Polygon Convexity: Another O(n) Test
Abstract
An n-gon is defined as a sequence \P=(V_0,...,V_{n-1}) of n points on the plane. An n-gon \P is said to be convex if the boundary of the convex hull of the set {V_0,...,V_{n-1}} of the vertices of \P coincides with the union of the edges [V_0,V_1],...,[V_{n-1},V_0]; if at that no three vertices of \P are collinear then \P is called strictly convex. We prove that an n-gon \P with n\ge3 is strictly convex if and only if a cyclic shift of the sequence (\al_0,...,\al_{n-1})\in[0,2\pi)^n of the angles between the x-axis and the vectors V_1-V_0,...,V_0-V_{n-1} is strictly monotone. A ``non-strict'' version of this result is also proved.
Cite
@article{arxiv.cs/0701045,
title = {Polygon Convexity: Another O(n) Test},
author = {Iosif Pinelis},
journal= {arXiv preprint arXiv:cs/0701045},
year = {2007}
}
Comments
14 pages; changes: (i) a test for non-strict convexity is added; (ii) the proofs are gathered in a separate section; (iii) a more detailed abstract is given