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Quaternions are also used in one of the proofs of Lagrange's four-square theorem in number theory, which states that every nonnegative integer is the sum of four integer squares. As well as being an elegant theorem in its own right, Lagrange's four square theorem has useful applications in areas of mathematics outside number theory, such as ...
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]
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 mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called.
This article describes Hamilton's original treatment of quaternions, using his notation and terms. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties. Mathematically, quaternions discussed differ from the modern definition only by the terminology which is used.
It can also be realized as the subgroup of unit quaternions generated by [10] = / and =. The generalized quaternion groups have the property that every abelian subgroup is cyclic. [ 11 ] It can be shown that a finite p -group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined ...
A principled construction of the planar quaternions can be found by first noticing that they are a subset of the dual-quaternions. There are two geometric interpretations of the dual-quaternions, both of which can be used to derive the action of the planar quaternions on the plane: As a way to represent rigid body motions in 3D space. The ...
Similar to the way that rotations in 3D space can be represented by quaternions of unit length, rigid motions in 3D space can be represented by dual quaternions of unit length. This fact is used in theoretical kinematics (see McCarthy [3]), and in applications to 3D computer graphics, [4] robotics [5] [6] and computer vision. [7]