A characterization of the convexity of cyclic polygons in terms of the central angles
Document Type
Article
Publication Date
2007
Department
Department of Mathematical Sciences
Abstract
Let P be a cyclic n-gon with n ≥ 3, the central angles θ 0 (-π, π], ... , θ n-1 (-π,π], and the winding number w := (θ 0 +...+ θ n-1)/(2π). The vertices of P are assumed to be all distinct from one another. It is then proved that P is convex if and only if one of the following four conditions holds: (I) w = 1 and θ 0,..., θ n-1 > 0; (II) w = -1 and θ 0,..., θ n-1 < 0; (III) w = 0 and exactly one of the angles θ 0,...,θ n-1 is negative; (IV) w = 0 and exactly one of the angles θ 0,..., θ n-1 is positive.
Publication Title
Journal of Geometry
Recommended Citation
Pinelis, I.
(2007).
A characterization of the convexity of cyclic polygons in terms of the central angles.
Journal of Geometry,
87(1-2), 106-119.
http://doi.org/10.1007/s00022-007-1799-9
Retrieved from: https://digitalcommons.mtu.edu/michigantech-p/4628