Showing papers on "Regular polygon published in 1983"

Solving the find-path problem by good representation of free space

Abstract: Free space is represented as a union of (possibly overlapping) generalized cones. An algorithm is presented which efficiently finds good collision-free paths for convex polygonal bodies through space littered with obstacle polygons. The paths are good in the sense that the distance of closest approach to an obstacle over the path is usually far from minimal over the class of topologically equivalent collision-free paths. The algorithm is based on characterizing the volume swept by a body as it is translated and rotated as a generalized cone, and determining under what conditions one generalized cone is a subset of another.

Solving geometric problems with the rotating calipers

Abstract: Shamos [1] recently showed that the diameter of a convex n-sided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once. In this paper we show that this simple idea can be generalized in two ways: several sets of calipers can be used simultaneously on one convex polygon, or one set of calipers can be used on several convex polygons simultaneously. We then show that these generalizations allow us to obtain simple O(n) algorithms for solving a variety of problems defined on convex polygons. Such problems include (1) finding the minimum-area rectangle enclosing a polygon, (2) computing the maximum distance between two polygons, (3) performing the vector-sum of two polygons, (4) merging polygons in a convex hull finding algorithms, and (5) finding the critical support lines between two polygons. Finding the critical support lines, in turn, leads to obtaining solutions to several additional problems concerned with visibility, collision, avoidance, range fitting, linear separability, and computing the Grenander distance between sets.

Some NP-hard polygon decomposition problems

Abstract: The inherent computational complexity of polygon decomposition problems is of theoretical interest to researchers in the field of computational geometry and of practical interest to those working in syntactic pattern recognition. Three polygon decomposition problems are shown to be NP-hard and thus unlikely to admit efficient algorithms. The problems are to find minimum decompositions of a polygonal region into (perhaps overlapping) convex, star-shaped, or spiral subsets. We permit the polygonal region to contain holes. The proofs are by transformation from Boolean three-satisfiability, a known NP-complete problem. Several open problems are discussed.

Traditional Galleries Require Fewer Watchmen

Abstract: Chvatal’s watchman theorem shows if the walls of an art gallery form an n-sided polygon then at most $[ n /3 ]$ watchmen are needed to guard it, and that this number is best possible. In this paper it is shown that if every pair of adjacent sides of the polygon form a right angle then at most $[ n / 4 ]$ guards are needed, and again this result is best possible. Our proof depends on showing that any finite region bounded by a finite number of edges, each of which lies parallel to one of a fixed pair of perpendicular axes, has a partition into convex quadrilaterals.

Generating an integrated graphic display of the safety status of a complex process plant

Abstract: The real time actual and reference values of parameters pertinent to the key safety concerns of a pressurized water reactor nuclear power plant are used to generate an integrated graphic display representative of the plant safety status. This display is in the form of a polygon with the distances of the vertices from a common origin determined by the actual value of the selected parameters normalized such that the polygon is regular whenever the actual value of each parameter equals its reference value despite changes in the reference value with operating conditions, and is an irregular polygon which visually indicates deviations from normal otherwise. The values of parameters represented in analog form are dynamically scaled between the reference value and high and low limits which are displayed as tic marks at fixed distances along spokes radiating from the common origin and passing through the vertices. Multiple, related binary signals are displayed on a single spoke by drawing the associated vertice at the reference value when none of the represented conditions exist and at the high limit when any such condition is detected. A regular polygon fixed at the reference values aids the operator in detecting small deviations from normal and in gauging the magnitude of the deviation. One set of parameters is selected for generating the display when the plant is at power and a second set reflecting wide range readings is used the remainder of the time such as following a reactor trip. If the quality of the status, reference or limit signals associated with a particular vertice is "bad", the sides of the polygon emanating from that vertice are not drawn to appraise the operator of this condition.

Optimal Algorithms for the Intersection and the Minimum Distance Problems Between Planar Polygons

Abstract: Two planar geometric problems relating to a convex n-gon P and a simple nonconvex m-gon Q are considered.

