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observed that every finite point set with no three in a line forms the vertices of a simple polygon. However, requiring no three to be in a line is unnecessarily strong. Instead, all that is required for the existence of a polygonalization (allowing 180° angles) is that the points do not all lie on one line. If they do not, then they have a polygonalization that can be constructed in
It is unknown whether it is possible for the system of all polygonalizations to form a connected state space under local moves that change a bounded number of the edges of the polygonalizations. If this were possible, it could be used as part of an algorithm for generating all polygonalizations, by applying a graph traversal to the state space. For this problem, it is insufficient to consider ''flips'' that remove two edges of a polygonalization and replace them by two other edges, or ''VE-flips'' that remove three edges, two of which share a vertex, and replace them by three other edges. There exist polygonalizations for which no flip or VE-flip is possible, even though the same point set has other polygonalizations.
The ''polygonal wraps'', weakly simple polygons that use each given point one or more times as a vertex, include all polygonalizations and are connected by local moves. Another more general class of polygons, the ''surrounding polygons'', are simple polygons that have some of the given points as vertices and enclose all of the points. They are again locally connected, and can be listed in polynomial time per polygon. The algorithm constructs a tree of polygons, with the
In computational geometry
Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ar ...
, a polygonalization of a finite set of points in the Euclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
is a simple polygon with the given points as its vertices. A polygonalization may also be called a polygonization, simple polygonalization, Hamiltonian polygon, non-crossing Hamiltonian cycle, or crossing-free straight-edge spanning cycle.
Every point set that does not lie on a single line has at least one polygonalization, which can be found in polynomial time. For points in convex position, there is only one, but for some other point sets there can be exponentially many. Finding an optimal polygonalization under several natural optimization criteria is a hard problem, including as a special case the travelling salesman problem. The complexity of counting all polygonalizations remains unknown.
Definition
A polygonalization is a simple polygon having a given set of points in theEuclidean plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...
as its set of vertices. A polygon may be described by a cyclic order on its vertices, which are connected in consecutive pairs by line segments, the edges of the polygon. A polygon, defined in this way, is "simple" if the only intersection points of these line segments are at shared endpoints.
Some authors only consider polygonalizations for points that are in general position, meaning that no three are on a line. With this assumption, the angle between two consecutive segments of the polygon cannot be 180°. However, when point sets with collinearities are considered, it is generally allowed for their polygonalizations to have 180° angles at some points. When this happens, these points are still considered to be vertices, rather than being interior to edges.
Existence
observed that every finite point set with no three in a line forms the vertices of a simple polygon. However, requiring no three to be in a line is unnecessarily strong. Instead, all that is required for the existence of a polygonalization (allowing 180° angles) is that the points do not all lie on one line. If they do not, then they have a polygonalization that can be constructed in polynomial time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
. One way of constructing a polygonalization is to choose any point in the convex hull
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 may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of (not necessarily one of the given points). Then radially ordering the points around (breaking ties by distance from q) produces the cyclic ordering of a star-shaped polygon
In geometry, a star-shaped polygon is a polygonal region in the plane that is a star domain, that is, a polygon that contains a point from which the entire polygon boundary is visible.
Formally, a polygon is star-shaped if there exists a poin ...
through all the given points, with in its kernel. The same idea of sorting points radially around a central point is used in some versions of the Graham scan
Graham's scan is a method of finding the convex hull of a finite set of points in the plane with time complexity O(''n'' log ''n''). It is named after Ronald Graham, who published the original algorithm in 1972. The algorithm finds all vertices o ...
convex hull algorithm, and can be performed in time. Polygonalizations that avoid 180° angles do not always exist. For instance, for and square grids, all polygonalizations use 180° angles.
As well as star-shaped polygonalizations, every non-collinear set of points has a polygonalization that is a monotone polygon
In geometry, a polygon ''P'' in the plane is called monotone with respect to a straight line ''L'', if every line orthogonal to ''L'' intersects the boundary of ''P'' at most twice.
Similarly, a polygonal chain ''C'' is called monotone with respec ...
. This means that, with respect to some straight line (which may be taken as the -axis) every perpendicular line to the reference line intersects the polygon in a single interval, or not at all. A construction of begins by sorting the points by their -coordinates, and drawing a line through the two extreme points. Because the points are not all in a line, at least one of the two open halfplane
In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space.
If the space is two-dimensional, then a half-space is called a half-plane (open or closed). A half-space in a one-dimensional sp ...
s bounded by this line must be non-empty. Grünbaum forms two monotone polygonal chains connecting the extreme points through sorted subsequences of the points: one for the points in this non-empty open halfplane, and the other for the remaining points. Their union is the desired monotone polygon. After the sorting step, the rest of the construction may be performed in linear time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
.
It is NP-complete to determine whether a set of points has a polygonalization using only axis-parallel edges. However, polygonalizations with the additional constraint that they make a right turn at every vertex, if they exist, are uniquely determined. Each axis-parallel line through a point must pass through an even number of points, and this polygonalization must connect alternating pairs of points on this line. The polygonalization may be found in time by grouping the points by equal coordinates and sorting each group by the other coordinate. For any point set, at most one rotation can have a polygonalization of this form, and this rotation can again be found in polynomial time.
