Showing papers on "Regular polygon published in 1989"
05 Jun 1989
TL;DR: This thesis introduces Epsilon Geometry, a framework for the development of robust geometric algorithms using inaccurate primitives based on a very general model of imprecise computations, which includes floating-point and rounded-integer arithmetic as special cases.
Abstract: This thesis introduces Epsilon Geometry, a framework for the development of robust geometric algorithms using inaccurate primitives. Epsilon Geometry is based on a very general model of imprecise computations, which includes floating-point and rounded-integer arithmetic as special cases. The algorithms in this framework produce exact solutions for perturbed versions of their input and return a bound on the size of these implicit perturbations.
The thesis begins with a formal description of the Epsilon Geometry framework. It introduces the notions of an epsilon predicate, a geometric predicate that can be satisfied by a perturbed version of its arguments; and a critical distance, the size of the perturbation required to make an epsilon predicate true. It also introduces epsilon and delta boxes, the implementations of these mathematical concepts as computer programs. The thesis describes some general rules for turning mathematical lemmas about epsilon predicates and critical distances into implementations of epsilon and delta boxes. Next, it presents a basic set of two-dimensional geometric predicates that are used for all of the algorithms in the sequel.
The second half of the thesis examines how Epsilon Geometry can be applied to various geometric objects and algorithms in the plane. It defines the notions of an $\varepsilon$-convex polygon, a polygon that can be made convex by perturbing each of its vertices by $\varepsilon$ or less; and a ($-\varepsilon)$-convex polygon, a polygon that remains convex even if its vertices are all displaced in arbitrary directions by a distance of $\varepsilon$ or less. The thesis develops robust algorithms for testing point inclusion in both kinds of polygons, and for testing a polygon's degree of convexity. Finally, the thesis investigates the existence of approximate hulls for sets of points. It proves that for every point set there exists a ($-\varepsilon$)-convex polygon H such that every point of the set is at most 4$\varepsilon$ away from H, and it describes robust algorithms for computing such hulls.
259 citations
TL;DR: A class of surface patch representations that unify and generalize triangular and tensor product Bézier surfaces by allowing patches to be defined over any convex polygonal domain; hence, S-patches may have any number of boundary curves.
Abstract: In this paper we introduce a class of surface patch representations, called S-patches, that unify and generalize triangular and tensor product Bezier surfaces by allowing patches to be defined over any convex polygonal domain; hence, S-patches may have any number of boundary curves. Other properties of S-patches are geometrically meaningful control points, separate control over positions and derivatives along boundary curves, and a geometric construction algorithm based on de Casteljau's algorithm. Of special interest are the regular S-patches, that is, S-patches defined on regular domain polygons. Also presented is an algorithm for smoothly joining together these surfaces with Ck continuity.
172 citations
TL;DR: These algorithms show that the mesh computer provides significantly better solutions to a variety of area and intersection problems, including the all-nearest neighbor problems for points and for sets of points.
Abstract: Asymptotically optimal parallel algorithms are presented for use on a mesh computer to determine several fundamental geometric properties of figures. For example, given multiple figures represented by the Cartesian coordinates of n or fewer planar vertices, distributed one point per processor on a two-dimensional mesh computer with n simple processing elements, Theta (n/sup 1/2/>or=-time algorithms are given for identifying the convex hull and smallest enclosing box of each figure. Given two such figures, a Theta (n/sup 1/2/>or=-time algorithm is given to decide if the two figures are linearly separable. Given n or fewer planar points, Theta (n/sup 1/2/>or=-time algorithms are given to solve the all-nearest neighbor problems for points and for sets of points. Given n or fewer circles, convex figures, hyperplanes, simple polygons, orthogonal polygons, or iso-oriented rectangles, Theta (n/sup 1/2/>or=-time algorithms are given to solve a variety of area and intersection problems. Since any serial computer has worst-case time of Omega (n) when processing n points, these algorithms show that the mesh computer provides significantly better solutions to these problems. >
124 citations
01 Jan 1989
82 citations
TL;DR: This work considers the problem of finding a polygon nested between two given convex polygons that has a minimal number of vertices, and presents an O(n log k) algorithm for solving the problem, where n is the total number of Vertices of the given polygons.
Abstract: We consider the problem of finding a polygon nested between two given convex polygons that has a minimal number of vertices. Our main result is an O(n log k) algorithm for solving the problem, where n is the total number of vertices of the given polygons, and k is the number of vertices of a minimal nested polygon. We also present an O(n) sub-optimal algorithm, and a simple O(nk) optimal algorithm.
