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A rotor is an object in the geometric algebra (also called Clifford algebra) of a vector space that represents a rotation about the origin. [1] The term originated with William Kingdon Clifford, [2] in showing that the quaternion algebra is just a special case of Hermann Grassmann's "theory of extension" (Ausdehnungslehre). [3]
In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar, then the function's value is multiplied by some power of this scalar; the power is called the degree of homogeneity, or simply the degree.
In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. [ 1 ]
The Lie algebra bracket captures the essence of the Lie group product in a sense made precise by the Baker–Campbell–Hausdorff formula. The elements of s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} are the "infinitesimal generators" of rotations, i.e., they are the elements of the tangent space of the manifold SO(3) at the identity element.
Connecting the Lie algebra to the Lie group is the exponential map, which is defined using the standard matrix exponential series for e A [14] For any skew-symmetric matrix A, exp(A) is always a rotation matrix. [nb 3] An important practical example is the 3 × 3 case.
In terms of Lie theory, the Rodrigues' formula provides an algorithm to compute the exponential map from the Lie algebra so(3) to its Lie group SO(3). This formula is variously credited to Leonhard Euler, Olinde Rodrigues, or a combination of the two. A detailed historical analysis in 1989 concluded that the formula should be attributed to ...
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A homogeneous subspace refers to a linear subspace of the algebraic space. The terms for objects: point , line , circle , sphere , quasi-sphere etc. are used to mean either the geometric object in the base space, or the homogeneous subspace of the representation space that represents that object, with the latter generally being intended unless ...