rotationer

Rotationer are rigid motions of Euclidean space that preserve distances and angles, transforming points through a rotation about a fixed point or axis. In two dimensions, a rotation centers on a point and rotates every location by a specified angle around that point. In three dimensions, a rotation occurs about a straight line, called the axis, by a given angle.

In 2D, the rotation by angle θ is represented by the matrix [[cos θ, -sin θ], [sin θ, cos θ]]. This matrix is orthogonal and has determinant +1, so it preserves lengths and orientation. In 3D, rotations can be described by orthogonal 3×3 matrices with determinant +1. Any 3D rotation has an invariant axis: a unit vector along the axis remains fixed, and points rotate around it by θ. Rodrigues’ formula gives a direct way to compute the matrix from an axis and angle: R = I + sin θ K + (1 − cos θ) K^2, where K is the cross-product matrix of the axis.

Beyond matrices, rotations are commonly represented by quaternions, Euler angles, or axis-angle triples. Quaternions offer a compact, numerically stable way to compose rotations. A rotation about a point p can be written as a composition of a translation, a rotation about the origin, and the inverse translation.

Rotationer form a mathematical group: the special orthogonal group SO(n). In two dimensions, all rotations commute; in three dimensions, the order of rotations matters. The inverse of a rotation by θ is a rotation by −θ; repeated rotation by 360 degrees returns to the identity.

Applications of rotationer include computer graphics, robotics, aerospace, astronomy, crystallography, and navigation, where orientation, pose, and object manipulation rely on precise rotational transformations.