Rodriguesképlet

Rodriguesképlet is a mathematical formula used in geometry and physics to describe the rotation of a rigid body. It relates the angular velocity of a body to its orientation in space. Specifically, it expresses the orientation of a rotated vector as a function of the original vector and the axis and angle of rotation. The formula is named after the French mathematician Olinde Rodrigues.

The Rodrigues' rotation formula is often expressed using the cross product. If a vector v is rotated by an angle theta around an axis represented by a unit vector k, the rotated vector v' is given by v' = v cos(theta) + (k x v) sin(theta) + k (k . v) (1 - cos(theta)). This formula accounts for both the component of the vector parallel to the axis of rotation and the component perpendicular to it.

In the context of rigid body dynamics, Rodrigues' formula can be used to determine the final orientation of a body after a series of rotations. It is particularly useful in robotics and computer graphics, where precise control of object orientation is crucial. The formula provides a compact and elegant way to represent rotations, avoiding the complexities associated with Euler angles or rotation matrices in certain applications. It is a fundamental tool for understanding and manipulating rotations in three-dimensional space.