Title:

The geometry of immobilizing sets of objects

A new proof of Czyzowicz, Stojmenovic and Urrutia’s theorem giving necessary and sufficient geometric conditions for immobilizing a triangle is obtained. The same method of proof is employed to obtain proofs of statements on immobilizing sets of polygonal planar objects. In three dimensions, a detailed study of immobilizing sets of a tetrahedron is carried out. A 3 x 3 matrix R is defined for each quadruple of points, one from the interior of each face of the tetrahedron using a good choice of outward normal vectors to the faces of the tetrahedron. A necessary and sufficient condition on the quadruple of points to immobilize the tetrahedron is that the matrix R is symmetric. An analysis of the eigenvalues of symmetric matrix R leads to a new proof of Bracho, Mayer, Fetter and Montejano’s theorem. This proof is adapted to give another treatment of necessary and sufficient conditions characterizing immobilizing sets of a triangle. The set of centroids, set of circumcenters and set of orthocenters of the faces of a tetrahedron are shown to immoblize it is appropriate cases. It is shown that a set of four immobilizing points one in each face of the tetrahedron has five degrees of freedom and immoblizing sets of a tetrahedron having two fixed points have one degree of freedom. An analysis of the orientation of the tetrahedron whose vertices are the points in an immobilizing set of a given tetrahedron reveals the existence of immobilizing sets of a regular tetrahedron which are coplanar. In higher dimensions, a method of generating sets of points for which the matrix R is symmetric from another such set is presented and some geometrical properties arising from the symmetry of R are analysed.
