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3D visualization of a sphere and a rotation about an Euler axis (^) by an angle of In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle about a fixed axis (called the Euler axis) that runs through the fixed point. [6]
The Euler parameters can be viewed as the coefficients of a quaternion; the scalar parameter a is the real part, the vector parameters b, c, d are the imaginary parts. Thus we have the quaternion Thus we have the quaternion
The real quaternion 1 is the identity element. The real quaternions commute with all other quaternions, that is aq = qa for every quaternion q and every real quaternion a. In algebraic terminology this is to say that the field of real quaternions are the center of this quaternion algebra.
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: ‖ ‖ =
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 .
The conjugate of that eigenvalue is also unity, yielding a pair of eigenvectors which define a fixed plane, and so the rotation is simple. In quaternion notation, a proper (i.e., non-inverting) rotation in SO(4) is a proper simple rotation if and only if the real parts of the unit quaternions Q L and Q R are equal in magnitude and have the same ...