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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 polynomials^{6.6} over the field of rational numbers
. Let
denote a polynomial.

**Subsections**
Steven M LaValle
2020-08-14