# Chapter 14
Nef Polyhedra embedded on the Sphere

Peter Hachenberger, Lutz Kettner, and Michael Seel

## Summary

Nef polyhedra are defined as a subset of the d-dimensional space obtained by a finite number of set complement and set intersection operations on halfspaces.

Due to the fact that all other binary set operations like union, difference and symmetric difference can be reduced to intersection and complement calculations, Nef polyhedra are also closed under those operations. Also, Nef polyhedra are closed under topological unary set operations. Given a Nef polyhedron one can determine its interior, its boundary, and its closure.

Additionally, a d-dimensional Nef polyhedron has the property, that its boundary is a (d-1)-dimensional Nef polyhedron. This property can be used as a way to represent 3-dimensional Nef polyhedra by means of planar Nef polyhedra. This is done by intersecting the neighborhood of a vertex in a 3D Nef polyhedron with an -sphere. The result is a planar Nef polyhedron embedded on the sphere.

The intersection of a halfspace going through the center of the -sphere, with the -sphere, results in a halfsphere which is bounded by a great circle. A binary operation of two halfspheres cuts the great circles into great arcs.

The incidence structure of planar Nef polyhedra can be reused. The items are denoted as svertex, shalfedge and sface, analogous to their counterparts in Nef_polyhedron_S2. Additionally, there is the shalfloop representing the great circles.

- Nef_polyhedron_S2<Traits>

- Nef_polyhedron_S2<Traits>::Sphere_point

- Nef_polyhedron_S2<Traits>::Sphere_segment

- Nef_polyhedron_S2<Traits>::Sphere_circle

- Nef_polyhedron_S2<Traits>::SVertex

- Nef_polyhedron_S2<Traits>::SHalfedge

- Nef_polyhedron_S2<Traits>::SHalfloop

- Nef_polyhedron_S2<Traits>::SFace

- Nef_polyhedron_S2<Traits>::SFace_cycle_iterator

- Qt_widget_Nef_S2<Nef_polyhedron_S2>