From Henrici’s flexible hyperboloid to snapping spatial four-bars

Abstract

The rods of Henrici's flexible hyperboloid are generators of a one-sheeted hyperboloid with spherical joints at each crossing point between two rods. Thus, the hyperboloid can vary within a confocal family terminated by two flat poses. The restriction to a quadrangle with sides along generators yields a one-parameter variation of this quadrangle. When we pick out two sufficiently close poses, then it is possible to find appropriate revolute joints at the vertices such that a physical model of this spatial four-bar with mutually skew revolute axes can snap from one pose into the other, though both poses are theoretically rigid. Also the converse is true: For each snapping spatial four-bar we find a hyperboloid such that the two poses originate from a Henrici flex. Consequently, additional generators of the hyperboloid in form of taut strings are compatible with the snapping of the quadrangular frame. We present an algorithm for the synthesis of snapping spatial four-bars and conclude with their geometric characterizations.