Immobilization of Solids and Mondriga Quadratic Forms

Immobilization of smooth convex figures

“…The proof follows the geometric ideas developed in Lemma 1, which were first introduced in [1]. For i = 1, 2, 3, let…”

“…Given a C ~ pointed curve (a, X), n _> 1, consider the set (1) defined in a neighborhood of the identity in g (or of the origin in ~3). We prove that S is a surface of class C '~.…”

“…Necessary conditions are known for immobilizing plane convex figures with three points ( [2]) (see Theorem 2 below), and convex bodies in 3-space with four points ( [1]). They are also known to be sufficient for the case of triangles and tetrahedra respectively.…”

Rotors in triangles and tetrahedra

“…In dimension three, immobilization results are much more complicated. See [2]. To characterize when four points in the faces of a tetrahedron T immobilize T we require the following definition.…”

“…The proof of Theorem 4 now follows straightforward from the proof or Theorem 3 in [2], but this time we consider, instead of a rigid triangle sliding along three fixed planes, the dual situation of a 3-dimensional rigid sector (the angle between three planes H a , H b and H c ) sliding along three fixed points a, b, c.…”