Yanxi Liu, and Robin J. Popplesone
"A Group Theoretic Formalization of Surface Contact"
*International Journal of Robotics Research,*
13-2, pp. 148-161, The MIT Press. April 1994.

## Abstract

The surface contacts between solids are always associated with a set of
symmetries of the contacting surfaces. These symmetries form a group,
known as the {\em symmetry group} of the surface.
In this article we develop a group theoretic formalization for describing
surface contact between solids. In particular we define:
\begin{itemize}
\item primitive and compound features of a solid;
\item a topological characterization of these features; and
\item the symmetry groups of primitive and compound features.
\end{itemize}

The symmetry group of a feature is a descriptor of the feature that is at
once abstract and quantitative. We show how to use group theory concepts to
describe the exact relative motion (position) of solids under surface
contacts, which can be either rigid or articulated.
The central result of this article is to prove the following:

\begin{itemize}
\item when primitive features of a solid are mutually
distinct, 1-congruent or 2-congruent,
the symmetry group of a compound feature can be expressed
in terms of the intersection of the symmetry groups of its primitive
features;
\item when two solids have surface contact, their relative positions
can be expressed as a coset of their common symmetry group, which in turn
can be expressed in terms of the intersections of the symmetry groups of the primitive
features involved in this contact.
\end{itemize}

These results show that using group theory to formalize surface contacts
is a general approach for specifying spatial relationships
and forms a sound basis for the automation of robotic task planning.
One advantage of this formulation is
its ability to express continuous motions between two
surface-contacting solids
in a computational manner, and to avoid combinatorics
arising from multiple relationships, especially from
discrete symmetries in the assembly parts and their features.
At the end of this article a geometric representation for
symmetry groups and an efficient group intersection algorithm using
characteristic invariants are described.