Convex Hull - most ubiquitous structure in computational geometry -useful to construct other structures -many applications: robot motion planning, shape analysis etc. - a beautiful object, one of the early success stories in computational geometry that sparked interest among Computer Scientists by the invention of O(nlogn) algorithm rather than a O(n**3) algorithm. - intimately related to sorting algorithm for both lower and upper bound.

Convex hulls Preliminaries and definitions Intuitive definition 1 Given a set S = {p1, p2, …, pN} of points in the plane, the convex hull H(S) is the smallest convex polygon in the plane that contains all of the points of S. Imagine nails pounded halfway into the plane at the points of S. The convex hull corresponds to a rubber band stretch around them.

Convex hulls Preliminaries and definitions Convex polygon A polygon is convex iff for any two points in the polygon (interior  boundary) the segment connecting the points is entirely within the polygon. convex not convex

Convex hulls Preliminaries and definitions Vertices A polygon vertex is convex if its interior angle  It is reflex if its interior angle >  reflex convex In a convex polygon, all the vertices are convex. In other words, any polygon with a reflex vertex is not convex.

Convex hulls Preliminaries and definitions Convex hull, definition 2 The convex hull H(S) of a subset of points S in a plane is the set of all convex combinations of the points of S. It should be intuitively clear that a hull defined in this way can not have a “dent” (reflex vertex). Note now that S in this definition is an infinite set. The convex hull is the smallest convex set that contains S. To be more precise, it is the intersection of all convex sets that contain S. These infinite convex sets could be limited to a specific infinite subset of halfplanes The latter three definitions are illustrated in the next three slides.

Convex hulls Preliminaries and definitions Convex hull, definition 3 The convex hull H(S) is the intersection of all convex sets that contain S. This is an intersection of an infinite number of Convex sets.

Convex hulls Preliminaries and definitions Convex hull, definition 4 The convex hull H(S) of a set of points S in the plane is the smallest convex polygon P that encloses S, smallest in the sense that there is no other polygon P such that P  P  S. P P

Convex hulls Preliminaries and definitions Convex hull, definition 5 The convex hull H(S) is the intersection of all halfspaces (for d = 2, halfplanes) that contain S. Actually, the number of such half planes is infinite but it can be defined as the intersection of a finite number of half planes. The hull construction algorithm essentially identifies these half planes.

Convex hulls Preliminaries and definitions Convex hull, definition 6 The convex hull H(S) of a set of points S in d dimensions is the set of all convex combinations of d + 1 or fewer points of S. This is different from definition 1 in that only d + 1 or fewer points are needed to get any point of H(S). S is again a finite set as in definition 1. For example, in the plane d = 2, convex polygons can be composed as the union of all points contained by the triangles of the given points, which are the convex combination of all d + 1 = 3 points. 3 1 8 9 4 5 2 7 6

Convex hulls Preliminaries and definitions Convex hull, definition 7 The convex hull H(S) of a set of points S in the plane is the union of all the triangles defined by points in S. This is a restatement of definition 6. Many triangles are not shown in the figure.

Convex hulls Preliminaries and definitions Convex hull, definition 8 The convex hull H(S) of a set of points S in the plane is the enclosing convex polygon P with the smallest area. Convex hull, definition 9 is the enclosing convex polygon P with the smallest perimeter. Extreme Points E A point p of a convex set is an extreme point if no two points a,b S exist such that p is between the line segment ab. Thus in Definition 6 example, the points (1,2 ,3,4,5,6,7) are extreme points but 8 and 9 and others are not. Alternately, the extreme points of S is the smallest subset of S having the property that H(E)=H(S). Thus E defines the vertices on H(S) but does not define the convex hull H(S) which requires the sequence of points on the hull viz. (1,3,4,2,6,7,5) for our example. Thus, there are two distinct problems.

Convex hulls Preliminaries and definitions Problem definitions CONVEX HULL INSTANCE. A set S = {p1, p2, …, pN} of d-dimensional points. QUESTION. Construct the convex hull H(S) for S. (The construction must give the vertices and their sequence, that is, obtain a description of the boundary which is a convex ploygon.) EXTREME POINTS QUESTION. Identify the points of S that are vertices of the convex hull H(S). (Here the ordering is not required.) We will assume d = 2 unless otherwise noted. It turns out that both have the same asymptotic complexity (nlogn) for d=2. Thus the second problem is not any easier than the first problem.