**n**-section is an

**n**-dimensional subsection. A shell of a region is all regions attached to the region.

##
**Complement Region Theorem**

The complement of each outside region is outside, and the complement of each inside region is empty.

Consider a 1-dimensional subsection

**L**with outside region isomorphic to an outside region

**R**. The hops of

**L**are ordered. The first hop of

**L**is a hop across a boundary of the shell of

**R**. Each hop reverses the sidedness of another boundary, and

**L**hops across each boundary once and only once. The last hop of

**L**reverses the last boundary, bringing us to the complement of

**R**. Thus the other outside region of

**L**is isomorphic to the complement of

**R**. By the Complement Region Lemma (unproven), the complement of

**R**is outside.

Suppose regions

**and**R

_{0}

**are complementary, and**R

_{1}

**is inside. By the Subsection Theorem (unproven), there is a 1-dimensional subsection**R

_{0}

**L**with regions isomorphic to

**and**R

_{0}

**. Since**R

_{1}

**is inside, the number of boundary reversals between**R

_{0}

**and**R

_{0}

**is not all, so they are not complementary.**R

_{1}

Consider a set of regions as a polytope. In the section by a boundary, of the space with the boundary removed, facet regions are those isomorphic to regions with both polytope and non-polytope subregions.

A perspective is determined by a choice of outside region, the region containing the focal point. Assume the regions in a polytope have "nearness" of "true". For each region

**in the polytope with neighbor**R

_{0}

**not in the polytope separated by boundary**R

_{1}

**B**, find whether the perspective region is on the same side of

**B**as

**, and xor that with nearness of**R

_{1}

**. Then take the section by**R

_{0}

**B**of the space with

**B**removed, and use the xor result as nearness for the facet region isomorphic to

**and**R

_{0}

**. Choose any outside region for perspective in the section. Nearness of the only region in a 0-section determines whether it is an entry or exit. Thus, 1-sections get direction, 2-sections get clockwisedness. A necessary condition for visibility of a facet is that the facet is nearer than its polytope region. By the Perspective Theorem below, facets may be ordered by a topological sort of 1-sections from the perspective region to its complement. This is useful in computer graphics.**R

_{1}

##
**Perspective Theorem**

The orderings of regions determined by 1-dimensional subsections from an outside region to its complement form a partial ordering of all regions.

Consider two 1-dimensional subsections between outside region

**and region**R

_{0}

**. Boundary**R

_{1}

**B**separates

**and**R

_{0}

**in one subsection iff it does in the other. Therefore a region ordering set up in both subsections must be the same in both subsections.**R

_{1}