Perspective

A perspective is a maximal set of paths through a focal region, such that each path crosses each boundary once and only once.

Because paths cross boundaries only once, regions and vertices after the boundary all have the same sidedness wrt the boundary. Thus path sidedness is well defined and equivalent to space sidedness.

Sidedness of a boundary wrt a vertex and perspective is whether a path reaches the boundary or one of the vertice’s regions first.

Note that the point of convergence of lines perpendicular to the base in the numeric space is in some particular outside region. Thus, there is a symbolic perspective that makes numeric sidedness wrt vertices equivalent to symbolic sidedness wrt vertices.

Write algorithms to go back and forth between a numeric space and a numeric cospace, with one boundary per vertex, and one boundary per vertex at infinity.

Complete the construction proof by noting that the algorithms do not use the choice function.