63 citations
62 citations
TL;DR: In this article, the authors present a survey of the classification theorems involving highly symmetric tilings by regular polygons and give drawings of these tilings, many of which were not shown in the original papers.
Abstract: Several classification theorems involving highly symmetric tilings by regular polygons have been established recently. This paper surveys that work and gives drawings of these tilings—many of which were not shown in the original papers. Included are all tilings with at most three symmetry classes (orbits) of tiles, vertices or edges and those tilings which satisfy certain homogeneity criteria; i.e. tilings where locally congruent portions of the tiling are always equivalent under a global symmetry of the tiling.
62 citations
TL;DR: It is shown that the entropy of a simple polygon is maximal if and only if it is convex and a similar but less computationally burdensome measure is also proffered.
54 citations
Patent•
31 Oct 1989
TL;DR: In this paper, a non-repetitive modular design is created by assembling a plurality of substantially identical modules to cover a surface, and the appearance of the resulting tiles can be varied by changing the orientation of one or more of the modules.
Abstract: This invention is a method making a non-repetitive modular design. The design is created by assembling a plurality of substantially identical modules to cover a surface. Each module has the shape of a polygon, especially a regular polygon, such as a square. The design of each module is created in the following manner. First, one selects a set of points, disposed symmetrically around the midpoint of a side of the polygon, and duplicates the same pattern of points for the remaining sides. Then, one connects every pair of points with a line, such that the lines so drawn form a pattern which is not symmetrical around any imaginary straight line joining any pair of vertices of the polygon. The spaces between lines, or between one or more lines and one or more sides of the polygon, can be filled in with a color, or with any other design element. To make the final design, one provides a plurality of such modules, and arranges them, with random orientations, to cover a surface. The design is non-repetitive, and any orientation of the individual modules will produce a valid design. The appearance of the design is varied by changing the orientation of one or more of the modules. In general, the appearance of the overall design is quite different from that of each of the modules. The modules made according to the invention can be used as floor tiles, or they can be otherwise secured permanently to a solid substrate for decorative purposes.
40 citations
TL;DR: It is shown that the polygon covering problem can be reduced to the problem of covering a weakly triangulated graph with a minimum number of cliques.
Abstract: We consider the problem of covering simple orthogonal polygons with convex orthogonal polygons. In the case of horizontally or vertically convex polygons we show that the polygon covering problem can be reduced to the problem of covering a permutation graph with minimum number of cliques. In general, orthogonal polygons can have concavities (dents) with four possible orientations. In the case where the polygon has three dent orientations, we show that the polygon covering problem can be reduced to the problem of covering a weakly triangulated graph with a minimum number of cliques. Since weakly triangulated graphs are perfect, we obtain the following duality relationship: the minimum number of orthogonally convex polygons needed to cover an orthogonal polygon P with at most three dent orientations is equal to the maximum number of points of P, no two of which can be contained together in an orthogonally convex covering polygon. Finally, we show that in th case of orthogonal polygons with all four dent orientations, the above duality relationship fails to hold.
37 citations
TL;DR: In this article, it was shown that for every smooth curve in Rn, there is a quadrilateral with equal sides and equal diagonals whose vertices lie on the curve.
Abstract: We show that for every smooth curve in Rn, there is a quadrilateral with equal sides and equal diagonals whose vertices lie on the curve. In the case of a smooth plane curve, this implies that the curve admits an inscribed square, strengthening a theorem of Schnirelmann and Guggenheimer. “Smooth” means having a continuously turning tangent. We give a weaker condition which is still sufficient for the existence of an inscribed square in a plane curve, and which is satisfied if the curve is convex, if it is a polygon, or (with certain restrictions) if it is piecewise of class C1. For other curves, the question remains open.
TL;DR: A solution method, based on nonlinear programming techniques, is presented for the problem of matching two series of observations of a 2-D contour, in which each sampling of the contour has different hidden parts.
01 Jan 1989
TL;DR: In this paper, the authors discuss the irregularities of point distribution relative to Convex Polygons and the problem of finding the right point distribution for a set of graphs.
