Schroefgroepen

Schroefgroepen are a concept from screw theory used to describe the group of rigid-body motions in three-dimensional space that can be generated by screw motions. A screw motion combines a rotation about an axis with a translation along the same axis. It is defined by a unit direction vector, a point on the axis, and a pitch, which is the ratio of translation to rotation. The collection of all such motions forms a Lie group isomorphic to the Special Euclidean group SE(3), with the corresponding Lie algebra se(3) consisting of twists, the infinitesimal generators of screw motions.

Mathematically, screw motions can be represented in several ways. Plücker coordinates describe the screw axis as a line in space, while dual quaternions provide a compact algebraic form for combining rotation and translation. The exponential map links a twist (an element of se(3)) to a finite screw motion in SE(3).

Schroefgroepen are used in various fields to model and analyze the movement of rigid bodies and kinematic chains. In robotics, they offer a compact description of joint motion and link configurations; in computer graphics and CAD/CAM, they aid interpolation and motion planning; in biomechanics and mechanical design, they help model complex motions of articulated systems.

Special cases include pure rotation (pitch zero) and pure translation (rotation angle zero). The concept underpins practical methods for composing motions, interpolating poses, and understanding the geometry of rigid-body movement. See also SE(3), screw theory, twists, wrenches, dual quaternions, and Plücker coordinates.