## 5 Introduction  
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 this article we will review, clarify and extend this extensive theory using a projective perspective. From at least the time when Mobius developed barycentric coordinates with weighted points (projective homogeneous coordinates) to write their text on statics [1,2], scientists, engineers and mathematicians (who were often the same individuals in the 1800s) have worked with static rigidity within a projective perspective, sometimes implicitly. James Clerk Maxwell explored static stresses in frameworks with planar graphs via projections of 3-dimensional spherical polyhedra, building on drafting-table graphical statics techniques of engineers [3]. In the same issue of the Philosophical Magazine, the engineer Rankine describes attending a lecture at the Royal Society on \"the new geometry\" (projective geometry from the continent). He immediately jotted down a short note for publication observing that statics was projectively invariant [4]. At the time, Rankine was writing their text on statics for engineers [5]. Throughout the remainder of the 1800s, various authors implicitly, and sometimes explicitly, connected work on static rigidity, and sometimes on the companion infinitesimal rigidity, to projective geometry. Klein, as a student of Plucker who developed Plucker coordinates for lines in projective geometry, understood that statics and static rigidity lived within the projective world, and therefore, implicitly, this would extend to all the metrics in their geometric hierarchy which draw on projective geometry: spherical; hyperbolic; Minkowski; de Sitter [6]. Throughout the last decades of the 1800s and the first few decades of the 1900s a number of authors recognized that statics, and therefore infinitesimal rigidity, were projective invariants [7, 8, 9, 10]. As mathematics separated from engineering, and projective geometry faded from basic undergraduate education, these connections were lost, though they were kept alive in some places, such as Russia and Austria [11,12]. As we look at a variety of rigidity-related topics, we can connect results, methods, and even new conjectures through shared underlying projective geometry. This survey is an opportunity to pull out those connections, and observe shared similarities. One of the ways of making the connections is to recast some of the concepts in projective language. Another way is to examine the underlying projectively embedded transformations: change of metrics to connect examples in Euclidean, spherical, Minkowski, hyperbolic, de Sitter spaces; projection and lifting as projective techniques; polarity as a connection between what might appear as distinct concepts; transformations which place critical geometric objects (e.g., points in 2D) at 'infinity' to bring in additional examples which were implicitly covered by previous results. There is much to be learned by moving methods and results among the concepts, results, and settings with a projective lens. We note that within Klein's Hierarchy of Geometries [6] projective geometry contains both combinatorics (counting) and topology as conceptual contexts. As we move through the sections below, there will be critical results based on counting of edges, vertices, faces, etc., and simple topological results--starting with the connectivity of a graph, on to results based on topological surfaces such as planar graphs and combinatorial spheres. Part I pulls together results which apply directly to rigidity of bar-joint frameworks. These offer a surprising sweep of rigidity results and applications which are, in their core, projective. The concepts, and a number of the techniques, are abstracted from questions arising in civil and structural engineering where building with iron bars and rivets started the study of pin-jointed frameworks.