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A spatial rotation around a fixed point of radians about a unit axis that denotes the Euler axis is given by the quaternion , where and . Compared to rotation matrices, quaternions are more compact, efficient, and numerically stable. Compared to Euler angles, they are simpler to compose.
The set of quaternions is a 4-dimensional vector space over the real numbers, with as a basis, by the component-wise addition. and the component-wise scalar multiplication. A multiplicative group structure, called the Hamilton product, denoted by juxtaposition, can be defined on the quaternions in the following way:
Euler angles (in 3-2-1 sequence) to quaternion conversion. By combining the quaternion representations of the Euler rotations we get for the Body 3-2-1 sequence, where the airplane first does yaw (Body-Z) turn during taxiing onto the runway, then pitches (Body-Y) during take-off, and finally rolls (Body-X) in the air.
Like rotation matrices, quaternions must sometimes be renormalized due to rounding errors, to make sure that they correspond to valid rotations. The computational cost of renormalizing a quaternion, however, is much less than for normalizing a 3 × 3 matrix. Quaternions also capture the spinorial character of rotations in three dimensions.
In mechanics and geometry, the 3D rotation group, often denoted SO (3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. [1] By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation ...
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.
Rotation matrix. In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix. rotates points in the xy plane counterclockwise through an angle θ about the origin of a two-dimensional Cartesian coordinate system.
In mathematics, the group of rotations about a fixed point in four-dimensional Euclidean space is denoted SO (4). The name comes from the fact that it is the special orthogonal group of order 4. In this article rotation means rotational displacement. For the sake of uniqueness, rotation angles are assumed to be in the segment [0, π] except ...