Monotone polygon

In geometry, a polygon P in the plane is called monotone with respect to a straight line L, if every line orthogonal to L intersects P at most twice.

Similarly, a polygonal chain C is called monotone with respect to a straight line L, if every line orthogonal to L intersects C at most once.

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 completely belongs to P.

Following the terminology for monotone functions, the former definition describes polygons strictly monotone with respect to L. The "with respect to" part is necessary for drawing the strict/nonstrict distinction: a polygon nonstrictly monotone with respect to L is strictly monotone with respect to a line L₁ rotated from L by a sufficiently small angle.

Properties

Assume that L coincides with the x-axis. Then the leftmost and rightmost vertices of a monotone polygon decompose its boundary into two monotone polygonal chains such that when the vertices of any chain are being traversed in their natural order, their X-coordinates are monotonically increasing or decreasing. In fact, this property may be taken for the definition of monotone polygon and it gives the polygon its name.