Orienteringsmatrise

Orienteringsmatrise, or orientation matrix, is a square matrix that encodes the orientation of one coordinate frame relative to another. In general, the columns (or rows, depending on convention) are the coordinates of the basis vectors of the rotated frame expressed in the reference frame. When the matrix is orthogonal and has determinant +1, it represents a proper rotation and belongs to the special orthogonal group SO(3) in three dimensions (or SO(n) in n dimensions). If the determinant is -1, it represents an improper rotation, i.e., a rotation combined with a reflection, reversing orientation.

The orienteringsmatrise is used to transform coordinates between frames. For a vector v expressed in the original frame, the transformed vector in the new frame is v' = R v, where R is the orienteringsmatrise. It is widely used in robotics, aerospace, computer graphics, and computer vision to rotate points and vectors without translation. In three dimensions, the matrix can be derived from Euler angles, axis-angle representations, or quaternions; converses allow conversion back to these representations. In practice, orientation matrices are often embedded in larger homogeneous transformation matrices to combine rotation and translation.

In 2D, the orienteringsmatrise is a 2x2 rotation matrix, typically written as [ [cosθ, -sinθ], [sinθ, cosθ] ]. In 3D, common special cases include rotations about the coordinate axes: Rx, Ry, Rz. Core properties include R^T R = I and det(R) = ±1; checking the determinant helps determine whether the transformation preserves orientation.