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The representation of a rotation as a quaternion (4 numbers) is more compact than the representation as an orthogonal matrix (9 numbers). Furthermore, for a given axis and angle, one can easily construct the corresponding quaternion, and conversely, for a given quaternion one can easily read off the axis and the angle.
A quaternion of the form a + 0 i + 0 j + 0 k, where a is a real number, is called scalar, and a quaternion of the form 0 + b i + c j + d k, where b, c, and d are real numbers, and at least one of b, c, or d is nonzero, is called a vector quaternion.
The quaternions are a non-commutative extension of the complex numbers which have numerous applications in mathematics, physics, and computer graphics The main article for this category is Quaternion .
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.
Rotations are not commutative (for example, rotating R 90° in the x-y plane followed by S 90° in the y-z plane is not the same as S followed by R), making the 3D rotation group a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable, so it is in fact a ...
The rotation group is a Lie group of rotations about a fixed point. This (common) fixed point or center is called the center of rotation and is usually identified with the origin. The rotation group is a point stabilizer in a broader group of (orientation-preserving) motions. For a particular rotation: The axis of rotation is a line of its ...
A quaternion has 4 real values: q w (the real part or the scalar part) and q x q y q z (the imaginary part). Defining the norm of the quaternion as follows: ‖ ‖ = + + + A unit quaternion satisfies: ‖ ‖ =
As a quaternion consists of two independent complex numbers, they form a four-dimensional vector space over the real numbers. The multiplication of quaternions is not quite like the multiplication of real numbers, though; it is not commutative – that is, if p and q are quaternions, it is not always true that pq = qp .