The Dirichlet problem for the equation of prescribed Gauss curvature

Abstract: We treat necessary and sufficient conditions for the classical solvability of the Dirichlet problem for the equation of prescribed Gauss curvature in uniformly convex domains in Euclidean n space. Our methods simultaneously embrace more general equations of Monge-Ampere type and we establish conditions which ensure that solutions have globally bounded second derivatives.

Sur les equations de Monge-Ampere. I

Abstract: In this paper we study the real Monge-Arapere equations: det(D2u)= f(x) in 0, u convex in 0, u=0 on ∂0, and we introduce a new method for solving these equations which enables us to show the existence of regular solutions. This method uses only p.d.e. techniques and does not use any geometrical results. Furthermore, it enables us to solve quasilinear Monge-Ampere equations.

On the X-Y convex hull of a set of X-Y polygons

Abstract: We study the class of rectilinear polygons, calledX – Y polygons, with horizontal and vertical edges, which are frequently used as building blocks for very large-scale integrated (VLSI) circuit layout and wiring. In the paper we introduce the notion of convexity within the class ofX – Y polygons and present efficient algorithms for computing theX – Y convex hulls of anX – Y polygon and of a set ofX – Y polygons under various conditions. Unlike convex hulls in the Euclidean plane, theX – Y convex hull of a set ofX – Y polygons may not exist. The condition under which theX – Y convex hull exists is given and an algorithm for testing if the given set ofX – Y polygons satisfies the condition is also presented.

The almost fixed point property for nonexpansive mappings

Abstract: It is shown that a closed convex subset of a reflexive Banach space has the almost fixed point property for nonexpansive mappings if and only if it is linearly bounded. Let C be a closed convex subset of a Banach space (E, I I). Recall that a mapping T: C --E is said to be nonexpansive if I Tx Ty I d2 for all n > 1. (Note that by Banach's fixed point theorem, the accretive operator I T does indeed satisfy the range condition.) It is also known [10, Proposition 4.3] that limn 00 I T'x I/n = d. Let a subsequence of {T'x/n} converge weakly to w. Clearly Received by the editors May 18, 1982. Presented to the Society by title (see Abstracts Amer. Math. Soc. 3 (1982), 402). 1980 Mathematics Subject Classification. Primary 47H09.

Non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons

Abstract: This paper proves the existence of non-periodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons. For anyn≧3 there exists a corresponding convexn-agon (forn=3 this will be a right triangle with a small acute angle), while in three-dimensional space it will be a prism, then-agon serving as the base. The results are applied for investigating a mechanical system of two absolutely elastic balls on a segment, and also for proving the existence of an infinite number of periodic trajectories in the given polygons. The exchange transformation of two intervals is used for proving the theorems. An arbitrary exchange transformation of any number of intervals can also be modeled by a billiard trajectory in some convex polygon with many sides.

Stochastic iteration for a constrained optimization problem

Abstract: A primal dual method of Kushner and Sanvicente for a constrained optimization problem with convex regression functions is investigated without a priori bounds. For the stochastic approximation sequence almost sure convergence to a random optimal solution and a random Kuhn-Tucker vector is shown, and for the uniqueness case, a functional central limit theorem is given.

Approximating Any Regular Polygon by Folding Paper

Abstract: By 350 B.C. the Greeks knew Euclidean constructions for the regular polygons of 4, 8, 16,... sides and for those of 3 and 5 sides-the equilateral triangle and the regular pentagon. From these it was easy to construct regular polygons of 2c x 3, 2c x 5, 2c x 3 x 5 sides, where c is any positive integer, and the Greeks in effect showed how. They got no farther. Young Gauss proved that if N is of the form 2c or 2c times a product of different Fermat primes F,,, then there is a Eucidean construction for a regular polygon of N sides. This form of N is both necessary and sufficient for the possibility of a Eucidean construction....

Shaping of the head of and the wrench for a screw

Abstract: It is proposed, to transmit a torque between a tool and a screw, a form having force application surfaces (11 to 16) shaped as convex circle arcs with the same radius. The angles ( alpha , beta ) formed by two force applying surfaces (11 to 16) are alternatingly larger and smaller than 180 , so that the convex points (17 to 19, 21 to 23) between the force applying surfaces are located alternatingly on two concentric circles (20, 24) and form the corners of a regular polygon.

