On the geometry of point-plane constraints on rigid-body displacements

In this paper the rigid-body displacements that transform a point in such a way that it remains on a particular plane are studied. These sets of rigid displacements are referred to as point-plane constraints and are given by the intersection of the Study quadric of all rigid displacements with another quadric in 7-dimensional projective space. The set of all possible point-plane constraints comprise a Segre variety. Two different classes of problems are investigated. First instantaneous kinematics, for a given rigid motion there are points in space which, at some instant, have no torsion or have no curvature to some order. The dimension and degrees of these varieties are found by very simple computations. The corresponding problems for point-sphere constraints are also found. The second class of problems concern the intersections of several given constraints. Again the characteristics of these varieties for different numbers of constraints are found using very simple techniques. © 2011 Springer Science+Business Media B.V.