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Rotation of an object in two dimensions around a point O. Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point.
In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which the origin is kept fixed and the x′ and y′ axes are obtained by rotating the x and y axes counterclockwise through an angle .
This is Rodrigues' formula for the axis of a composite rotation defined in terms of the axes of the two component rotations. He derived this formula in 1840 (see page 408). [3] The three rotation axes A, B, and C form a spherical triangle and the dihedral angles between the planes formed by the sides of this triangle are defined by the rotation ...
Rodrigues' rotation formula rotates v by an angle θ around vector k by decomposing it into its components parallel and perpendicular to k, and rotating only the perpendicular component. Vector geometry of Rodrigues' rotation formula, as well as the decomposition into parallel and perpendicular components.
A rotation can be represented by a unit-length quaternion q = (w, r →) with scalar (real) part w and vector (imaginary) part r →. The rotation can be applied to a 3D vector v → via the formula = + (+). This requires only 15 multiplications and 15 additions to evaluate (or 18 multiplications and 12 additions if the factor of 2 is done via ...
The set of all reflections in lines through the origin and rotations about the origin, together with the operation of composition of reflections and rotations, forms a group. The group has an identity: Rot(0). Every rotation Rot(φ) has an inverse Rot(−φ). Every reflection Ref(θ) is its own inverse. Composition has closure and is ...
Rotation matrices provide an algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics. In some literature, the term rotation is generalized to include improper rotations, characterized by orthogonal matrices with a determinant of −1 (instead of +1).
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. [ 1 ] By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry ), and orientation ...