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Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure
The convex hull of a simple polygon is divided by the polygon into pieces, one of which is the polygon itself and the rest are pockets bounded by a piece of the polygon boundary and a single hull edge. Although many algorithms have been published for the problem of constructing the convex hull of a simple polygon, nearly half of them are ...
A demo of Graham's scan to find a 2D convex hull. Graham's scan is a method of finding the convex hull of a finite set of points in the plane with time complexity O(n log n). It is named after Ronald Graham, who published the original algorithm in 1972. [1] The algorithm finds all vertices of the convex hull ordered along its boundary.
The convex hull of a simple polygon (blue). Its four pockets are shown in yellow; the whole region shaded in either color is the convex hull. In discrete geometry and computational geometry, the convex hull of a simple polygon is the polygon of minimum perimeter that contains a given simple polygon.
The convex-hull operation is needed for the set of convex sets to form a lattice, in which the "join" operation is the convex hull of the union of two convex sets = = ( ()). The intersection of any collection of convex sets is itself convex, so the convex subsets of a (real or complex) vector space form a complete lattice .
The polynomially convex hull contains the holomorphically convex hull. The domain is called holomorphically convex if for every compact subset , ^ is also compact in G. Sometimes this is just abbreviated as holomorph-convex.
This theorem has a variant in which the convex hull is replaced by the conical hull. [10]: Thm.2.2 Let X 1, ..., X d be sets in R d and let x be a point contained in the intersection of the conical hulls of all these d sets. Then there is a set T = {x 1, ..., x d}, where x 1 ∈ X 1, ..., x d ∈ X d, such that the conical hull of T contains ...
A 2D demo for Chan's algorithm. Note however that the algorithm divides the points arbitrarily, not by x-coordinate. In computational geometry, Chan's algorithm, [1] named after Timothy M. Chan, is an optimal output-sensitive algorithm to compute the convex hull of a set of points, in 2- or 3-dimensional space.