The *cylindrical decomposition* is very similar to the vertical
decomposition, except that when any of the cases in Figure
6.2 occurs, then a vertical line slices through all
faces, all the way from
to
. The result
is shown in Figure 6.18, which may be considered as a
singular complex. This may appear very inefficient in comparison to
the vertical decomposition; however, it is presented here because it
generalizes nicely to any dimension, any C-space topology, and any
semi-algebraic model. Therefore, it is presented here to ease the
transition to more general decompositions. The most important
property of the cylindrical decomposition is shown in Figure
6.19. Consider each vertical strip between two events.
When traversing a strip from to , the points
alternate between being
and
. For example, between
events and , the points below edge are in
. Points
between and lie in
. Points between and lie in
, and so forth. The cell decomposition can be defined so that
2D cells are also created in
. Let denote the logical
predicate (3.6) from Section
3.1.1. When traversing a strip, the value of
also alternates. This behavior is the main reason to construct a
cylindrical decomposition, which will become clearer in Section
6.4.2. Each vertical strip is actually considered to be a
*cylinder*, hence, the name
cylindrical decomposition (i.e., there are not necessarily any
cylinders in the 3D geometric sense).

Steven M LaValle 2020-08-14