Abstract
In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar–joint framework in projective d-space and places particular emphasis on the projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body–hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas.
Highlights
In the same issue of the Philosophical Magazine, the engineer Rankine describes attending a lecture at the Royal Society on “the new geometry”
Given a bar–joint framework (G, q) on the sphere Sd, we may rotate the whole framework in Sd so that all points are moved off the equator, and invert all points that lie on the lower hemisphere to obtain a spherical framework (G, q ) that lies on the strict upper hemisphere Sd>0
If we develop the pure condition for an isostatic block and hole polyhedron P, as a bar–joint framework, there will be a factor for each block, which may depend on which generically isostatic graph was inserted
Summary
The study of the rigidity and flexibility properties of bar–joint structures can be traced back to work of Cauchy and Euler on Euclidean polyhedra. In a companion paper Projective Geometry of Scene Analysis, Parallel Drawing and Reciprocal Drawing [27], we will continue the exploration of related topics which have a deep projective basis: (i) scene analysis: the lifting of pictures in dimension d to scenes in dimension d + 1; (ii) the parallel drawings of configurations with fixed normals to faces (polar to scene analysis); and (iii) reciprocal diagrams developed by Maxwell, Rankine and Cremona [3,4], as well as engineers working on examples at their drafting tables This field was called graphical statics in both Europe and the US in the last half of the 19th century. Both projective geometry and rigidity theory are active fields for continuing research, and we invite you to join this work
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