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Topic: Convex hull

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Polygon triangulation

	hull - convex hulls, Delaunay triangulations, alpha shapes
		Hull is an ANSI C program that computes the convex hull of a point set in general (but small!) dimension.
		The input is a list of points, and the output is a list of facets of the convex hull of the points, each facet presented as a list of its vertices.
		If the convex hull is a single point, the algorithm will fail to report it.
	www.netlib.org /voronoi/hull.html (878 words)

	Convex hull - Wikipedia, the free encyclopedia
		This is equivalent to saying that the convex hull is the union of all simplexes with vertices in X. This is known as Carathéodory's theorem.
		Computing the convex hull means that a non-ambiguous and efficient representation of the required convex shape is constructed.
		For a finite set of points, the convex hull is a convex polyhedron in three dimensions, or in general a convex polytope for any number of dimensions.
	en.wikipedia.org /wiki/Convex_hull (1212 words)

	PlanetMath: convex combination
		This is version 9 of convex combination, born on 2001-10-19, modified 2006-08-25.
		The definition of convex hull as the intersection of all convex sets containing X most definately works.
		Also we need a definition of closed convex hull which is the intersection of all closed convex sets containing X. I'm now not sure if this is the same as the closure of the convex hull, need to check this out.
	planetmath.org /encyclopedia/ConvexCombination.html (210 words)

	PlanetMath: polynomially convex hull
		Polynomially convex hull is the same thing, but with polynomials.
		For example, hulls with respect to plurisubharmonic functions are very useful in multivariate complex analysis.
		This is version 3 of polynomially convex hull, born on 2004-05-05, modified 2005-03-07.
	planetmath.org /encyclopedia/PolynomiallyConvexHull.html (388 words)

	Convex Hull
		The primary convex hull algorithm in the plane is the Graham scan.
		Qhull [BDH97] appears to be the convex hull code of choice for general dimensions (in particular from 2 to about 8 dimensions).
		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].
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE185.HTM (1597 words)

	Wikinfo \| Convex hull
		The convex hull for a geometrical object or a set of geometrical objects is the minimal convex set containing the given objects.
		Computing the convex hull means that a non-ambiguous and efficient represesentation of the required convex shape is constructed.
		For a finite set of points, the convex hull is a convex polyhedron.
	www.wikinfo.org /wiki.php?title=Convex_hull (575 words)

	Carathéodory's theorem (convex hull) - Wikipedia, the free encyclopedia
		lies in the convex hull of a set P, there is a subset P′ of P consisting of no more than d+1 points such that x lies in the convex hull of P′.
		The convex hull of this set is a square.
		Consider now a point x=(1/4, 1/4), which is in the convex hull of P.
	en.wikipedia.org /wiki/Carath%C3%A9odory's_theorem_(convex_hull) (326 words)

	Convex Hull of a 2D Point Set or Polygon
		Computing a convex hull (or just "hull") is one of the first sophisticated geometry algorithms, and there are many variations of it.
		The convex hull of a geometric object (such as a point set or a polygon) is the smallest convex set containing that object.
		The lower or upper convex chain is constructed using a stack algorithm almost identical to the one used for the Graham scan.
	geometryalgorithms.com /Archive/algorithm_0109/algorithm_0109.htm (2504 words)

	qconvex -- convex hull
		The convex hull of a set of points is the smallest convex set containing the points.
		Compute the convex hull of 1000 points near the surface of a randomly rotated simplex.
		Convex hull of 8 points in 3-d: Number of vertices: 8 Number of facets: 6 Number of non-simplicial facets: 6 Statistics for: RBOX c
	www.qhull.org /html/qconvex.htm (1514 words)

	Chris Harrison - Convex Hull
		The next point in the convex hull must continue in the clockwise motion, or it would be a concave, and there would be a point with a lower angle relative to the prior point in the perimeter (see point labeled “impossible point”).
		Convex hulls are applicable to points in arbitrary dimensions, so most algorithms have been extended to operate in higher dimensions, which has also generated a lot of new and interesting problems.
		As Jarvis’ March progresses around the convex hull, it finds the point with the most minimal or maximal angle relative to itself; that is the next point in the convex hull.
	www.chrisharrison.net /projects/convexHull (2270 words)

	Search Tuna Report for convex hull (Site not responding. Last check: )
		Problem: Find the smallest convex polygon containing all the points of S. Excerpt from The Algorithm Design Manual: Finding the convex hull of a set of points is the most elementary interesting problem in computational geometry, just as minimum spanning tree is the most elementary interesting problem in graph algorithms.
		Convex Hull The convex hull of a set of points is the smallest convex set that includes the points....
		The convex hull of a set of points S in n dimensions is the intersection of all convex sets containing S...
	www.searchtuna.com /ftlive2/285.html (2320 words)