Optimization
Problems of finding an optimal polygonalization (for various criteria of optimality) are often computationally infeasible. For instance, the solution to the travelling salesman problem, for the given points, does not have any crossings. Therefore, it is always a polygonalization, the polygonalization with the minimum perimeter. It isNP-hard
In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard pr ...
to find. Similarly, finding the simple polygonalization with minimum or maximum area is known to be NP-hard, and has been the subject of some computational efforts. The maximum area is always more than half of the area of the convex hull
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 may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
, giving an approximation ratio
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
of 2. The exact complexity of the simple polygonalization with maximum perimeter, and the existence of a constant approximation ratio
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
for this problem, remain unknown. The polygonalization that minimizes the length of its longest edge is also NP-hard to find, and hard to approximate to an approximation ratio better than ; no constant-factor approximation is known.
A non-optimal solution to the travelling salesman problem may have crossings, but it is possible to eliminate all crossings by local optimization steps that reduce the total length. Using steps that also eliminate crossings at each step, this can be done in polynomial time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
, but without this restriction there exist local optimization sequences that instead use an exponential number of steps.
The shortest bitonic tour
In computational geometry, a bitonic tour of a set of point sites in the Euclidean plane is a closed polygonal chain that has each site as one of its vertices, such that any vertical line crosses the chain at most twice.
Optimal bitonic tours
The ...
(the minimum-perimeter monotone polygon through the given points) is always a polygonalization, and can be found in polynomial time.
Counting
The problem of counting all polygonalizations of a given point set belongs to #P, the class of counting problems associated with decision problems in NP. However, it is unknown whether it is #P-complete or, if not, what its computational complexity might be. A set of points has exactly one polygonalization if and only if it is in convex position. There exist sets of points for which the number of polygonalizations is as large as , and every set of points has at most polygonalizations. Methods applying the planar separator theorem to labeled triangulations of the points can be used to count all polygonalizations of a set of points in subexponential time, .Dynamic programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.
I ...
can be used to count all monotone polygonalizations in polynomial time, and the results of this computation can then be used to generate a random monotone polygonalization.
Generation
It is unknown whether it is possible for the system of all polygonalizations to form a connected state space under local moves that change a bounded number of the edges of the polygonalizations. If this were possible, it could be used as part of an algorithm for generating all polygonalizations, by applying a graph traversal to the state space. For this problem, it is insufficient to consider ''flips'' that remove two edges of a polygonalization and replace them by two other edges, or ''VE-flips'' that remove three edges, two of which share a vertex, and replace them by three other edges. There exist polygonalizations for which no flip or VE-flip is possible, even though the same point set has other polygonalizations.
The ''polygonal wraps'', weakly simple polygons that use each given point one or more times as a vertex, include all polygonalizations and are connected by local moves. Another more general class of polygons, the ''surrounding polygons'', are simple polygons that have some of the given points as vertices and enclose all of the points. They are again locally connected, and can be listed in polynomial time per polygon. The algorithm constructs a tree of polygons, with the convex hull
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 may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
as its root and with the parent of each other surrounding polygon obtained by removing one vertex (proven to be possible by applying the two ears theorem
In geometry, the two ears theorem states that every simple polygon with more than three vertices has at least two ears, vertices that can be removed from the polygon without introducing any crossings. The two ears theorem is equivalent to the exis ...
to the exterior of the polygon). It then applies a reverse-search algorithm Reverse-search algorithms are a class of algorithms for generating all objects of a given size, from certain classes of combinatorial objects. In many cases, these methods allow the objects to be generated in polynomial time per object, using only ...
to this tree to list the polygons. As a consequence of this method, all polygonalizations can be listed in exponential time ( for points) and polynomial space
In computational complexity theory, PSPACE is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space.
Formal definition
If we denote by SPACE(''t''(''n'')), the set of all problems that can b ...
.
Applications
Classicalconnect the dots
Connect the dots (also known as connect-the-dots, dot to dot, or join the dots) is a form of puzzle containing a sequence of numbered dots. When a line is drawn connecting the dots the outline of an object is revealed. The puzzles frequently c ...
puzzles involve connecting points in sequence to form some unexpected shape, often without crossings. The travelling salesman problem and its variants have many applications. Polygonalization also has applications in the reconstruction of contour line
A contour line (also isoline, isopleth, or isarithm) of a function of two variables is a curve along which the function has a constant value, so that the curve joins points of equal value. It is a plane section of the three-dimensional grap ...
s from scattered data points, and in boundary tracing
Boundary tracing, also known as contour tracing, of a binary digital region can be thought of as a segmentation technique that identifies the boundary pixels of the digital region. Boundary tracing is an important first step in the analysis of ...
in image analysis
Image analysis or imagery analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques. Image analysis tasks can be as simple as reading bar coded tags or as sophi ...
.
See also
* Denjoy–Riesz theorem, on sets of infinitely many points that can be connected by a Jordan arcReferences
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, title = Algorithmic enumeration of surrounding polygons
, url = http://cgm.cs.mcgill.ca/~avis/doc/avis/YAHOUY20.pdf
, volume = 303
, year = 2021
{{citation
, last1 = Zhu , first1 = Chong
, last2 = Sundaram , first2 = Gopalakrishnan
, last3 = Snoeyink , first3 = Jack
, last4 = Mitchell , first4 = Joseph S. B. , author4-link = Joseph S. B. Mitchell
, doi = 10.1016/0925-7721(95)00031-3
, issue = 5
, journal = Computational Geometry: Theory & Applications
, mr = 1408922
, pages = 277–290
, title = Generating random polygons with given vertices
, volume = 6
, year = 1996
Polygons
Computational geometry