## 6.4.1 Basic Definitions and Concepts

This section builds on the semi-algebraic model definitions from Section 3.1 and the polynomial definitions from Section 4.4.1. It will be assumed that , which could for example arise by representing each copy of or in its or matrix form. For example, in the case of a 3D rigid body, we know that , which is a six-dimensional manifold, but it can be embedded in , which is obtained from the Cartesian product of and the set of all matrices. The constraints that force the matrices to lie in or are polynomials, and they can therefore be added to the semi-algebraic models of and . If the dimension of is less than , then the algorithm presented below is sufficient, but there are some representation and complexity issues that motivate using a special parameterization of to make both dimensions the same while altering the topology of to become homeomorphic to . This is discussed briefly in Section 6.4.2.

Suppose that the models in are all expressed using polynomials from , the set of polynomials6.6 over the field of rational numbers . Let denote a polynomial.

Subsections
Steven M LaValle 2012-04-20