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Quaternions also avoid a phenomenon called gimbal lock which can result when, for example in pitch/yaw/roll rotational systems, the pitch is rotated 90° up or down, so that yaw and roll then correspond to the same motion, and a degree of freedom of rotation is lost.
Expression of the rotation matrix in terms of quaternion parameters involves no trigonometric functions; It is simple to combine two individual rotations represented as quaternions using a quaternion product; Like rotation matrices, quaternions must sometimes be renormalized due to rounding errors, to make sure that they correspond to valid ...
Spatial rotations in three dimensions can be parametrized using both Euler angles and unit quaternions.This article explains how to convert between the two representations. Actually this simple use of "quaternions" was first presented by Euler some seventy years earlier than Hamilton to solve the problem of magic squar
Pairs of unit quaternions represent a rotation in 4D space (see Rotations in 4-dimensional Euclidean space: Algebra of 4D rotations). The set of all unit quaternions forms a 3-sphere S 3 and a group (a Lie group) under multiplication, double covering the group (,) of real orthogonal 3×3 matrices of determinant 1 since two unit quaternions ...
Isoclinic rotations with like signs are denoted as left-isoclinic; those with opposite signs as right-isoclinic. Left- and right-isoclinic rotations are represented respectively by left- and right-multiplication by unit quaternions; see the paragraph "Relation to quaternions" below. The four rotations are pairwise different except if α = 0 or ...
Expressing rotations in 3D as unit quaternions instead of matrices has some advantages: Concatenating rotations is computationally faster and numerically more stable. Extracting the angle and axis of rotation is simpler. Interpolation is more straightforward. See for example slerp. Quaternions do not suffer from gimbal lock as Euler angles do.
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. A single multiplication by a versor, either left or right, is itself a rotation, but in four dimensions.
The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations. The map from S 3 onto SO(3) that identifies antipodal points of S 3 is a surjective homomorphism of Lie groups, with kernel {±1}. Topologically, this map is a two-to-one covering map. (See the plate trick.)