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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.
The matrix and the vector can be represented with respect to a right-handed or left-handed coordinate system. Throughout the article, we assumed a right-handed orientation, unless otherwise specified. Vectors or forms The vector space has a dual space of linear forms, and the matrix can act on either vectors or forms.
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 ...
The quaternion formulation of the composition of two rotations R B and R A also yields directly the rotation axis and angle of the composite rotation R C = R B R A. Let the quaternion associated with a spatial rotation R is constructed from its rotation axis S and the rotation angle φ this axis. The associated quaternion is given by,