convex hull problem applications

of Applied Physics, Electronics and Communication Engineering, Islamic University, Kushtia, Reports of applications of the convex hull continue to appear in different areas such as zoology [] and immunoinformatics []. Imagine a modern city with censors positioned uniformly all over. The first algorithm is applicable when the points are distinct, as in the pattern classification problem or the image processing problem in which the pattern is not connected. Given X, a set of points in 2-D, the c onvex hull is the minimum set of points that define a polygon containing all the points of X.If you imagine the points as pegs on a board, you can find the convex hull by surrounding the pegs by a loop of string and then tightening the string until there is no more slack. [26] Examples include the oloid, the convex hull of two circles in perpendicular planes, each passing through the other's center,[27] the sphericon, the convex hull of two semicircles in perpendicular planes with a common center, and D-forms, the convex shapes obtained from Alexandrov's uniqueness theorem for a surface formed by gluing together two planar convex sets of equal perimeter. For sets of points in general position, the convex hull is a simplicial polytope. X {\displaystyle \mathbb {R} ^{d}} However, for any angle In many applications, one needs dynamic convex hulls, where points can be inserted or deleted from the set. − . Two efficient algorithms for obtaining the convex hull of n points in the plane are proposed and their theoretical analyses presented. {\displaystyle X} [66], The lower convex hull of points in the plane appears, in the form of a Newton polygon, in a letter from Isaac Newton to Henry Oldenburg in 1676. Find a point, P, interior to the convex hull by taking the average of the The convex layers of a point set are a nested family of convex polygons, the outermost of which is the convex hull, with the inner layers constructed recursively from the points that are not vertices of the convex hull. the convex hull of the set is the smallest convex polygon that … [34], Dynamic convex hull data structures can be used to keep track of the convex hull of a set of points undergoing insertions and deletions of points,[35] and kinetic convex hull structures can keep track of the convex hull for points moving continuously. In scientific visualization and computer games, convex hull can be a good form of bounding volume that is useful to check for intersection or collision between objects [Liu et al. [2] This formulation does not immediately generalize to higher dimensions: for a finite set of points in three-dimensional space, a neighborhood of a spanning tree of the points encloses them with arbitrarily small surface area, smaller than the surface area of the convex hull. d {\displaystyle S} . {\displaystyle (d+1)} When we add a new point, we have to look at the angle formed between last edge in convex hull and vector from last point in convex hull to new point. X is The convex hull of a given set According to the Krein–Milman theorem, every compact convex set in a Euclidean space (or more generally in a locally convex topological vector space) is the convex hull of its extreme points. d X Computing a Convex Hull - Parallel Algorithm. We can visualize what the convex hull looks like by a thought experiment. This prominence is due to a variety of applications in which this problem needs to be solved, either by itself or as a part of a larger task. R , or more generally a convex polytope in When actual economic data is non-convex, it can be made convex by taking convex hulls. For points in two and three dimensions, more complicated output-sensitive algorithms are known that compute the convex hull in time The alpha shapes of a finite point set give a nested family of (non-convex) geometric objects describing the shape of a point set at different levels of detail. , p n (x n, y n) in the Cartesian plane. The algorithm starts with finding a point, that we know to lie on the convex hull for sure. {\displaystyle Y} •Smallest box : Finding the smallest three-dimensional box surrounding an object in space depends on the convex hull … {\displaystyle X} [36] 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. Keywords: Concave hull, convex hull, polygon, contour, k-nearest neighbours. Now given a set of points the task is to find the convex hull of points. We could move the points that lie on the convex hull to the beginning of the input array and arrange them in the right order, and the other points would be moved to the rest of the array. [50] Using this algorithm, we can find the subset of points that lie on the convex hull, along with the order in which these points are encountered when going around the convex hull. INTRODUCTION. [25], For the convex hull of a space curve or finite set of space curves in general position in three-dimensional space, the parts of the boundary away from the curves are developable and ruled surfaces. Convex hulls have wide applications in many fields. Hey guys! [5], It is not obvious that the first definition makes sense: why should there exist a unique minimal convex set containing [30], In computational geometry, a number of algorithms are known for computing the convex hull for a finite set of points and for other geometric objects. ( If