Monotone polygon

For many practical purposes this definition may be extended to allow cases when some edges of P are orthogonal to L, and a simple polygon may be called monotone if a line segment that connects two points in P and is orthogonal to L lies completely in P. Following the terminology for monotone functions, the former definition describes polygons strictly monotone with respect to L. Assume that L coincides with the x-axis.

A linear time algorithm is known to report all directions in which a given simple polygon is monotone.

The minimum perimeter bitonic tour for a given point set with respect to a fixed direction may be found in polynomial time using dynamic programming.

[5] It is easily shown that such a minimal bitonic tour is a simple polygon: a pair of crossing edges may be replaced with a shorter non-crossing pair while preserving the bitonicity of the new tour.

A monotone polygon is sweepable by a line which does not change its orientation during the sweep.