Incommensurability Proofs: A Pattern that Peters Out

Abstract: The concept of incommensurable magnitudes can be traced back to the ancient Greeks. Recall that two magnitudes are incommensurable if their ratio is not given by a pair of positive integers. Our main interest here is not to make a historical study, but rather to examine the scope of a proof technique, which, though sometimes attributed to the Greeks, may well be of more recent origin. It is a method of infinite descent which can be applied to show the incommensurability of the diagonal and side of a square [2, p. 270]; [11, p. 44]; [13, p. 23]. This technique can also be employed to prove incommensurable the diagonal and a side of a regular pentagon and similarly the shortest diagonal and a side of a regular hexagon. Thus we have a method which works for three cases in a row, and we may well believe that it will apply to n-sided regular polygons for n > 6. Alas, such a hope is unfounded. While the shortest diagonal and a side of a regular n-gon may be incommensurable, the pattern that seems to emerge in establishing the result for n = 4, 5, 6 fails for n > 6. In this paper, we first discuss the reason for this failure. In doing so, we utilize some elementary algebraic facts about the roots of unity. Later, we consider the question of when the second shortest diagonal and a side of a regular polygon are incommensurable. Finally, we note the connection of these questions with the Euclidean algorithm.

Dissections of Polygons

Abstract: Publisher Summary This chapter discusses the dissections of polygons. A polygon P in the Euclidean plane E 2 is said to be dissected into the polygons P 1 , P2,…P K . The two polygons with the same area are not necessarily equivalent by dissection with respect to the group T of translations (for example, one can prove that an equilateral triangle and a square with the same area are never equivalent by dissection with respect to T). The chapter proves that the only convex rep-5 polygons are (up to a similarity) the right triangles with lengths of sides 1,2, and the parallelograms with lengths of sides 1.

Configuration of head and tool-wrench for a screw

Abstract: It is proposed, to transmit a torque between a tool and a screw, a form having force application surfaces (11 to 16) shaped as convex circle arcs with the same radius. The angles ($g(a), $g(b)) formed by two force applying surfaces (11 to 16) are alternatingly larger and smaller than 180?o, so that the convex points (17 to 19, 21 to 23) between the force applying surfaces are located alternatingly on two concentric circles (20, 24) and form the corners of a regular polygon.

Distortion in a 2-D scan pattern generated by combining a plane mirror and a regular polygon scanner

Abstract: The distortion in a 2-D scan pattern generated by a system of a plane mirror and a regular polygon scanner was investigated using the matrix formulation of reflection by plane mirrors. A particular configuration involving the two scan elements was identified which does not introduce any distortion in its 2-D scan pattern.

Arc displaying system

Abstract: PURPOSE:To optionally control the relation of surface coarseness and picture drawing speed in accordance with purpose by converting arc-shaped definition data into vector data in accordance with external specification indicating the relation of the surface coarseness and the picture drawing speed, and storing the data. CONSTITUTION:The length L0 of one side of a regular polygon approximate to an arc to be displayed is inputted from a keyboard KB 3 to a CRT 2 in a graphic display device 1 and stored in a screen control table TB6 through an I/O controlling part 5. The radius gamma and rotation angle theta of the arc are stored in an arc shape definiton data 7. A vector decomposing part 8 inputs the data 7, converts the L0 from a physical coordinate system to a logical coordinate system value l0 while referring the TB 6, calculates the center angle theta0 for one side of a regular polygon from the gamma and l0 by cosine law, finds theta/theta0 to find the number (n) of sides of the regular polygon and compensates Z an error between the end point of the n sides and that of the arc to prepare a vector data 9. A control part 5 inputs the data 9, rewrites the contents of a picture data writing area 10 and transfers the rewritten contents to the device 1 to display it at the specified coarseness.

A Geometric Application of Majorization

Abstract: We ‘know’ (intuitively?) that the area (or perimeter) of a (convex) polygon with n + 1 sides, inscribed in a circle, is greater than that of a polygon with n sides. This, of course, is not true, in general. It clearly is true for regular polygons (maybe that is all we ‘knew’ in the first place).

