Thursday, November 6, 2014


A complement of a region is the region separated by all boundaries. A subsection is a section or a section of a subsection. An 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 R0 and R1 are complementary, and R0 is inside. By the Subsection Theorem (unproven), there is a 1-dimensional subsection L with regions isomorphic to R0 and R1. Since R0 is inside, the number of boundary reversals between R0 and R1 is not all, so they are not complementary.

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 R0 in the polytope with neighbor R1 not in the polytope separated by boundary B, find whether the perspective region is on the same side of B as R1, and xor that with nearness of R0. Then take the section by B of the space with B removed, and use the xor result as nearness for the facet region isomorphic to R0 and R1. 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.

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 R0 and region R1. Boundary B separates R0 and R1 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.

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