In mathematics Nef polygons and Nef polyhedra are the sets of
polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain.
The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
s and
polyhedra
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
which can be obtained from a finite set of
halfplanes (
halfspaces) by
Boolean operations of set intersection and set complement. The objects are named after the Swiss mathematician
Walter Nef (1919–2013), who introduced them in his 1978 book on polyhedra.
Since other Boolean operations, such as union or difference, may be expressed via intersection and complement operations, the sets of Nef polygons (polyhedra) are closed with respect to these operations as well.
In addition, the class of Nef polyhedra is closed with respect to the topological operations of taking closure, interior, exterior, and boundary. Boolean operations, such as difference or intersection, may produce non-regular sets. However the class of Nef polyhedra is also closed with respect to the operation of
regularization.
Convex polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
s are a special subclass of Nef polyhedra, being the set of polyhedra which are the intersections of a finite set of half-planes.
Terminology
In the language of Nef polyhedra you can refer to various objects as 'faces' with different dimensions. What would normally be called a 'corner' or 'vertex' of a shape is called a 'face' with dimension of 0. An 'edge' or 'segment' is a face with dimension 1. A flat shape in 3D space, like a triangle, is called a face with dimension 2 – or a 'facet'. A shape in 3D space, like a cube, is called a face with dimension 3 – or a 'volume'.
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Implementations
The Computational Geometry Algorithms Library, or CGAL, represents Nef Polyhedra by using two main data structures. The first is a 'Sphere map' and the second is a 'Selective Nef Complex' (or SNC). The 'sphere map' stores information about the polyhedron by creating an imaginary sphere around each vertex, and painting it with various points and lines representing how the polyhedron divides space. The SNC basically stores and organizes the sphere maps. Each face contains a 'label' or 'mark' telling whether it is part of the object or not.[
]
See also
* CGAL
References
{{reflist , refs =
[
{{cite CiteSeerX , citeseerx = 10.1.1.73.157 , title = Boolean Operations on 3D Selective Nef Complexes: Data Structure, Algorithms, Optimized Implementation and Experiments , first1 = Peter , last1 = Hachenberger , first2 = Lutz , last2 = Kettner , first3 = Kurt , last3 = Mehlhorn , year = 2006
]
Types of polygons
Polyhedra