	Convex Hull
		A convex polygon is a polygon that for every two points (on the boundary or inside the polygon) you pick, the segments that connects those points is entirely inside the polygon.
		The Convex Hull is the smallest polygon that satisfies this rule.
		Given a set of n points, the Convex Hull of this set of points is the smallest polygon that includes all the points (either on the boundary or inside the polygon).
	www.eecs.tufts.edu /GK-12/algebra/convex_hull.html (1096 words)

	The Incredible Hull
		For a set of two-dimensional objects, their convex hull is the convex polygon of least area that completely encloses all of the objects.
		Determining the convex hull of a set of objects is a fundamental problem in both computer graphics and computational geometry.
		For example, supposing that (10,1), (10,7) and (10,12) are consecutive points on the convex hull, the point (10,7) should not be removed but rather reported as part of the convex hull.
	acm.uva.es /p/v5/596.html (553 words)

	Glossary: Convex Hull (Site not responding. Last check: )
		The convex envelope of a set is the boundary of its convex hull.
		For example, the convex envelope of three points in space is the union of the three line segments joining the points in pairs (this is the boundary of the planar triangle that has the three points as its vertices).
		The convex envelope of a surface in space is always an embedded, convex, topological sphere.
	www.maa.org /cvm/1998/01/tprppoh/article/Glossary/ConvexEnvelope.html (67 words)

	VB Helper: HowTo: Find the convex hull of a set of points in Visual Basic 2005
		A convex hull is a smallest convex polygon that surrounds a set of points.
		It adds that point to the hull and repeats the sweep from the new point starting with the sweep angle it used in the last test.
		This rectangle is guaranteed to lie within the convex hull.
	www.vb-helper.com /howto_2005_convex_hull.html (354 words)

	Convex Hull
		The convex hull of a set of points is the smallest convex set that includes the points.
		For a two dimensional finite set the convex hull is a convex polygon.
		My convex hull applet demonstrates four algorithms for computing the convex hull.
	www.cse.unsw.edu.au /~lambert/java/3d/ConvexHull.html (104 words)

	Qhull code for Convex Hull, Delaunay Triangulation, Voronoi Diagram, and Halfspace Intersection about a Point
		Barber, C.B., Dobkin, D.P., and Huhdanpaa, H.T., "The Quickhull algorithm for convex hulls," ACM Trans.
		The convex hull of a set of points is the smallest convex set that contains the points.
		It is similar to the randomized, incremental algorithms for convex hull and Delaunay triangulation.
	www.qhull.org (545 words)

	Arbitrary dimensional convex hull, Voronoi diagram, Delaunay triangulation
		Arbitrary dimensional convex hull, Voronoi diagram, Delaunay triangulation
		Arbitrary dimensional convex hulls, Delaunay triangulations, alpha shapes, volumes of Voronoi cells; no non-degeneracy assumptions.
		Dual convex hull - computes the vertices of a polytope defined as an intersection of halfspaces (nice because it's easier to solve convex hull given a program for the dual than visa versa).
	www.geom.uiuc.edu /software/cglist/ch.html (379 words)

	55:148,55:247 Chapter 6, Part 3
		The convex hull can be used to describe region shape properties and can be used to build a tree structure of region concavity.
		If D is outside of the current convex hull, it must become a new convex hull vertex and based on the current convex hull shape, either none, one, or several vertices must be removed from the current convex hull.
		A convex hull of the whole region is constructed first, and convex hulls of concave residua are found next.
	www.icaen.uiowa.edu /~dip/LECTURE/Shape3.html (2543 words)

	3D Convex Hulls (Site not responding. Last check: )
		Both versions accept a range of input iterators defining the set of points whose convex hull is to be computed and a traits class defining the geometric types and predicates used in computing the hull.
		Then the number of points of the convex hull are obtained by counting the number of triangulation vertices incident to the infinite vertex.
		Notice that the vertices incident to the infinite vertex of the triangulation are on the convex hull but it may be that not all of them are vertices of the hull.
	www.cgal.org /Manual/3.2/doc_html/cgal_manual/Convex_hull_3/Chapter_main.html (727 words)

