3x3Rotationsmatrix

A 3x3 rotation matrix is a mathematical tool used in three-dimensional space to represent a rotation around an arbitrary axis. It is a square matrix of order three with specific properties. The determinant of a 3x3 rotation matrix is always +1, signifying that it preserves orientation and does not include reflections. Its columns and rows form an orthonormal basis, meaning they are mutually perpendicular unit vectors.

These matrices are fundamental in computer graphics, robotics, physics, and engineering for tasks such as orienting objects in a 3D environment, describing the movement of rigid bodies, or transforming coordinate systems. A rotation matrix is defined by the angle of rotation and the direction of the axis of rotation.

There are various methods to construct a 3x3 rotation matrix. One common approach involves using Euler angles, which represent a sequence of rotations around the primary axes. Another method utilizes the axis-angle representation, where a unit vector defines the axis of rotation and a scalar value represents the angle. Rodrigues' rotation formula provides a direct way to compute the rotation matrix from the axis and angle. The inverse of a rotation matrix is its transpose, meaning that rotating by an angle $\theta$ is undone by rotating by $-\theta$ around the same axis.