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using the Hamilton product, where the vector part of the pure quaternion L(p ′) = (0, r x, r y, r z) is the new position vector of the point after the rotation. In a programmatic implementation, the conjugation is achieved by constructing a pure quaternion whose vector part is p, and then performing
The Rodrigues vector (sometimes called the Gibbs vector, with coordinates called Rodrigues parameters) [3] [4] can be expressed in terms of the axis and angle of the rotation as follows: = ^ This representation is a higher-dimensional analog of the gnomonic projection , mapping unit quaternions from a 3-sphere onto the 3-dimensional pure ...
Even though every quaternion can be viewed as a vector in a four-dimensional vector space, it is common to refer to the vector part as vectors in three-dimensional space. With this convention, a vector is the same as an element of the vector space R 3 . {\displaystyle \mathbb {R} ^{3}.} [ b ]
where q is the versor, q −1 is its inverse, and x is the vector treated as a quaternion with zero scalar part. The quaternion can be related to the rotation vector form of the axis angle rotation by the exponential map over the quaternions, = /, where v is the rotation vector treated as a quaternion.
which is a quaternion of unit length (or versor) since ‖ ‖ = + + + = Most importantly, the above equations for composition of rotations are precisely the equations for multiplication of quaternions =. In other words, the group of unit quaternions with multiplication, modulo the negative sign, is isomorphic to the group of rotations with ...
Vector geometry of Rodrigues' rotation formula, as well as the decomposition into parallel and perpendicular components. Let k be a unit vector defining a rotation axis, and let v be any vector to rotate about k by angle θ ( right hand rule , anticlockwise in the figure), producing the rotated vector v rot {\displaystyle \mathbb {v} _{\text ...
A rotor is an object in the geometric algebra (also called Clifford algebra) of a vector space that represents a rotation about the origin. [1] The term originated with William Kingdon Clifford, [2] in showing that the quaternion algebra is just a special case of Hermann Grassmann's "theory of extension" (Ausdehnungslehre). [3]
Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called. As with complex and real analysis , it is possible to study the concepts of analyticity , holomorphy , harmonicity and conformality in the context of quaternions.