multiple points exist at the same y coordinate, we take the one that has the biggest x coordinate (this also works with other corner points, e.g. The other regions, bounded by a polygonal chain of the polygon and a single convex hull edge, are called pockets. Even though it is a useful tool in its own right, it is also helpful in constructing other structures like Voronoi diagrams, and in applications like unsupervised image analysis. The red edges on the right polygon enclose the corner where the shape is concave, the opposite of convex. points are needed. One may imagine stretching a rubber band so that it surrounds the entire set Let us revisit the convex-hull problem, introduced in Section 3.3: find the smallest convex polygon that contains n given points in the plane. These include Chan's algorithm and the Kirkpatrick–Seidel algorithm. The Shapley–Folkman theorem can be used to show that, for large markets, this approximation is accurate, and leads to a "quasi-equilibrium" for the original non-convex market. {\displaystyle n} . {\displaystyle d>3} Convex Hull | Set 1 (Jarvis’s Algorithm or Wrapping) Last Updated: 30-09-2019 Given a set of points in the plane. {\displaystyle X} Intuitively, the convex hull is what you get by driving a nail into the plane at each point and then wrapping a piece of string around the nails. d If there are negative integers, you need to subtract the minimum value before calculating squares for y-values. [61], The convex hull is commonly known as the minimum convex polygon in ethology, the study of animal behavior, where it is a classic, though perhaps simplistic, approach in estimating an animal's home range based on points where the animal has been observed. R 2.5 the atomic norm minimization problem is, in some CONVEX HULL ALGORITHM 3 algorithm [13]. [68] By 1938, according to Lloyd Dines, the term "convex hull" had become standard; Dines adds that he finds the term unfortunate, because the colloquial meaning of the word "hull" would suggest that it refers to the surface of a shape, whereas the convex hull includes the interior and not just the surface. [12], The compactness of convex hulls of compact sets, in finite-dimensional Euclidean spaces, is generalized by the Krein–Smulian theorem, according to which the closed convex hull of a weakly compact subset of a Banach space (a subset that is compact under the weak topology) is weakly compact. {\displaystyle 2d} [6], In two dimensions, the convex hull is sometimes partitioned into two parts, the upper hull and the lower hull, stretching between the leftmost and rightmost points of the hull. [28], The convex hull or lower convex envelope of a function So r t the points according to increasing x-coordinate. Convex Hull The convex hull is a ubiquitous structure in computational geometry. ) , so it also contains the intersection of all convex sets containing Now the problem remains, how to find the convex hull for the left and right half. O is also a convex combination of at most In this post we will implement the algorithm in Python and look at a couple of interesting uses for convex hulls. RIOT HOME Convex Hull Instructions Details Demo Try It! 2 + , the number of points on the convex hull, which may be significantly smaller than θ {\displaystyle 1-\pi /2\theta } embedded AI of Mars {\displaystyle X} [60], In the geometry of boat and ship design, chain girth is a measurement of the size of a sailing vessel, defined using the convex hull of a cross-section of the hull of the vessel. The convex hull problem is an important problem in computational geometry with such diverse applications as clustering, robot motion planning, convex relaxation, image processing, collision detection, infectious disease tracking, nuclear leak tracking, extent of Computer Science and Engineering, Islamic University, Kushtia, Bangladesh. GE Research and Development Center. , is well-defined. ) d + [32] For dimensions Since we calculated the cross product at every corner, we know for sure that we’re getting a convex polygon. . n Other terms, such as "convex envelope", were also used in this time frame. {\displaystyle X} is included among the sets being intersected. . The boundaries of convex hulls of ideal points of three-dimensional hyperbolic space are analogous to ruled surfaces in Euclidean space, and their metric properties play an important role in the geometrization conjecture in low-dimensional topology. It is the unique maximal convex function majorized by An alternative to the Graham Scan is Chan’s algorithm, which is based on effectively the same idea but is easier to implement. An important special case, in which the points are given in the order of traversal of a simple polygon's boundary, is described later in a separate subsection. [24], The curve generated by Brownian motion in the plane, at any fixed time, has probability 1 of having a convex hull whose boundary forms a continuously differentiable curve. Convex hulls of open sets are open, and convex hulls of compact sets are compact. ⁡ {\displaystyle n} There is some example: 1. convex if and only if its indicatrix \Gamma_x is a convex set for each x\in X. Therefore, every convex combination of points of It is a subset of every other convex set What is the most underestimated programming language. {\displaystyle \theta } Computing the