![]() The other regions, bounded by a polygonal chain of the polygon and a single convex hull edge, are called pockets.Floating Solutions for Challenges Facing Humanity (C. The convex hull of a simple polygon encloses the given polygon and is partitioned by it into regions, one of which is the polygon itself. Special cases Finite point sets Ĭonvex hull ( in blue and yellow) of a simple polygon (in blue) The projective dual operation to constructing the convex hull of a set of points is constructing the intersection of a family of closed halfspaces that all contain the origin (or any other designated point). This provides a step towards the Shapley–Folkman theorem bounding the distance of a Minkowski sum from its convex hull. The operations of constructing the convex hull and taking the Minkowski sum commute with each other, in the sense that the Minkowski sum of convex hulls of sets gives the same result as the convex hull of the Minkowski sum of the same sets. When applied to a finite set of points, this is the closure operator of an antimatroid, the shelling antimatroid of the point set.Įvery antimatroid can be represented in this way by convex hulls of points in a Euclidean space of high-enough dimension. ![]() They can be solved in time O ( n log n ). The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of computational geometry. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. Every compact convex set is the convex hull of its extreme points. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.Ĭonvex hulls of open sets are open, and convex hulls of compact sets are compact. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. ![]() In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull of the red set is the blue and red convex set.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |