2. Polyhedral Convex Cones

FixtureNet: interactive computer-aided design via the World Wide Web

“…Lakshminarayana (1978) showed that seven frictionless contacts are necessary to hold a three-dimensional part in form closure; Mishra, Schwartz and Sharir (1987) showed that seven frictionless contacts are also sufficient. Goldman and Tucker (1956), in a purely mathematical paper on linear algebra. described the necessary and sufficient conditions for positively spanning an n-dimensional Euclidean space, which coincidentally describes the necessary and sufficient conditions for form closure.…”

Automatic Generation of Mechanical Assembly Sequences

“…This system erf linear inequalities defines a polyhedral convex cone. It has been shown [16] that such a polyhedral convex cone can be built up from its (unique) d-dimensional face and its Oi-f l)-dimensional faces (if any), where <i=3-rank(Af), and M is the matrix of the coefficients n^j. If d is greater than zero, then the polyhedral convex cone has a face of dimension greater than zero and therefore the system of inequalities has a nonzero solution.…”

“…Goldman and Tucker [16] present important theoretical results that have been used as basis for the formulation presented in this paper. Those results alone have been used by Ohwovoriole and Roth [27], in the context of mechanical assembly, to solve a system of inequalities in a five dimensional space.…”

The practical behavior of the homogeneous self-dual formulations in interior point methods

“…Jansen, Terlaky and Roos [9] presented a similar self-dual model in a symmetric form. Xu, Hung and Ye [35,36] considered a homogeneous self-dual feasibility (HLF) model, which was already studied by Goldman and Tucker [6,29] and proved by numerical experiments that a long-step path following algorithm can solve the HLF model efficiently [33,34].…”