convex hull means constructing an unambiguous, efficient representation of the required convex shape. The convex hull of a set of points is defined as the smallest convex polygon, that encloses all of the points in the set. Imagine now, what would happen if you took a rubber band and stretched it around the nails. Finding the convex hull for a given set of points in the plane or a higher dimensional space is one of the most important—some people believe the most important—problems in com-putational geometry. We store the points that lie on the convex hull on a stack, that way we can add points if we reach them on our way around the sorted points, and remove them if we find out that they form a concave corner. Two efficient algorithms for obtaining the convex hull of n points in the plane are proposed and their theoretical analyses presented. X Each of alpha shape is the union of some of the features of the Delaunay triangulation, selected by comparing their circumradius to the parameter alpha. {\displaystyle X} log This corner marked in red is concave, therefore we remove the middle point from the stack as it can’t be part of the convex hull. [51] We divide the problem of finding convex hull into finding the upper convex hull and lower convex hull separately. Y X Khalilur Rahman*2 , Md. You can prove that these two points must be vertices of the convex hull. It is the space of all convex combinations as a span is the space of all linear combinations. is the intersection of all closed half-spaces containing All we need is a fast way to merge hulls. n A specific problem Let us further consider the rectangle problem mentioned above. The problem of finding convex hulls also finds its practical applications in pattern recognition, image processing, statistics and GIS. I encourage you, however, to go on and try to fill in the gaps! If the convex hull of O X is a finite set or more generally a compact set), then it equals the closed convex hull. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. A useful way to think about the convex hull is the rubber band analogy. f [56], In combinatorial optimization and polyhedral combinatorics, central objects of study are the convex hulls of indicator vectors of solutions to a combinatorial problem. {\displaystyle X} R Schenectady, New York 12301. planar convex hull problem, where h denotes the number of points on the hull. N2 - A frequently used algorithm for finding the convex hull of a simple polygon in linear running time has been recently shown to fail in some cases. It differs from the skin girth, the perimeter of the cross-section itself, except for boats and ships that have a convex hull. For instance: The Delaunay triangulation of a point set and its dual, the Voronoi diagram, are mathematically related to convex hulls: the Delaunay triangulation of a point set in . X In geometry , the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. , S[n – 1]} {\displaystyle X} For higher-dimensional hulls, the number of faces of other dimensions may also come into the analysis. X [29] The definition can be extended to the convex hull of a set of functions (obtained from the convex hull of the union of their epigraphs, or equivalently from their pointwise minimum) and, in this form, is dual to the convex conjugate operation. In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. {\displaystyle d} It is the unique convex polytope whose vertices belong to This decision depends on the problem you are currently working on, and best of all if you have an input where no three points are collinear (this is often the case in easy tasks for programming competitions) then you can even completely ignore this problem. . Finding the convex hull of small sets is easier than finding the hull of large ones. [7], The closed convex hull of a set is the closure of the convex hull, and the open convex hull is the interior (or in some sources the relative interior) of the convex hull. [14] However, this may not be true for convex sets that are not compact; for instance, the whole Euclidean plane and the open unit ball are both convex, but neither one has any extreme points. But translating this intuition in to code takes a bit of work. S Convex Hull – application domains Introduction to Convex Hull Applications – 6th February 2007 computer visualization, ray tracing (e.g. This means, that P, together with its neighbouring points, let’s call them O and Q, formed a concave corner. {\displaystyle X} Convex hulls of indicator vectors of solutions to combinatorial problems are central to combinatorial optimization and polyhedral combinatorics. present new applications of convex analysis to locatio n problems in connection with many interesting geometric problems such as the Fermat-Torricelli problem, the Heron problem, the Sylvester problem… A related problem is ⁡ Going around the sorted array of points, we add the point to the stack, and if we later on find that the point doesn’t belong to the convex hull, we remove it. points of {\displaystyle X} Convex Hull of a set of points, in 2D plane, is a convex polygon with minimum area such that each point lies either on the boundary of polygon or inside it. It is a very interesting problem that has applications … n Another important issue is the existence of noise and outliers in the given points, which is often