4 - Convex Hulls in Three Dimensions
Published online by Cambridge University Press: 05 June 2012
Summary
The focus of this chapter is algorithms for constructing the convex hull of a set of points in three dimensions (Section 4.2). We will also touch on related issues: properties of polyhedra (Section 4.1), how to represent polyhedra (Section 4.4), and a brief exploration of higher dimensions (Section 4.6). Finally, several related topics will be explored via a series of exercises (Section 4.7). The centerpiece of the chapter is the most complex implementation in the book: code for constructing the three-dimensional hull via the incremental algorithm (Section 4.3).
POLYHEDRA
Introduction
A polyhedron is the natural generalization of a two-dimensional polygon to three-dimensions: It is a region of space whose boundary is composed of a finite number of flat polygonal faces, any pair of which are either disjoint or meet at edges and vertices. This description is vague, and it is a surprisingly delicate task to make it capture just the right class of objects. Since our primary concern in this chapter is convex polyhedra, which are simpler than general polyhedra, we could avoid a precise definition of polyhedra. But facing the difficulties helps develop three-dimensional geometric intuition, an invaluable skill for understanding computational geometry.
We concentrate on specifying the boundary or surface of a polyhedron. It is composed of three types of geometric objects: zero-dimensional vertices (points), one-dimensional edges (segments), and two-dimensional faces (polygons). It is a useful simplification to demand that the faces be convex polygons, which we defined to be bounded (Section 1.1.1).
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- Computational Geometry in C , pp. 101 - 154Publisher: Cambridge University PressPrint publication year: 1998
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