On complexity of envelopes of piecewise linear functions, unions and intersections of polygons
Pavel Kozhevnikov (Moscow Institute of Physics, Technology)
TL;DR
This paper establishes tight upper bounds on the number of vertices in polygons formed by unions or intersections of simpler polygons, and explores similar bounds for envelopes of piecewise linear functions.
Contribution
It provides the first tight bounds for the complexity of envelopes of piecewise linear functions and polygon unions/intersections with specified convex and concave vertices.
Findings
Derived tight upper bounds for polygon vertex counts.
Analyzed complexity of envelopes of piecewise linear functions.
Extended results to graphs of envelopes of two functions.
Abstract
We prove tight upper bounds for the number of vertices of a simple polygon that is the union or the intersection of two simple polygons with given numbers of convex and concave vertices. The similar question on graphs of the lower (or upper) envelope of two continuous piecewise linear functions is considered.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Optimization and Packing Problems · Advanced Graph Theory Research