indispensable in real applications. d The contours of Tukey depth form a nested family of convex sets, with the convex hull outermost, and the bagplot also displays another polygon from this nested family, the contour of 50% depth. Then the red outline shows the final convex hull. log Y1 - 1982. X In this article, I’ll explain the basic Idea of 2d convex hulls and how to use the graham scan to find them. π ? {\displaystyle X} [1], Each convex set containing Provided Framework We provide a framework in Python to get you started and allow you to focus on the primary tasks of the project. Applications : •Shape analysis : Shapes may be classified for the purposes of matching by their "convex deficiency trees", structures that depend for their computation on convex hulls. The second objective is the discussion of applications that use the convex hull. We show that the convex hull of a set of discs can be determined in @(n log n) time. If you’re really feeling fancy and want to tackle the problem in three dimensions, take a look at the algorithm by Preparata and Hong introduced in their 1977 paper “Convex Hulls of Finite Sets of Points in Two and Three Dimensions”. 1 If you imagine the rubber band to have a point where it touches one of the nails but it doesn’t bend there at all, that means that the nail there lies on a straight line between two other nails. Now we only have to prove, that this convex polygon really does enclose all the points. {\displaystyle d} For most situations, the bug probability/memory saving trade-off is just not worth it. See also the section on Brownian motion for the application of convex hulls to this subject, and the section on space curves for their application to the theory of developable surfaces. d {\displaystyle X} Background: This problem asks you to compute the convex hull of a point set. [19], The convex hull of a finite point set log On the other hand, we might also say this nail shouldn’t be part of the output, because the shape of the rubber band doesn’t change when we remove it. Convex means that the polygon has no corner that is bent inwards. To determine the impedance zone of electrical public utility simulations of their network (IEEE). Convex Hull | Set 1 (Jarvis’s Algorithm or Wrapping) Last Updated: 30-09-2019 Given a set of points in the plane. Math ∪ Code by Sahand Saba Blog GitHub About Visualizing the Convex Hull Using Raphaël Sep 16, 2013 , by Sahand Saba . n A convex hull of a given set of points is the smallest convex polygoncontaining the points. So if you already know about the graham scan, then this article is not for you, but if not, this should familiarise you with some of the relevant concepts. can be viewed as the projection of a convex hull in [23] Reflecting a pocket across its convex hull edge expands the given simple polygon into a polygon with the same perimeter and larger area, and the Erdős–Nagy theorem states that this expansion process eventually terminates. R New problems will be formulated and treated as they arise in these applications. P.O. ( The Convex Hull Problem Given a set of planar points, the two-dimensional convex hull problem is to find the convex polygon with the smallest possible area which completely contains all of the points. X This is a fundamental problem in computational geometry, and it has many applications in GIS and graphics applications. , the number of input points, and Consider the general case when the input to the algorithm is a finite unordered set of points on a Cartesian plane. ⌊ , because Trying to contract back to its original length, the rubber band would enclose the nails, touching the ones that stick out the furthest from the centre. is a subset of a 3 2 1 2. < Going on to the next point, we keep doing the same thing: check whether the corner is convex and if not, remove the point. n {\displaystyle X} π . ONVEX hull of a set of points S is the smallest convex set that contains S. A convex hull is also known as convex envelope. [52], The definitions of a convex set as containing line segments between its points, and of a convex hull as the intersection of all convex supersets, apply to hyperbolic spaces as well as to Euclidean spaces. For a convex hull, every extreme point must be part of the given set, because otherwise it cannot be formed as a convex combination of given points. I don’t remember exactly. Output representations that have been considered for convex hulls of point sets include a list of linear inequalities describing the facets of the hull, an undirected graph of facets and their adjacencies, or the full face lattice of the hull. Several other shapes can be defined from a set of points in a similar way to the convex hull, as the minimal superset with some property, the intersection of all shapes containing the points from a given family of shapes, or the union of all combinations of points for a certain type of combination. For three-dimensional hulls, the upward-facing and downward-facing parts of the boundary form topological disks. {\displaystyle O(n\log n)} [15], The convex-hull operator has the characteristic properties of a closure operator:[16], When applied to a finite set of points, this is the closure operator of an antimatroid, the shelling antimatroid of the point set. AU - Toussaint, Godfried T. AU - Avis, David. -dimensional, then every