rotationoperation

A rotation operation is a transformation that rotates a geometric object in a Euclidean space around a fixed point or axis by a specified angle. It is an isometry that preserves distances and angles, and, when orientation is preserved, a proper rotation.

In two dimensions, a rotation about the origin by angle θ maps a point (x, y) to (x cos θ − y sin θ, x sin θ + y cos θ). In three dimensions, rotations are defined around a unit axis vector u by angle θ, and can be expressed via Rodrigues’ formula, a rotation matrix, Euler angles, or quaternions. Quaternions provide a compact, non-singular representation that can be composed via multiplication.

Rotations form a mathematical group known as the special orthogonal group, denoted SO(n) for n-dimensional space. They preserve inner products and therefore distances and angles. In 2D the relevant group is SO(2); in 3D it is SO(3). Rotations are linear when the rotation is applied about the origin; otherwise, they can be described using homogeneous coordinates with 4×4 matrices to combine rotation with translation.

There are active and passive interpretations of a rotation. An active rotation moves the object within a fixed coordinate system, while a passive rotation redefines the coordinate frame itself. Both viewpoints yield consistent results when used with appropriate transformation conventions.

Applications of rotation operations appear across computer graphics, robotics, aerospace, physics, and crystallography. Common tasks include reorienting objects, transforming coordinate systems, and composing multiple rotations to achieve a desired final orientation. The choice of representation (matrix, Euler angles, or quaternions) depends on the context and computational needs.