## ABSTRACT

Rigidity and flexibility of frameworks (motions preserving lengths of bars) and scene analysis (liftings from plane polyhedral pictures to spatial polyhedra) are two core examples of a general class of geometric problems:

Given a discrete configuration of points, lines, planes, … $ \ldots $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315119601/b3720761-bf99-43a0-9a78-d7389452e8b8/content/inline-math61_1.tif"/> in Euclidean space, and a set of geometric constraints (fixed lengths for rigidity, fixed incidences, and fixed projections of points for scene analysis), what is the set of solutions and what is its local form: discrete? k-dimensional?

Given a structure satisfying the constraints, is it unique, or at least locally unique, up to trivial changes, such as congruences for rigidity, or vertical scale for liftings?

How does this answer depend on the combinatorics of the structure and how does it depend on the specific geometry of the initial data or object?

The rigidity of frameworks examines points constrained by fixed distances between pairs of points, using vocabulary and linear techniques drawn from structural engineering: bars and joints, first-order rigidity and first-order flexes, and static rigidity and static self-stresses (Section 61.1). Section 61.2 describes some extended structures and their applications. Scene analysis and the dual concept of parallel drawings are described in Section 61.3. Finally, Section 61.4 describes reciprocal diagrams which form a fundamental geometric connection between liftings of polyhedral pictures and self-stresses in frameworks which continue to be used in structural engineering.