Inscribed and circumscribed histogons of a convex polygon

Constant-Time Convexity Problems on Reconfigurable Meshes

Recently the authors have shown that the versatility of the reconfigurable mesh can be exploited to devise 0(1) time algorithms for a number of important computational tasks relevant to image processing, computer graphics, and computer vision. Specifically, we have shown that if one or two n-vertex (convex) polygons are pretiled, one vertex per processor, onto a reconfigurable mesh of size sqrt n X sqrt n, then a number of geometric problems can be solved in 0(1) time. These include testing an arbitrary polygon for convexity, the point location problem, the supporting lines problem, the stabbing problem, constructing the common tangents of two separable convex polygons, deciding whether two convex polygons intersect, and computing the smallest distance between the boundaries of two convex polygons. The novelty of these algorithms is that the problems are solved in the dense case. The purpose of this paper is to add to the list of problems that can be solved in 0(1) time in the dense case. The problems that we address are: determining the minimum area corner triangle for a convex polygon, determining the k-maximal vertices of a restricted class of convex polygons, updating the convex hull of a convex polygon in the presence of a set of query points, and determining a point that belongs to exactly one of two given convex polygons.

The problem of cutting a convex polygon P out of a piece of paper Q with minimum total cutting length is a well studied problem in computational geometry. Researchers studied several variations of the problem, such as P and Q are convex or non-convex polygons and the cuts are line cuts or rays cuts. In this paper we consider yet another variation of the problem where Q is a circle and P is convex polygon such that P is bounded by a half circle of Q and all the cuts are line cuts. We give a simple linear time O(log n)-approximation algorithm for this problem where n is the number of vertices of P.

The purpose of this paper is to demonstrate that the versatility of the reconfigurable mesh can be exploited for the purpose of devising constant time algorithms for a number of important computational geometry tasks relevant to image processing, computer graphics, and computer vision. In all our algorithms we assume that one or two n-vertex (convex) polygons are pretiled, one vertex per processor, onto a reconfigurable mesh of size √n × √n. In this context, we propose constant-time solutions for testing an arbitrary polygon for convexity, solving the point location problem, the supporting lines problem, the stabbing problem, constructing the common tangents for separable convex polygons, deciding whether two convex polygons intersect, and computing the smallest distance between the boundaries of two convex polygons. To the best of our knowledge this is the first time that O(1) time algorithms are proposed for these problems on this architecture for the “dense” case. The proposed algorithms translate immediately into constant-time algorithms that work on binary images.

Analytical expressions for the distribution of the length of chords corresponding to the affine invariant measure on the set of chords are given for convex polygons. These analytical expressions are a computational improvement over other expressions published in 2011. The correlation function of convex polygons can be computed from the results obtained in this work, because it is determined by the distribution of chords. An analytical expression for the multi-chord distribution of the length of chords corresponding to the affine invariant measure on the set of chords is found for non convex polygons. In addition we give an algorithm to find this multi-chord distribution which, for many concave polygons, is computationally more efficient than the said analytical expression. The results also apply to non simply connected polygons.

Strip-packing problem is apparent in textile industry where a set of items, i.e. cutting parts (2D convex or non-convex polygons) need to be placed on a rectangular container (fabric with an mxn area) so that cutting parts do not overlap and do not exceed the boundaries of the container. The goal is to find a placement that utilizes the area of a container. In this paper three methods (random search, greedy algorithm and genetic algorithm) are tested on sets of regular (convex polygons) and irregular (cutting parts) items. The goal is to find the optimal items placement that minimizes the cover area. In this paper a no-fit polygon (NFP) is used to assure two items touch without overlapping. NFP is constructed by rotating polygon B around a static polygon A in a way their edges always touch and never overlap. The result is a polygonal area enclosed by trajectory of rotating polygon's reference point which represents the overlapping area of A and B. Items touch if polygon B is placed on the NFP's border. Non-convex cutting parts are approximated with their convex hull since a NFP version for convex polygons is used in this paper.