Abstract: 1. Irregularities of Point Distribution Relative to Convex Polygons.- 2. Balancing Matrices with Line Shifts II.- 3. A Few Remarks on Orientation of Graphs and Ramsey Theory.- 4. On a Conjecture of Roth and Some Related Problems I.- 5. Discrepancy of Sequences in Discrete Spaces.- 6. On the Distribution of Monochromatic Configurations.- 7. Covering Complete Graphs by Monochromatic Paths.- 8. Canonical Partition Behavior of Cantor Spaces.- 9. Extremal Problems for Discrepancy.- 10. Spectral Studies of Automata.- 11. A Diophantine Problem.- 12. A Note on Boolean Dimension of Posets.- 13. Intersection Properties and Extremal Problems for Set Systems.- 14. On an Imbalance Problem in the Theory of Point Distribution.- 15. Problems.
05 Jun 1989
TL;DR: This method combines a new notion of Delaunay triangulation for points and edges with the well-known functions based on Davenport-Schinzel sequences producing an almost quadratic algorithm for the problem.
Abstract: Given a convex polygon P and an environment consisting of polygonal obstacles, we find the largest similar copy of P that does not intersect any of the obstacles. Allowing translation, rotation, and change-of-size, our method combines a new notion of Delaunay triangulation for points and edges with the well-known functions based on Davenport-Schinzel sequences producing an almost quadratic algorithm for the problem. Namely, if P is a convex k-gon and if Q has n corners and edges then we can find the placement of the largest similar copy of P in the environment Q in time O(k4n l4(kn) log n), where l4 is one of the almost-linear functions related to Davenport-Schinzel sequences. If the environment consists only of points then we can find the placement of the largest similar copy of P in time O(k2n l3(kn) log n).
01 Apr 1989-Graphical Models \/graphical Models and Image Processing \/computer Vision, Graphics, and Image Processing
TL;DR: The visual potential (also called the aspect graph) is a mathematical representation of the stable views of a polygon and the abrupt changes in stable views brought about by moving the viewpoint and an optimal algorithm is obtained for finding the maximal aspects.
Abstract: We examine the visual potential of a single convex polygon with n faces. The visual potential (also called the aspect graph) is a mathematical representation of the stable views of a polygon and the abrupt changes in stable views brought about by moving the viewpoint. Stable views, aspects , are described by the topological structure-incidence relations-of the visible vertices and faces which are projected onto the retina of a cyclopian eye. While the metrical structure may vary, the topological structure is invariant to most small changes in viewpoint. Abrupt changes in aspect, visual events , correspond to the appearance and disappearance of vertices and faces. For a convex polygon visual events occur when the viewpoint crosses any line containing a face of the polygon. These lines partition the space around the object into regions. All viewpoints within a region see the same aspect. Aspects and visual events correspond respectively to the nodes and edges of a graph that represents the visual potential. A given visual potential describes a class of object instances which have the same shape or surface topology. When metrical attributes such as scale, length, and angle are added to the class description a particular instance of the object class is realized. We examine the local and global structure of the visual potential and prove that a cylindrical chain link fence can serve as its underlying manifold. In addition we obtain an optimal algorithm, O ( n ), for finding the maximal aspects , a set of aspects which encode the entire visual potential, and an O ( n 2 ) algorithm for generating the entire visual potential from these maximal aspects. We also give bounds on the number of isomorphism classes (i.e., number of shapes) for convex polygons with a fixed number of faces. We discuss realizability of visual potentials, relationship to other representations, extension to three dimensions, and an extension for a class of non-convex polygons.
05 Jun 1989
TL;DR: A data structure is built that lets us in O (log n ) time determine the range of directions in which the robot can move from a query point to the goal in a single step, and can solve single-step problems allowing uncertainty in control and position sensing.
Abstract: We consider motion planning under the compliant motion model, in which a robot directed to walk into a wall may slide along it. We examine several variants of compliant motion planning for a point robot inside a simple polygon with n sides, where the goal is a fixed vertex or edge. For the case in which the robot moves with perfect control, we build a data structure that lets us in O(log n) time determine the range of directions in which the robot can move from a query point to the goal in a single step. This structure lets us solve a variety of other problems: we can find a similar query data structure for multi-step paths; we can solve single-step problems allowing uncertainty in control and position sensing; and we can explicitly compute the set of all points that can reach the goal in a single step, even allowing uncertainty in control. Our algorithms run in O(n log n) time and linear space; they use a novel method for maintaining convex hulls of simple paths that may be of independent interest.
TL;DR: This approach incorporates the techniques of Thomas, Monsky, and Mead, in particular, the use of Sperner's lemma and non-Archimedean valuations, but also makes use of affine transformations to distort a given regular polygon into one to which those techniques apply.
