Convex Hull

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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(n3)
algorithm. - intimately related to sorting
algorithm for both lower and upper bound.
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Convex hullsPreliminaries 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.
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Convex hullsPreliminaries 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.
not convex
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Convex hullsPreliminaries and definitions
Vertices A polygon vertex is convex if its
interior angle ??????????? It is reflex if its
interior angle gt ?????????
reflex
convex
In a convex polygon, all the vertices are
convex. In other words, any polygon with a reflex
vertex is not convex.
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Convex hullsPreliminaries 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.
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Convex hullsPreliminaries 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.
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Convex hullsPreliminaries 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
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Convex hullsPreliminaries 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.
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Convex hullsPreliminaries 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.
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1
8
9
4
5
2
7
6
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Convex hullsPreliminaries 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.
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Convex hullsPreliminaries 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 The convex hull H(S) of a set
of points S in the plane 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.
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Convex hullsPreliminaries 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.) EXTRE
ME POINTS INSTANCE. A set S p1, p2, , pN of
d-dimensional 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 d2. Thus the second
problem is not any easier than the first problem.