Approximation

approximate(R::Rectification; apply_convex_hull::Bool=false)

Approximate a rectification of a polytopic set with a convex polytope.

Input

• R – rectification of a polytopic set
• apply_convex_hull – (optional; default: false) option to remove redundant vertices

Output

A polytope in vertex representation (VPolygon in 2D, VPolytope otherwise). There is no guarantee that the result over- or underapproximates R.

Algorithm

Let $X$ be the set that is rectified. We compute the vertices of $X$, rectify them, and return the convex hull of the result.

Notes

Let $X$ be the set that is rectified and let $p$ and $q$ be two vertices on a facet of $X$. Intuitively, an approximation may occur if the line segment connecting these vertices crosses a coordinate hyperplane and if the line segment connecting the rectified vertices has a different angle.

As a corollary, the approximation is exact for the special cases that the original set is contained in either the positive or negative orthant or is axis-aligned.