Abstract: This paper answers the question, "If a regular polygon withn sides is dissected intom triangles of equal areas, mustm be a multiple ofn?" Forn=3 the answer is "no," since a triangle can be cut into any positive integral number of triangles of equal areas. Forn=4 the answer is again "no," since a square can be cut into two triangles of equal areas. However, Monsky showed that a square cannot be dissected into an odd number of triangles of equal areas.
We show that ifn is at least 5, then the answer is "yes." Our approach incorporates the techniques of Thomas, Monsky, and Mead, in particular, the use of Sperner's lemma and non-Archimedean valuations, but also makes use of affine transformations to distort a given regular polygon into one to which those techniques apply.
TL;DR: This algorithm yields a linear time solution to the symmetric all-farthest neighbors problem for simple polygons, thereby settling an open question raised by Toussaint in 1983.
TL;DR: This work develops a method of analysis based on dent diagrams for orthogonal polygons and is able to show that Keil's O(n2) algorithm for covering horizontally convex polygons is asymptotically optimal, but it can be improved to O( n) for counting the number of polygons required for a minimal cover.
05 Jun 1989
TL;DR: A linear algorithm is developed that determines a linear number of elementary steps to deform a regular polygon into any other regularpolygon with the same winding number.
Abstract: We consider a discrete version of the Whitney-Graustein theorem concerning regular equivalence of closed curves. Two regular polygons P and P', i.e. polygons without overlapping adjacent edges, are called regularly equivalent if there is a continuous one-parameter family Ps, O ≥ s ≤ 1 of regular polygons with Po = P and P1 = P'. Geometrically the one-parameter family is a kink-free deformation transforming P into P'. The winding number of a polygon is a complete invariant of its regular equivalence class. We develop a linear algorithm that determines a linear number of elementary steps to deform a regular polygon into any other regular polygon with the same winding number.
TL;DR: In this article, the authors considered taxicab geometry using the isometric grid and derived a distance function for the two points A = (xi, Y1) and B = (x2, Y2) of a square grid.
Abstract: As anyone who has played with a bit of screen wire knows, a square is not a rigid geometrical figure. A slight push to the right or left will deform a square into a nonsquare rhombus. If such a transformation is performed on the points of a plane which has been coordinatized by a square grid so that the positive y-axis makes an angle of 600 with the positive x-axis and the shorter diagonals of each rhombus are drawn, an isometric grid results. Points will still be named by ordered pairs of real numbers with respect to the x-axis and the transformed y-axis (FIGURE la). There are only three regular polygons which will tessellate the plane: the equilateral triangle, the square, and the regular hexagon. The first two of these can be subdivided into smaller similar polygons (yielding the isometric grid and the square grid). Work has already been done on taxicab geometry using the square grid; this note considers taxicab geometry using the isometric grid. Square-taxi geometry arises because, for the two points A = (xi, Y1) and B = (x2, Y2), a new distance function is chosen:
05 Jun 1989
TL;DR: This paper presents a strategy that discovers the exact shape of a simple polygon with no colinear edges by means of at most 3 probes, which is shown to be optimal in the worst-case.
Abstract: We show, in this paper, how one can probe a class of non convex polyhedra and scenes of disjoint such polyhedra. A polyhedron of that class has convex faces; any two faces are not coplanar and any two edges are not colinear. The basic step of our method is a strategy for probing a single simple polygon with no colinear edges. When each probe outcome consists of a contact point and the normal to the object at the point, we present a strategy that discovers the exact shape of a simple polygon with no colinear edges by means of at most 3n - 3 probes, which is shown to be optimal in the worst-case. This strategy can be extended to probe a family of disjoint polygons. It can also be applied in the supporting planes of the faces of a scene of polyhedra of the class above. If the scene consists of k polyhedra with altogether n faces, we show that 8n2 - 6n + k probes are sufficient to discover the exact shapes of the polyhedra.
30 Oct 1989
TL;DR: In this paper, the complete graph induced by a set of 2n points on the boundary of a polygon is considered, and edges are assigned weights equal to the Euclidean distance between their endpoints if the endpoints see each other in the polygon, and + infinity otherwise.