We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of n real numbers (for short, grid). First, we prove that every such grid contains a convex polygon with Omega(log n) vertices and that this bound is tight up to a constant factor. We generalize this result to d dimensions (for a fixed d in N), and obtain a tight lower bound of Omega(log^{d-1}n) for the maximum number of points in convex position in a d-dimensional grid. Second, we present polynomial-time algorithms for computing the longest convex polygonal chain in a grid that contains no two points with the same x- or y-coordinate. We show that the maximum size of such a convex polygon can be efficiently approximated up to a factor of 2. Finally, we present exponential bounds on the maximum number of convex polygons in these grids, and for some restricted variants. These bounds are tight up to polynomial factors.

The problem of cutting a convex polygon P out of a planar piece of material Q (P is already drawn on Q) with minimum total cutting cost is a well studied problem in computational geometry that has been studied with several variations such as P and Q are convex or non-convex polygons, Q is a circle, and the cuts are line cuts or ray cuts. In this paper, we address this problem without the restriction that P is fixed inside Q and consider the variation where Q is a circle and the cuts are line cuts. We show that if P can be placed inside Q such that P does not contain the center of Q, then placing P in a most cornered position inside Q gives a cutting cost of 6.48 times the optimal. We also give an O(n 2)-time algorithm for finding such a position of P, a problem that may be of independent interest. When any placement of P must contain the center of Q, we show that P can be cut of Q with cost 6.054 times the optimal.KeywordsPolygon cuttingline cutcutting costmost cornered positioncornerable and non-cornerable polygon

Polygons appear as constructors in many applications and deciding if two polygons match under similarity transformations and noise is a fundamental problem. Solutions in the literature consider only matching pairs of polygons, implying a sequential comparison when we have a collection. In this paper, we present the first algorithm allowing indexed retrieval of polygons under similarities. We reduce the problem to searching points in the plane, exact searching in the absence of noise, and approximate searching for similar noisy polygons. The above gives a \(O(n+\log (m))\) time algorithm to find the matching polygons under noise and O(1) time for exact similar polygons. We tested our heuristic for indexed polygons in an extensive collection of convex, star-shaped, simple, and self-intersecting polygons. For small amounts of noise, we achieve perfect recall for all polygons. For large amounts of noise, the lowest recall is for convex polygons, while attaining the highest recall is for general (self-intersecting) polygons. The above is not a significant limitation. To recover convex polygons efficiently before indexing, we define a random permutation of the vertices, converting all input polygons to a general polygon and achieving the same successful recovery rates, which is a perfect recall for high noise levels.

Objectives It is known that at most 20 convex polygons can be formed using all seven pieces of the puzzle, but this limit is imposed by the omission of the premise that the puzzle is a square divided into seven pieces. This study analyzes a convex polygon that is conjectured to be impossible to create using the puzzle and then uses the puzzle to prove that it cannot actually be constructed as the convex polygon.
 Methods The proof is based on the assumption that the given convex polygon cannot be formed using sev-en-piece puzzles. It proceeds in two different ways, depending on how the pieces are arranged within the square shape.
 Results We accomplished this in two ways: either by placing in the pieces having the largest area or by initially arranging the pieces on an angle and the diagonal of a square. Consequently, we have proved that the convex poly-gon cannot be formed by seven-piece puzzles.
 Conclusions The actual number of convex polygons that can be formed by seven-piece puzzles was 19, not 20. This study is significant because it provides a corollary for the number of convex polygons that can be constructed using all the pieces of the puzzle.

This paper proposes a shape recognition method for detecting convex polygons (CVPs) in image planes, which is characterized by using Hough transformation (HT) and designing a set classifier. HT is used to extract the sides of a CVP by searching the peaks of accumulators in the Hough Parametric Space (HPS). A total set of the intersection points formed by all sides of CVP and their extending lines can be established based on the peaks in HPS, which includes a subset only containing the vertex of CVP. Based on the gradient distribution of the elements in the subset, the classifier for extracting the subset is designed. Compared with conventional method, the detection method has advantage of detecting the CVP shape with discontinuity and broken edges, and thus worthy of being promoted.

