). In other words, rotates the vector part of ''v'' around the vector r. See quaternions and spatial rotation for details. A quaternionic versor expressed in the is an element of the special unitary group SU(2). Spin(3) and SU(2) are the same group. Left multiplication ''qv'' of a quaternion ''v'' to a versor ''q'' is another kind of quaternion rotation as a 4-dimensional real vector space, identical to the SU(2) action on the 2-dimensional complex space identical to quaternions (''v'' = ''A'' + ''Bj''). Angles of rotation in this ''λ'' = 1/2 representation are equal to ''a''; there is no "2" factor in angles unlike the ''λ'' = 1 adjoint representation mentioned above; see representation theory of SU(2) for details. For a fixed r, versors of the form exp(''a''r) where ''a'' ∈ , form a subgroup isomorphic to the circle group. Orbits of the left multiplication action of this subgroup are fibers of a fiber bundle over the 2-sphere, known as Hopf fibration in the case r = ''i''; other vectors give isomorphic, but not identical fibrations. In 2003 David W. Lyons wrote "the fibers of the Hopf map are circles in S3" (page 95). Lyons gives an elementary introduction to quaternions to elucidate the Hopf fibration as a mapping on unit quaternions. ==Presentation on 3- and 2-spheres== Hamilton denoted the versor of a quaternion ''q'' by the symbol U''q''. He was then able to display the general quaternion in polar coordinate form : ''q'' = T''q'' U''q'', where T''q'' is the norm of ''q''. The norm of a versor is always equal to one; hence they occupy the unit 3-sphere in H. Examples of versors include the eight elements of the quaternion group. Of particular importance are the right versors, which have angle π/2. These versors have zero scalar part, and so are vectors of length one (unit vectors). The right versors form a sphere of square roots of −1 in the quaternion algebra. The generators ''i'', ''j'', and ''k'' are examples of right versors, as well as their additive inverses. Other versors include the twenty-four Hurwitz quaternions that have the norm 1 and form vertices of a 24-cell polychoron. Hamilton defined a quaternion as the quotient of two vectors. A versor can be defined as the quotient of two unit vectors. For any fixed plane Π the quotient of two unit vectors lying in Π depends only on the angle (directed) between them, the same ''a'' as in the unit vector–angle representation of a versor explained above. That's why it may be natural to understand corresponding versors as directed arcs that connect pairs of unit vectors and lie on a great circle formed by intersection of Π with the unit sphere, where the plane Π passes through the origin. Arcs of the same direction and length (or, the same, its subtended angle in radians) are equivalent, i.e. define the same versor. Such an arc, although lying in the three-dimensional space, does not represent a path of a point rotating as described with the sandwiched product with the versor. Indeed, it represents the left multiplication action of the versor on quaternions that preserves the plane Π and the corresponding great circle of 3-vectors. The 3-dimensional rotation defined by the versor has the angle two times the arc's subtended angle, and preserves the same plane. It is a rotation about the corresponding vector r, that is perpendicular to Π. On three unit vectors, Hamilton writes〔''Elements of Quaternions'', 2nd edition, v. 1, p. 146〕 : and : imply : Multiplication of quaternions of norm one corresponds to the (non-commutative) "addition" of great circle arcs on the unit sphere. Any pair of great circles either is the same circle or has two intersection points. Hence, one can always move the point ''B'' and the corresponding vector to one of these points such that the beginning of the second arc will be the same as the end of the first arc. An equation : implicitly specifies the unit vector–angle representation for the product of two versors. Its solution is an instance of the general Campbell–Baker–Hausdorff formula in Lie group theory. As the 3-sphere represented by versors in ℍ is a 3-parameter Lie group, practice with versor compositions is a step into Lie theory. Evidently versors are the image of the exponential map applied to a ball of radius π in the quaternion subspace of vectors. Versors compose as aforementioned vector arcs, and Hamilton referred to this group operation as "the sum of arcs", but as quaternions they simply multiply. The geometry of elliptic space has been described as the space of versors.〔H. S. M. Coxeter (1950) (Review of "Quaternions and Elliptic Space" ) (by Georges Lemaître) from Mathematical Reviews〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Versor」の詳細全文を読む スポンサード リンク
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