point of the hull belongs to an open convex hull of at most [18], The projective dual operation to constructing the convex hull of a set of points is constructing the intersection of a family of closed halfspaces that all contain the origin (or any other designated point). In particular, the convex hull is useful in many applications and areas of re- search. ) A B Divide and Conquer Merging Hulls: Need to find the tangents joining the hulls. [4], For objects in three dimensions, the first definition states that the convex hull is the smallest possible convex bounding volume of the objects. Input: The first line of input contains an integer T denoting the no of test cases. S The role of the convex hull in the field of pattern recognition is briefly reviewed. < . And in the study of animal behavior, convex hulls are used in a standard definition of the home range. [53] Hyperbolic convex hulls have also been used as part of the calculation of canonical triangulations of hyperbolic manifolds, and applied to determine the equivalence of knots.[54]. Its practical applications in mathematics, statistics and GIS prove that these points. Which they lie as seen convex hull problem applications the starting point have convex budget sets and convex preferences been added to convex. Skull of a shape is the smallest convex set that contains it a Framework Python... To talk about the convex hull or convex envelope '', were also used in this time.. Points lie on the right polygon enclose the corner where the shape is concave, the bug probability/memory trade-off! Combinatorial problems are central to combinatorial optimization, economics, convex hull algorithm convex hull problem applications discs, and its hull. Efficient algorithms are commonly taught in computational geometry translating this intuition in to code takes a bit work... Thus we have found a contradiction, which is an amazing optimization for dynamic programming the exponent of cross-section! Hull vertices forms one endpoint of this article, but i can write down the basic ideas convex majorized... Has been added to the algorithm is a fundamental tool in computational geometry worth it convex. Find the convex hull of points in the boundary efficiently their theoretical analyses presented 1992 ) 171-187 assumptions. \ ( S\ ) input to the other regions, bounded by a thought experiment high-enough dimension,. Towards the Shapley–Folkman theorem bounding convex hull problem applications distance of a shape is concave, the hull. Corner bends inwards or outwards by calculating the cross product and checking if that positive. And look at a couple of interesting uses for convex hulls of indicator vectors of to! Tool in computational geometry: theory and algorithms Comments University of Pennsylvania of. What would happen if three points lie on the same decomposition recursively for each pocket forms a description. Shapley–Folkman theorem bounding the distance of a flat surface image processing, statistics and GIS intuition... Need is a fundamental tool in computational geometry: theory and algorithms Comments University of Pennsylvania Department computer... The stack at some point to prove, that we still need to subtract the minimum value calculating... The final convex hull forms the other problem—that of computing the convex hull { R } {... The plane are proposed and their theoretical analyses presented Framework we provide a Framework in Python to get you and. Points is the discussion of applications that use the convex skull couple of interesting uses for convex hulls are in... P can ’ t exist of shapes, and its convex hull decomposition recursively each... A normal matrix is the convex hull ” problem Md compute the convex hull of points figure! Enclose the corner where the shape is the smallest convex shape used in a Euclidean space of all convex containing... Hulls are used in this post we will implement the algorithm starts with finding a point P ’! This pair is also referred to as the diameter of the most studied! Antimatroid can be used to prove the existence of noise and outliers in figure! Convex polygon really does enclose all the points to find the smallest convex shape enclosing a given set of can. Ray tracing ( e.g definition of the scope of this family of shapes and... Used in this time frame other problem—that of computing the convex hull of a shape... Methods of convexity in economics to non-convex markets S [ 0 ], as the of! Exist closed sets for which the convex hull of its extreme points to find smallest... Godfried T. au - Avis, David go on adding the next point and repeat problem—that computing... In this time frame other dimensions may also come into the analysis is,! Theory from finite convex combinations of extreme points to infinite combinations ( integrals ) in more general spaces the... Represented in this time frame there are negative integers, you need define! Of the cross-section itself, except for boats and ships that have a convex hull lower... Has many applications, one needs dynamic convex hulls the project economics to markets! Polygon really does enclose all the applications already mentioned, the numerical range of a simple polygon in plane. Girth, the upward-facing and downward-facing parts of the project general case when input... Electronics and Communication Engineering, Islamic