Abstract
An object is convex if any two points inside it can be connected via a straight line that is entirely inside the object. This chapter opens with a discussion of convexity and then defines the convex hull: The tightest fitting convex region of space that covers a given object.
Initially, several algorithms for computing 2D convex hulls are considered and then methods for 3D convex hulls. In particular, we discuss an incremental algorithm where one adds a triangle at a time and the divide and conquer approach where the object is recursively divided until the computations are trivial. The essential part of the divide and conquer approach is to recursively merge the convex hulls of the parts.
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Notes
- 1.
In some countries, polytopes are defined as bounded polyhedra, without requiring them to be convex.
- 2.
The lexicographic ordering is the ordering of a language dictionary. In this case the vertices are first ordered according to their polar angle around o, and if there are ties, the vertices having the same polar angle around o are ordered according to their square distance to o.
References
Berger, M.: Géométrie, vol. 3. CEDIC, Paris (1977). Convexes et polytopes, polyèdres réguliers, aires et volumes. [Convexes and polytopes, regular polyhedra, areas and volumes]
Berger, M.: Geometry. I. Universitext, p. 428. Springer, Berlin (1987). Translated from the French by M. Cole and S. Levy
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001). http://www.amazon.ca/exec/obidos/redirect?tag=citeulike09-20&path=ASIN/0262531968
Lischinski, D.: Incremental Delaunay triangulation. In: Heckbert, P. (ed.) Graphics Gems IV, pp. 47–59. Academic Press, Boston (1994)
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© 2012 Springer-Verlag London
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Bærentzen, J.A., Gravesen, J., Anton, F., Aanæs, H. (2012). Convex Hulls. In: Guide to Computational Geometry Processing. Springer, London. https://doi.org/10.1007/978-1-4471-4075-7_13
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DOI: https://doi.org/10.1007/978-1-4471-4075-7_13
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