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Special positions of frameworks and the Grassmann-Cayley algebra. (English) Zbl 1404.15022

Sitharam, Meera (ed.) et al., Handbook of geometric constraint systems principles. Boca Raton, FL: CRC Press (ISBN 978-1-4987-3891-0/hbk; 978-1-4987-3892-7/ebook). Discrete Mathematics and its Applications, 85-106 (2019).
From the introduction: Given a framework determined by a system of geometric constraints, a fundamental question is to determine whether the framework is rigid or flexible. If the framework is a bar-and-joint or body-and-bar framework, the constraints are all given by fixing the distance between selected pairs of points with bars. The theorems by [G. Laman, J. Eng. Math. 4, 331–340 (1970; Zbl 0213.51903)], (actually discovered first by H. Pollaczek-Geiringer, Z. Angew. Math. Mech. 7, 58–72 (1927; JFM 53.0743.02), T.-S. Tay, Adv. Appl. Math. 23, No. 1, 14–28 (1999; Zbl 0936.05027), and N. White and W. Whiteley, SIAM J. Algebraic Discrete Methods 8, 1–32 (1987; Zbl 0635.51014)] characterize rigidity combinatorially in various setting, and these combinatorial characterizations only determine the behavior of sufficiently generic realzations of a framework with given combinatorics.
The pure condition of a generically rigid framework, introduced by White, Whiteley [loc. cit.] and by N. L. White and W. Whiteley, SIAM J. Algebraic Discrete Methods 4, 481–511 (1983; Zbl 0542.51022)], is a polynomial in brackets. The goal of this chapter is to give an introduction to the Grassmann-Cayley algebra, which can be used to give a geometric interpretation of the vanishing of bracket polynomials in order to better understand when a generically rigid framework admits nontrival internal motions.
In order to motivate the development of the Grassmann-Cayley algebra we present the simplest example of a generically rigid structure in the plane, two rigid bodies connected by three bars, which we represent by a multigraph with a vertex for each body and an edge for each bar.
Following White, Whiteley [loc. cit.], we embed the framework in the projective plane \(\mathbb{P}^2\) and label each bar with vector corresponding to the line in the direction of the bar. This vector is obtained by taking the cross product of the vectors corresponding to the endpoints of the bars. Equivalently, the bars are labele by their Plücker coordinates in a projective plane that is dual to the plane in which the framework embedded. The resulting body-and-bar framework is generically infinitesimally rigid and has a nontrivial infinitesimal motion exactly when the three bars are parallel or meet at a point.
In \(\mathbb{P}^2\) lines are parallel exactly when they meet at a point on the line at infinity, hence the framework has an infinitesimal motion exactly when the lines along the bars are coincident. Three lines in \(\mathbb{P}^2\) are coincident when their Plücker coordinates in the dual projective plane are collinear. This is precisely the condition that the \(3\times 3\) matrix \([abc]\) has determinant zero.
In Section 4.2 we give an introduction to projective space, the Grassmannian, and Plücker coordinates. In Section 4.3 we discuss the bracket ring as the ring of invariant polynomials on \(\mathbb{P}^n\) and as a quotient ring with relations given by Plücker relations and Van der Waerden syzygies. We treat the Grassmann-Cayley algebra in Section 4.4 and Cayley factorization in Section 4.5, illustrating the theory with examples from the rigidity theory of body-and-bar frameworks.
In the final section we include a new result, Theorem 4.1 due to the first author, Cai, St. John, and Theran, characterizing which body-and-bar frameworks have pure conditions which may be represented by a bracket monomial. We use the notation and conventions by B. Sturmfels [Algorithms in invariant theory. 2nd ed. Wien: Springer (2008; Zbl 1154.13003)] throughout, so that the reader who would like a fuller exposition may transition easily between this chapter and [Sturmfels, loc. cit.].
For the entire collection see [Zbl 1397.05005].

MSC:

15A75 Exterior algebra, Grassmann algebras
14M15 Grassmannians, Schubert varieties, flag manifolds
51M35 Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations)
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