ABSTRACT
This chapter discusses the geometric approach to infinitesimal rigidity in non-generic settings. One of the fundamental characterizations of infinitesimal rigidity is via the linear independence of the rows of the rigidity matrix. Namely the configuration is infinitesimally rigid if and only if the corresponding rigidity matrix is of full rank. Since the rank of the rigidity matrix is defined via algebraic equations, a rigidity matrix of a generic graph realization in the plane is always either full rank or not full rank. The chapter describes the geometric conditions of infinitesimal rigidity in terms of extended Cayley algebra. It employs the techniques of tensegrities that is very useful in the study of infinitesimal rigidity. The chapter also describes the objects that are involved in the geometric conditions for non-parallelizable tensegrities. In practice it is sufficient to choose even less cycles to get the corresponding geometric existence condition of a non-parallelizable tensegrity.
