ABSTRACT
This chapter begins with a generalized introduction to geometric constraint systems (GCS). A constraint can be expressed as a set of quadratic polynomials with real coefficients. The combinatorics of a GCS are usually captured separately in a constraint graph: a graph where each vertex represents a geometric primitive and each edge represents a constraint on the corresponding primitives. A realization of a GCS is a placement of the geometric primitives that satisfies the constraints. The realizations of a GCS can be found algebraically by solving a system of polynomial equations corresponding to the GCS, where the variables are the coordinates of the geometric primitives. The chapter also presents some key concepts present in the book. The book describes the bracket algebra and Grassmann-Cayley algebra for the plane with a view toward proving theorems in Projective and Euclidean geometry. It discusses geometric conditions for global rigidity of generic bar-joint frameworks in arbitrary dimensions.
