Screw axis
[1][2] Plücker coordinates are used to locate a screw axis in space, and consist of a pair of three-dimensional vectors.
The term centro is also used for a velocity pole, and the locus of these points for a planar movement is called a centrode.
[4] 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.
Unlike for rotations, a righthand and lefthand screw operation generate different groups.
When n/g screw operations have been performed, the displacement will be m/g, which since it is a whole number means one has moved to an equivalent point in the lattice, while carrying out a rotation by 360°/g.
These rigid motions are defined by transformations of x in R3 given by consisting of a three-dimensional rotation A followed by a translation by the vector d. A three-dimensional rotation A has a unique axis that defines a line L. Let the unit vector along this line be S so that the translation vector d can be resolved into a sum of two vectors, one parallel and one perpendicular to the axis L, that is, In this case, the rigid motion takes the form Now, the orientation preserving rigid motion D* = A(x) + d⊥ transforms all the points of R3 so that they remain in planes perpendicular to L. For a rigid motion of this type there is a unique point c in the plane P perpendicular to L through 0, such that The point C can be calculated as because d⊥ does not have a component in the direction of the axis of A.
A point C on the screw axis satisfies the equation:[9] Solve this equation for C using Cayley's formula for a rotation matrix where [B] is the skew-symmetric matrix constructed from Rodrigues' vector such that Use this form of the rotation A to obtain which becomes This equation can be solved for C on the screw axis P(t) to obtain, The screw axis P(t) = C + tS of this spatial displacement has the Plücker coordinates S = (S, C × S).
[9] The screw axis appears in the dual quaternion formulation of a spatial displacement D = ([A], d).
If these two vectors are constant and along one of the principal axes of the body, no external forces are needed for this motion (moving and spinning]]).
[10] In any single plane, the path formed by the locations of the moving instantaneous axis of rotation (IAR) is known as the 'centroid', and is used in the description of joint motion.