The problem Minimum Convex Cover of covering a given polygon with a minimum number of (possibly overlapping) convex polygons is known to be NP-hard, even for polygons without holes [3]. We propose a polynomial-time approximation algorithm for this problem for polygons with or without holes that achieves an approximation ratio of O(log n), where n is the number of vertices in the input polygon. To obtain this result, we first show that an optimum solution of a restricted version of this problem, where the vertices of the convex polygons may only lie on a certain grid, contains at most three times as many convex polygons as the optimum solution of the unrestricted problem. As a second step, we use dynamic programming to obtain a convex polygon which is maximum with respect to the number of “basic triangles” that are not yet covered by another convex polygon. We obtain a solution that is at most a logarithmic factor off the optimum by iteratively applying our dynamic programming algorithm. Furthermore, we show that Minimum Convex Cover is APX-hard, i.e., there exists a constant δ > 0 such that no polynomial-time algorithm can achieve an approximation ratio of 1 + δ. We obtain this result by analyzing and slightly modifying an already existing reduction [3].KeywordsApproximation AlgorithmApproximation RatioConvex PolygonClockwise OrderBasic TriangleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present a new technique for the decomposition of convex structuring elements for morphological image processing. A unique feature of our approach is the use of linear integer programming technique to determine optimal decompositions for different parallel machine architectures. This technique is based on Shephard's theorem for decomposing Euclidean convex polygons. We formulated the necessary and sufficient conditions to decompose a Euclidean convex polygon into a set of basis convex polygons. We used a set of linear equations to represent the relationships between the edges and the positions of the original convex polygon and those of the basis convex polygons. This is applied to a class of discrete convex polygons in the discrete space. Further, a cost function was used to represent the total processing time for performing dilations on different machine architectures. Then integer programming was used to solve the linear equations based on the cost function. Our technique is general and flexible, so that different cost functions could be used, thus achieving optimal decompositions for different parallel machine architectures. >

). If so then we assign one processor to each edge ofQ′ (respectively P′) and find one edge on upper and one edge on lower convex hull chain of Q′ (respectively P′) so that the x-coordinate P max.x (respectively Q max .x) is between the x-coordinates of the endpoints of the edge. Then P max (respectively Q max) is interior point of Q′ (respectively P′) if and only if it is below the upper edge and above the lower edge.If P max (or Q max) is interior point of Q′ (respectively P′) then we are done since the intersection is nonempty. Otherwise we assign one processor to each pair of one edge from P′ and one edge from Q′. Also, we assign one processor to each pair of one vertex on P′ and one edge on Q′ (also one edge on P′ and one vertex of Q′) and to each pair of vertices taken one from P′ and one from Q′. Then exactly one information appears on exactly one processor and it has one of the following “nature”the corresponding pair of edges assigned to the processor intersect and the intersection is the uppermost point of intersection,there is no vertex of P′ nor Q′ which lies in the area between normals on segment p(i)q(j) joining two assigned vertices p(i) from P′ and q(j) from Q′,there is no vertex of P′ nor Q′ which lies in the area between straight line passing through an edge p(i - √n)p(i) of P′ (similarly for Q′) assigned to the processor and straight line parallel to the edge passing by the vertex q(j) of Q′ (respectively P′) assigned to the considered processor, and the perpendicular from the point q(j) to the edge p(i - √n)p(i) separates p(i - √n) and p(i).Note that i) is easily unique case of P′ and Q′ intersect and P max and Q max are not interior points of Q′ (P′) since each edge has information about coming from upper or lower convex hull chain.If obtained information is of nature in case i) then we are done since P″ and Q″ intersect. However, in cases ii) and iii) there still can be an intersection. But, it is easy to show that the intersection can appear only between convex polygons P″ = (p(i - √n),p(i - √n + 1),…,p(i),…,p(i + √n)) and Q″ = (q(j - √n),q(j - √n + 1),…,q(j),…,q(j + √n). But, detecting this intersection is just algorithm considered above applied on polygons P″ and Q″ with O(√n) vertices each.Thus, the algorithm terminates in constant time on a CREW PRAM. The proof of correctness of explained algorithm is obvious and henceforth omitted.Together with constant time algorithm for finding the distance between two convex polygons [1], we obtain the following result:Given two convex polygons P and Q, in constant time we can determine if they intersect and, if no, determine the minimal distance between them.

On a partition into convex polygons