Abstract: The complete graph induced by a set of 2n points on the boundary of a polygon is considered. The edges are assigned weights equal to the Euclidean distance between their endpoints if the endpoints see each other in the polygon, and + infinity otherwise. An O(n log n)-time algorithm is obtained for finding a minimum-weight perfect matching in this graph if the polygon is convex, and an O(n log/sup 2/n)-time algorithm if the polygon is simple but nonconvex. The assignment problem for a convex polygon is solved in time O(n log n), and O(n alpha (n)) and O(n alpha (n) log n) time bounds are obtained for the verification problem on convex and nonconvex polygons, respectively, where alpha (n) is the functional inverse of the Ackermann function. >
17 Aug 1989
TL;DR: In this paper, the authors consider the piecewise curvi-linear or CL-environment and show that various techniques for computing Voronoi diagrams and for motion planning generalize in a satisfactory way: there is no asymptotic increase in complexity when the algebraic complexity is kept constant.
Abstract: Most motion planning work assumes the piecewise-linear or PL-environment. Here we consider the piecewise curvi-linear or CL-environment. We describe extensions to current techniques suitable for the CL-environment. In particular, we show that various techniques for computing Voronoi diagrams and for motion planning generalize in a satisfactory way: there is no asymptotic increase in complexity when the algebraic complexity is kept constant. An underlying premise of our approach is that in the CL-environment, convex chains play the role of convex polygons in PL-environments.
TL;DR: An Ω( n log n ) lower bound time complexity in the computational tree model for the maximum area and maximum perimeter k -gon problems is demonstrated using a reduction to the set-disjointness problem and is based on extremal properties of regular polygons.
14 May 1989
TL;DR: It is shown that 3n-3 probes are necessary and sufficient to discover the exact shape of a polygon with n noncollinear edges.
Abstract: The authors present a strategy for discovering the exact shape of a simple (but not necessarily convex) polygon by means of a minimal number of simple probes. When each probe outcome consists of a contact point, a ray measuring that point and the normal to the object at the point, it is shown that 3n-3 probes are necessary and sufficient to discover the exact shape of a polygon with n noncollinear edges. Each probe can be determined in O(log n) time, yielding on O(n log n)-time O(n)-space algorithm. >
TL;DR: In this paper, it was shown that a centrally symmetric regular polygon cannot be cut into an odd number of triangles of equal areas, even if the restriction "regular" is removed from the hypothesis.
Abstract: In 1970 Monsky proved that a square cannot be cut into an odd number of triangles of equal areas. In 1988 Kasimatis proved that if a regularn-gon,n ⩾ 5, is cut intom triangles of equal areas, thenm is a multiple ofn. These two results imply that a centrally symmetric regular polygon cannot be cut into an odd number of triangles of equal areas. We conjecture that the conclusion holds even if the restriction “regular” is deleted from the hypothesis and prove that it does forn = 6 andn = 8.
14 May 1989
TL;DR: A CAD-based grasp-synthesis system has been developed that precomputes valid, stable grasp sites on CAD models for use by a robot system, and a method of computing the rotational stability of edge contacts is given.
Abstract: A CAD-based grasp-synthesis system has been developed that precomputes valid, stable grasp sites on CAD models for use by a robot system. Extended Gaussian images, which are efficient data structures for mapping surfaces into Gaussian space based on surface normals, are shown. A modified version is introduced which allows the mapping of polygon, edge, and vertex normals. Using the modified extended Gaussian image, pairs of parallel surfaces are found in linear time. Pairs of graspable surfaces include polygon-polygon, polygon-edge, and polygon-point. A method of computing the rotational stability of edge contacts is given. The stability of each grasp point and its orientation are computed, and all grasps are ranked in descending order of stability. The grasp-synthesis program is demonstrated on three models, one concave and two convex; all three types of grasps are shown. >
TL;DR: In this paper, the robust stability problem for a class of uncertain neutral time-delay systems where the characteristic equations involve a polytope P of quasipoly nounials of neutral type was considered.
23 May 1989
TL;DR: The authors present a technique for translation-invariant binary convex polygon shape recognition based on a morphological shape decomposition that uses triangular shape primitives from the decomposition of convex shapes as features for shape recognition.
Abstract: The authors present a technique for translation-invariant binary convex polygon shape recognition based on a morphological shape decomposition. The triangular shape primitives from the decomposition of convex shapes are used as features for shape recognition. The shape primitives are smaller and simpler than the templates of shapes; thus they are more efficient in representing shapes for discrimination. Maximum entropy reduction is used as an optimization criterion for selecting features from among the shape primitives at each node of a decision tree. Experiments on the classification of ten classes of noisy polygon shapes, where five replications per class were used for training and fifty replications per class were used for testing, achieved a recognition rate of 98.80% on the test set. >