The.Algorithm.Design.Manual.Springer-Verlag.1998

Convex Hull
http://www.cs.att.com/netlib/voronoi/hull.html. It does not appear to be as widely used or actively
maintained as Qhull, but it also does alpha-shapes. For an excellent alpha-shapes code, originating from
the work of Edelsbrunner and Mucke [EM94], check out http://fiaker.ncsa.uiuc.edu/alpha/.
Fukuda's cdd program is the best choice for nonsimplicial polytopes in about 6D and higher. See
ftp://ifor13.ethz.ch/pub/fukuda/cdd/. It may be used for computing convex hulls and half-space
intersection.
XTango (see Section ) provides animations of the Graham scan and Jarvis march algorithms in the
plane.
A Pascal implementation of Graham scan appears in [MS91]. See Section . C++ implementations of
planar convex hulls includes LEDA (see Section ). Algorithm 523 [Edd77] of the Collected
Algorithms of the ACM is a Fortran code for planar convex hulls. It is available from Netlib (see Section
).
Notes: Constructing planar convex hulls plays a similar role in computational geometry as sorting does
in algorithm theory. Like sorting, convex hull is a fundamental problem for which a wide variety of
different algorithmic approaches lead to interesting or optimal algorithms. Preparata and Shamos [PS85]
give a good exposition of several such algorithms, including quickhull and mergehull, both inspired by
the sorting algorithms. In fact, a simple construction involving points on a parabola reduces sorting to
convex hull, so the information-theoretic lower bound for sorting implies that planar convex hull requires
time to compute. A stronger lower bound is established in [Yao81].
Good expositions of the Graham scan algorithm [Gra72] and the Jarvis march [Jar73] include [CLR90,
PS85]. The gift wrapping algorithm was introduced by Chand and Kapur [CK70]. Noteworthy among
planar convex hull algorithms is Seidel and Kirkpatrick [KS86], which takes time, where h is the
number of hull vertices, which captures the best performance of both Graham scan and gift wrapping and
is (theoretically) better in between.
Alpha-hulls, introduced in [EKS83], provide a useful notion of the shape of a point set. A generalization
to three dimensions, with an implementation, is presented in [EM94].
Reverse-search algorithms for constructing convex hulls are effective in higher dimensions [AF92],
although constructions demonstrating the poor performance of convex hull algorithms for nonsimplicial
polytopes are presented in [AB95]. Through a clever lifting-map construction [ES86], the problem of
building Voronoi diagrams in d-dimensions can be reduced to constructing convex hulls in (d+1)-
dimensions. See Section for more details.
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