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Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
An infinitesimal rotation matrix or differential rotation matrix is a matrix representing an infinitely small rotation. While a rotation matrix is an orthogonal matrix = representing an element of () (the special orthogonal group), the differential of a rotation is a skew-symmetric matrix = in the tangent space (the special orthogonal Lie ...
The product Aθ is the "generator" of the particular rotation, being the vector (x, y, z) associated with the matrix A. This shows that the rotation matrix and the axis-angle format are related by the exponential function. One can derive a simple expression for the generator G.
From linear algebra one knows that a certain matrix can be represented in another basis through the transformation ′ = where is the basis transformation matrix. If the vectors b {\displaystyle b} respectively c {\displaystyle c} are the z-axis in one basis respectively another, they are perpendicular to the y-axis with a certain angle t ...
The generating rotation matrix can be classified with respect to the values θ 1 and θ 2 as follows: If θ 1 = 0 and θ 2 ≠ 0 or vice versa, then the formulae generate simple rotations; If θ 1 and θ 2 are nonzero and θ 1 ≠ θ 2, then the formulae generate double rotations; If θ 1 and θ 2 are nonzero and θ 1 = θ 2, then the formulae ...
The product Aθ is the "generator" of the particular rotation, being the vector (x,y,z) associated with the matrix A. This shows that the rotation matrix and the axis–angle format are related by the exponential function.
In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform all three basis vectors to compute a rotation matrix in SO(3) , the group of all rotation matrices ...
Equivalently, it is the group of n × n orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has two connected components.