Geometric Poisson brackets on Grassmannians and conformal spheres

Three-Dimensional Projective Geometry with Geometric Algebra

“…The same inner product (1.1) also gives the 6-D space of twists the same signature R 3,3 . The interpretation of a null vector of R 3,3 in the setting of twists, is that it represents an infinitesimal pure rotation or pure translation.…”

“…The same inner product (1.1) also gives the 6-D space of twists the same signature R 3,3 . The interpretation of a null vector of R 3,3 in the setting of twists, is that it represents an infinitesimal pure rotation or pure translation. A positive vector represents an infinitesimal screw motion where the translation along the screw axis follow the right-hand rule with the orientation of the rotation, while a negative vector represents a left-handed infinitesimal screw motion.…”

“…The inner product (1.1) gives the 6-D space of wrenches a signature R 3,3 , where a pure force has zero inner product with itself, called a null vector. A positive (or negative) vector of R 3,3 is one having positive (or negative) inner product with itself.…”

“…The group SL(4) which acts in the 4-space of homogeneous coordinates of points, can be lifted to a group action in the 6-D space of wrenches by acting upon the Plücker coordinates of the lines representing the wrenches. The image of the lift is SO 0 (3,3), the connected component of SO (3,3) containing the identity [3]. As P SL(4) = SL(4)/Z 2 is the group of orientation-preserving projective transformations, the crossed inner product provides a 6-D orthogonal geometric model of wrenches to study 3-D projective geometry of points.…”

“…A positive vector represents an infinitesimal screw motion where the translation along the screw axis follow the right-hand rule with the orientation of the rotation, while a negative vector represents a left-handed infinitesimal screw motion. The lift of group SL(4) to SO 0 (3,3) then makes the space of infinitesimal rigid-body motions a 6-D orthogonal geometric model to study the orientation-preserving projective geometry of points.…”