Search results
Results from the WOW.Com Content Network
The Rodrigues vector (sometimes called the Gibbs vector, with coordinates called Rodrigues parameters) [3] [4] can be expressed in terms of the axis and angle of the rotation as follows: = ^ This representation is a higher-dimensional analog of the gnomonic projection , mapping unit quaternions from a 3-sphere onto the 3-dimensional pure ...
The case of θ = 0, φ ≠ 0 is called a simple rotation, with two unit eigenvalues forming an axis plane, and a two-dimensional rotation orthogonal to the axis plane. Otherwise, there is no axis plane. The case of θ = φ is called an isoclinic rotation, having eigenvalues e ±iθ repeated twice, so every vector is rotated through an angle θ.
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 ...
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 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]
The curl of a vector field F, denoted by curl F, or , or rot F, is an operator that maps C k functions in R 3 to C k−1 functions in R 3, and in particular, it maps continuously differentiable functions R 3 → R 3 to continuous functions R 3 → R 3. It can be defined in several ways, to be mentioned below:
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]
As an example, let the vector = (,), be a vector in the plane. A rotation of the vector through an angle θ in counterclockwise direction is given by the rotation matrix: = ( ), which can be viewed either as an active transformation or a passive transformation (where the above matrix will be inverted), as described below.