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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 ...
In geometry, the convex hull, convex envelope or convex closure [1] of a shape is the smallest convex set that contains it. 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.
Convex regular polygons can also form plane tilings that are not edge-to-edge. Such tilings can be considered edge-to-edge as nonregular polygons with adjacent colinear edges. There are seven families of isogonal figures , each family having a real-valued parameter determining the overlap between sides of adjacent tiles or the ratio between the ...
The collection of convex subsets of a vector space, an affine space, or a Euclidean space has the following properties: [9] [10] The empty set and the whole space are convex. The intersection of any collection of convex sets is convex. The union of a sequence of convex sets is convex, if they form a non-decreasing chain for inclusion.
The complete list of convex polygons that can tile the plane includes the above 15 pentagons, three types of hexagons, and all quadrilaterals and triangles. [5] A consequence of this proof is that no convex polygon exists that tiles the plane only aperiodically, since all of the above types allow for a periodic tiling.
An example of a convex polygon: a regular pentagon. In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon (not self-intersecting). [1]
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts [ 1 ] [ 2 ] use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object.
For the sake of simplicity, the description below assumes that the points are in general position, i.e., no three points are collinear.The algorithm may be easily modified to deal with collinearity, including the choice whether it should report only extreme points (vertices of the convex hull) or all points that lie on the convex hull [citation needed].