A screw axis (helical axis or twist axis) is a line that is simultaneously the axis of rotation and the line along which translation of a body occurs. Chasles' theorem shows that each Euclidean displacement in three-dimensional space has a screw axis, and the displacement can be decomposed into a rotation about and a slide along this screw axis.
Plücker coordinates are used to locate a screw axis in space, and consist of a pair of three-dimensional vectors. The first vector identifies the direction of the axis, and the second locates its position. The special case when the first vector is zero is interpreted as a pure translation in the direction of the second vector. A screw axis is associated with each pair of vectors in the algebra of screws, also known as screw theory.
The spatial movement of a body can be represented by a continuous set of displacements. Because each of these displacements has a screw axis, the movement has an associated ruled surface known as a screw surface. This surface is not the same as the axode, which is traced by the instantaneous screw axes of the movement of a body. The instantaneous screw axis, or 'instantaneous helical axis' (IHA), is the axis of the helicoidal field generated by the velocities of every point in a moving body.
When a spatial displacement specializes to a planar displacement, the screw axis becomes the displacement pole, and the instantaneous screw axis becomes the velocity pole, or instantaneous center of rotation, also called an instant center. The term centro is also used for a velocity pole, and the locus of these points for a planar movement is called a centrode.
The proof that a spatial displacement can be decomposed into a rotation around, and translation along, a line in space is attributed to Michel Chasles in 1830. Recently the work of Giulio Mozzi has been identified as presenting a similar result in 1763.
A screw displacement (also screw operation or rotary translation) is the composition of a rotation by an angle φ about an axis (called the screw axis) with a translation by a distance d along this axis.
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Screw theory is the algebraic calculation of pairs of vectors, such as forces and moments or angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies. The mathematical framework was developed by Sir Robert Stawell Ball in 1876 for application in kinematics and statics of mechanisms (rigid body mechanics). Screw theory provides a mathematical formulation for the geometry of lines which is central to rigid body dynamics, where lines form the screw axes of spatial movement and the lines of action of forces.
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. The rigid transformations include rotations, translations, reflections, or any sequence of these. Reflections are sometimes excluded from the definition of a rigid transformation by requiring that the transformation also preserve the handedness of objects in the Euclidean space.
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space ; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension n of the space, and is commonly denoted E(n) or ISO(n). The Euclidean group E(n) comprises all translations, rotations, and reflections of ; and arbitrary finite combinations of them.
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