Polygon Convexity: Another O(n) Test

Details Polygon Convexity: Another O(n) Test Polygon Convexity: Another O(n) Test

2007-01-08

arxiv.org/abs/cs/0701045v2

Computer Science

Computational Geometry

14 pages; changes: (i) a test for non-strict convexity is added; (ii) the proofs are gathered in a separate section; (iii) a m

Scientific paper

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.

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