	Convex Hull - Algorithmist
		Computing the convex hull in Computational Geometry is what Sorting in many problems - it is perhaps the most basic, elementary function on a set of points.
		One way to visualize a convex hull is to put a "rubber band" around all the points, and let it wrap as tight as it can.
		Any convex hull algorithm have the lower bound of θ(nlogn) through a reduction from Sorting, but it gives rises to Output Sensitive Algorithms, where the Complexity of the algorithm depends on the size of the output.
	www.algorithmist.com /index.php/Convex_Hull (466 words)

	Diameter Algorithm Based on Convex Hull
		The convex hull of a set of points is a polygon that has the shape of an elastic band streched around all the points.
		Computing the diameter of a convex polygon (In this case the polygon happens to be the convex hull of a set of points.
		Recall that a convex hull is a convex polygon).
	cgm.cs.mcgill.ca /~msuder/schools/mcgill/diameter/node2.html (398 words)

	On-Line Computer Graphics Notes
		convex combination of these points is also in the set.
		Since any convex combination of points from a convext set must lie in the set, then certainly the straight line joining any two points of the set must also be completely in the set.
		bounding box about the set of points, and since the bounding box is convex, we are insured that the convex-hull of the set of points is also contained in the bounding box.
	graphics.idav.ucdavis.edu /education/GraphicsNotes/Convex-Combinations/Convex-Combinations.html (441 words)

	Convex Hull of a 2D Point Set or Polygon
		Computing a convex hull (or just "hull") is one of the first sophisticated geometry algorithms, and there are many variations of it.
		The convex hull of a geometric object (such as a point set or a polygon) is the smallest convex set containing that object.
		The lower or upper convex chain is constructed using a stack algorithm almost identical to the one used for the Graham scan.
	www.geometryalgorithms.com /Archive/algorithm_0109/algorithm_0109.htm (2504 words)

	hull - convex hulls, Delaunay triangulations, alpha shapes (Site not responding. Last check: )
		Hull is an ANSI C program that computes the convex hull of a point set in general (but small!) dimension.
		The input is a list of points, and the output is a list of facets of the convex hull of the points, each facet presented as a list of its vertices.
		If the convex hull is a single point, the algorithm will fail to report it.
	cm.bell-labs.com /netlib/voronoi/hull.html (878 words)

	2D Convex Hulls and Extreme Points (Site not responding. Last check: )
		The linear-time algorithm of Melkman for producing the convex hull of simple polygonal chains (or polygons) is available through the function ch_melkman.
		In addition to the functions for producing convex hulls, there are a number of functions for computing sets and sequences of points related to the convex hull.
		Each of the functions used to compute convex hulls or extreme points is paramterized by a traits class, which specifies the types and geometric primitives to be used in the computation.
	www.cgal.org /Manual/3.2/doc_html/cgal_manual/Convex_hull_2/Chapter_main.html (816 words)

	Stella's Polyhedral Glossary
		The 13 convex semi-regular polyhedra, excluding prisms and antiprisms.
		A convex polygon or polyhedron is one where any line segment drawn from a point inside the shape to another point inside the shape, will lie entirely within the shape.
		The convex hull of a polygon or polyhedron is the smallest convex polygon or polyhedron which encloses the given shape.
	home.aanet.com.au /robertw/Glossary.html (5970 words)

	Relative Convex Hull (Site not responding. Last check: )
		On the left is the standard convex hull, while on the right is the convex hull relative to the larger (hatched) polygon.
		The relative convex hull is the same thing, except geodesic curves lie inside the complement of P relative to another bounded figure (say Q) rather than the whole plane.
		Figure 8 shows an example of a relative convex hull (the right figure) where the geodesics lie inside the hatched polygon.
	www.cs.mcgill.ca /~stever/pattern/MPP/node7.html (268 words)

	Geometry in Action: Convex Hulls (Site not responding. Last check: )
		Jason Everhart of Los Alamos uses convex hulls as part of a heuristic for estimating the percentage of lung volume occupied by a pneumonea infection.
		US Patents 5317681 and 5428717 cover methods for finding convex hulls of polyhedra based on flipping reflex edges, along with an animated version of this procedure that creates a smooth morph of the polyhedron to its hull.
		US Patent 5463721 describes the use of convex hulls in a method for finding a path for a radiation-beam scanner so it can get enough data to reconstruct object shapes.
	www.ics.uci.edu /~eppstein/gina/hull.html (309 words)

	qhull -- convex hull and related structures
		The Delaunay triangulation and furthest-site Delaunay triangulation are equivalent to a convex hull in one higher dimension.
		Halfspace intersection about a point is equivalent to a convex hull by polar duality.
		Compute the 3-d convex hull of 1000 random points.
	www.math.sunysb.edu /~sorin/online-docs/qhull/qhull.htm (709 words)

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