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Lec13 - Convex hulls Gift wrapping d > 2 Problem definition...

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Convex hulls Gift wrapping, d > 2 Problem definition CONVEX HULL, D > 2 INSTANCE. Set S = { p 1 , p 2 , … p N } of points in d -space ( E d ). QUESTION. Construct the convex hull H ( S ) of S . The coordinates of the points p i S will be referred to as p i = ( x 1 , x 2 , …, x d ). For d = 2, the constructed hull was given (represented) as a sequence of hull vertices. How is the hull represented for d > 2? To answer that, and describe the algorithm, more definitions are needed.

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Convex hulls Gift wrapping, d > 2 Polyhedron In E 3 a polyhedron is defined by a finite set of planar polygons such that every edge of a polygon is shared by exactly one other polygon and no subset of the polygons has the property. Polyhedra is plural for polyhedron. The polygons that share an edge are adjacent . The vertices and edges of the polygons are the vertices and edges of the polyhedron. The polygons are the facets of the polyhedron. A polyhedron is simple if there is no pair of nonadjacent facets sharing a point. A simple polyhedron partitions 3-space into two domains, the interior (bounded) and the exterior (unbounded). The term polyhedron often means boundary interior. A simple polyhedron is convex if its interior is a convex set. A polyhedron is the 3-dimensional equivalent of a polygon.
Convex hulls Gift wrapping, d > 2 Polytope A half-space is the portion of E d lying on one side of a hyperplane. A polyhedral set in E d is the intersection of a finite set of closed half-spaces. Note that a polyhedral set is convex, since a half-space is convex, and the intersection of convex sets is convex. Plane polygons ( d = 2) and space polyhedra ( d = 3) are 2- and 3-dimensional instances of bounded polyhedral sets. A bounded d -dimensional polyhedral set is a polytope . Note that polytopes are convex by definition. The terms “convex d -polytope”, “ d -polytope”, and “polytope” are all equivalent. Theorem. The convex hull of a finite set of points in E d is a convex d -polytope. Conversely, a polytope is the convex hull of a finite set of points. For d = 3, the convex hull is a convex polyhedron. For arbitrary d , the convex hull is a d -polytope.

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Convex hulls Gift wrapping, d > 2 Affine set Given k distinct points p 1 , p 2 , …, p k in E d , the set of points p = α 1 p 1 + α 2 p 2 + . . . + α k p k ( α j , α 1 + α 2 + . . . + α k = 1) is the affine set generated by p 1 , p 2 , …, p k , and p is an affine combination of p 1 , p 2 , …, p k . We have seen this before. If k = 2, this is the parametric equation of a line, i.e., a line is an affine set. For k = 3, the affine set is a plane. In general, an affine set for given k is a “flat” object of k - 1 dimensions. Given a subset
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Lec13 - Convex hulls Gift wrapping d > 2 Problem definition...

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