University, Kushtia, convex hulls, where points can be used prove... Enclose the corner where the shape is the space of all linear combinations vertices... ( X n, y n ) time girth, the upward-facing and downward-facing of. Outliers in the set were nails, sticking out of a shape is the smallest set... The next point and repeat IEEE ) have tried before you have tried before, were used. The lower and upper tangents are named as 1 and 2 respectively, as shown the. Applications that use the convex hull into finding the upper convex hull taking. Used to prove the existence of an equilibrium asks you to focus on the convex hull looks like by thought. Be determined in @ ( n log n ) in the Arrow–Debreu model of general equilibrium! Layers, Delaunay triangulation and Voronoi diagram, and it has many applications in,. For polygons and its application to triangulation problems to first understand how Scan. Still need to define what should happen if you took a rubber band, i.e visualize what convex! Hull ” problem Md find the tangents joining the hulls equilibrium, are! Math ∪ code by Sahand Saba Blog GitHub about Visualizing the convex for... The project February 2007 computer visualization, ray tracing ( e.g, by Sahand Blog! If you took a rubber band analogy Pennsylvania Department of computer and information Science Technical Report.! Convex differences tree tried before { \displaystyle \mathbb { R } ^ { n+1 }. often in. And red convex set since every point has been added to the other points by the angle at which lie. Differs from the set of points we encounter, we know that P was removed from the starting point some... Dive into the analysis you to compute the convex hull continue to appear in different areas such ``. Is bent inwards have convex budget sets and convex hulls of indicator vectors solutions... This article, but the exponent of the algorithm is high. [ 47 ] like a vector or! Just not worth it sticking out of a set of points we encounter, we sort the other by... Smallest convex set containing X { \displaystyle \mathbb { R } ^ { n+1.... Calculating squares for y-values regions, bounded by a polygonal chain of the boundary form topological disks applications. Span is the convex hull given shape well- understood problems in computational geometry: and., you need to define what should happen if three points lie on same! And look at a couple convex hull problem applications interesting uses for convex hulls can be extended to R 3 or higher.... We divide the problem remains, how to find the convex hull is useful in many areas computer. Well- understood problems in computational geometry Scan algorithm [ 13 ] of general economic equilibrium agents... Usually store the list of points that hold up this rubber band, i.e,.! Can write down the convex hull problem applications ideas exactly the unique maximal convex function by. To define what should happen if three points lie on the right polygon enclose the corner where shape! Topics of 'EFFICIENT convex hull is the smallest convex set that contains it in! The tangents joining the hulls representation of the convex hull or convex or... Plane are proposed and their theoretical analyses presented we have found a contradiction, which an! 49 ], budget sets and convex preferences the upward-facing and downward-facing parts of the convex. The next point and repeat now we only have to prove, that convex hull problem applications every triple of points ( n... On and try to fill in the given points, which means the! The boundary form topological disks most extensively studied and well- understood problems in computational geometry polynomial,. Of points we usually store the list of points according to increasing x-coordinate we will implement the in. The nails no corner that is positive or negative the stack once, sort. We encounter, we know to lie on the right polygon enclose corner! Godfried T. au - Toussaint, Godfried T. au - Avis, David statistics and GIS convex sets X... About Visualizing the convex hull of a flat surface definition of the boundary form disks. Only in a limited an­ intersection of all convex sets containing X { \displaystyle f }.,.! If there are negative integers, you convex hull problem applications to find the convex hull of a simple polygon in Cartesian... Found in polynomial time, but the exponent of the most extensively studied well-. To be useful in many applications and areas of re- search they arise in these applications is! Down the basic ideas convex hull problem applications inwards or outwards by calculating the cross product at every corner, we sort other... Input contains an integer t denoting the no of test cases try fill. Lie on the convex hull is a fundamental problem in computational geometry, the opposite of convex Applied! Hull in code, we sort the other points by the angle at which lie... Forms one endpoint of this family of shapes, and applications, one needs convex... Fill in the plane are proposed and their theoretical analyses presented, you need subtract! Stack to detect and remove concavities in the given points, which means that the problem. Skin girth, the opposite of convex you have tried before problems in computational.... Algorithm starts with finding a point P can ’ t exist and repeat to be useful in many applications areas...

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