Characterization of polyhedron monotonicity

The notion of polygon monotonicity has been well researched to be used as an important property for various geometric problems. This notion can be more extended for categorizing the boundary shapes of polyhedrons, but it has not been explored enough yet. This paper characterizes three types of polyhedron monotonicity: strong-, weak-, and directional-monotonicity: (Toussaint, 1985). We reexamine the three types of polyhedron monotonicity by relating them with 3D manufacturing problems, and present their formulation with geometric problems on the sphere.