The decomposition of polygons

Abstract: In this paper a number of interesting results are defined and explained that are related to the notions of piecewise congruence by addition or subtraction. Two polygons are piecewise congruent by addition if either polygon can be cut into a finite number of polygonal pieces that can be rearranged to cover the other polygon. Initially, the fundamental theorem, that two polygons equal in area are also piecewise congruent by addition, is treated. Then it is shown, additionally, that there always exists a decomposition in which the sides of corresponding parts are parallel. Furthermore, for certain pairs of polygons, decompositions can be found which require only translations in relating corresponding parts. The necessary conditions for this theorem are found by using the boundary descriptions of the polygons under consideration.

Use of a computer to find the number of regular pentagons that can simultaneously touch a given one

Abstract: Around an initial regular pentagon one describes a contour L on which one introduces a measure m. One investigates the difference S(M)=1/7m(L)−m(L∩M) where M is a pentagon touching the initial one and congruent to it. The geometric part of the investigation reduces the proof of the inequality S(M)<0 for all M to the proof of the negativity of two effectively computable functions F(u,v) and G(v) in the compact domain of the variation of the arguments. By the method of demonstrative computations, one calculates on a computer the values of these functions at the nodes of a rectangular net of the domain of the variation of the arguments by taking into account the monotonicity and one estimates the computational error. The results of the computation show that we have the inequality S(M)<0, from where it follows that the desired number is equal to six.

The settling of an arbitrary number of spherical particles arranged on the corners of a regular polygon in a viscous fluid

Abstract: T i t l e o f T h e s is : The S e t t l i n g o f an A r b i t r a r y Number o f S p h e r ic a l P a r t i c l e s A rran g ed on th e C o m ers o f a R e g u la r P o lygon in a V isco u s F lu id E r ic R. B ix o n , D o c to r o f E n g in e e r in g S c ie n c e , 1983 T h e s is d i r e c t e d b y : E r n e s t N. B a r t , A s s i s t a n t P r o f e s s o r The c r e e p in g m o tion e q u a tio n h a s b e e n s o lv e d f o r th e c a se o f p l a n a r a r r a y s o f s p h e re s s e t t l i n g u n d e r th e in f lu e n c e o f g r a v i t y in a v is c o u s f l u i d . The s o lu t i o n i s a g e n e r a l s o lu t i o n w hich a p p l i e s to an a r b i t r a r y num ber o f s p h e r e s . A l l p a r t i c l e s w i l l l i e a t th e c o m e rs o f a r e g u l a r p o ly g o n . T h u s , two p a r t i c l e s s id e by s i d e , th r e e p a r t i c l e s in an e q u i l a t e r a l t r i a n g u l a r a r r a y , o r f o u r s p h e re s i n a s q u a re a r r a y w i l l be s p e c i a l c a s e s o f th e g e n e r a l s o l u t i o n . The s o l u t i o n h a s been o b ta in e d by a u n iq u e a p p l i c a t i o n o f th e m ethod o f r e f l e c t i o n s . Only a f i r s t c o r r e c t i o n to th e d ra g h a s b een o b ta in e d w h ich p u ts an a d d i t i o n a l c o n s t r a i n t on th e s o lu t i o n s in c e th e h ig h e r o r d e r te rm s h av e b een n e g le c te d . As a r e s u l t , th e s o l u t i o n i s m ost a c c u r a te when th e s p h e re s a re f a r a p a r t . I n o r d e r t o v e r i f y th e g e n e r a l s o l u t i o n f o r th e c a se o f two s p h e r e s , th e r e s u l t h a s been com pared w i th th e l i t e r a t u r e v a lu e w hich e x i s t s f o r th e c a se o f two s p h e re s f a l l i n g p e r p e n d ic u la r to t h e i r l i n e o f c e n t e r s . The s o l u t i o n o b ta in e d in t h i s w ork f o r two s p h e re s i s i n e x a c t ag reem en t w i th th e l i t e r a t u r e s o l u t i o n f o r th e two s p h e re c a s e . The r e s u l t s o f th e g e n e ra l s o l u t i o n i n d i c a t e t h a t as th e num ber o f s p h e re s i n th e a r r a y i s i n c r e a s e d , th e te r m in a l s e t t l i n g v e lo c i ty i n c r e a s e s r a p i d l y .