Rotatsiooni

Rotatsioon, or rotatsiooni, is a geometric transformation that turns every point of an object around a fixed point or axis by a certain angle, preserving distances and angles and thus leaving the shape unchanged. In two dimensions a rotation is performed about a point; in three dimensions it is defined by an axis, with all points moved along circles in planes orthogonal to that axis.

Rotations can be described in several equivalent ways. In coordinates, a 2D rotation by angle θ about the origin is given by the matrix [[cos θ, -sin θ], [sin θ, cos θ]]. In 3D, rotations are represented by 3×3 orthogonal matrices with determinant 1, forming the group SO(3). Other representations include the axis-angle form (an axis and a rotation angle), Euler angles (three successive rotations about specified axes), and quaternions, which avoid gimbal lock and are efficient for computation.

Composition of rotations is associative but generally non-commutative in 3D. The inverse of a rotation is its transpose (or its negative angle about the same axis). Rotations can be active (rotating the object in space) or passive (rotating the coordinate frame).

Rotational concepts are central in physics (angular velocity, angular momentum, torque), engineering (machinery orientation, gyroscopes), computer graphics and robotics (orientation of cameras and end-effectors), astronomy (rotation of planets and stars), and navigation (attitude). Understanding rotatsiooni involves both geometric intuition and algebraic representations, enabling accurate description of orientation and motion in space.