Showing papers on "Regular polygon published in 1984"

Area cutting method

Abstract: The invention relates to an area cutting method in which a tool is moved along a plurality of paths (CP3-CPo, LP5-LP1) successively offset in an inward direction from a closed curve (OFC) specifying an area (AR), or in other words, in which the tool is moved in a spider web-like pattern, to cut the interior of the area. More particularly, the invention relates to an area cutting method in which, when an area is divided into a plurality of convex polygons, the number of cutting paths in each convex polygonal portion is decided in dependence upon the size of the convex polygon and the cut-in pitch (path interval) of each convex polygonal portion is made substantially the allowable cut-in pitch.

Efficient nesting of congruent convex figures

Abstract: Optimal nesting is the arrangement of two- dimensional polygons within a rectangular board so that waste is minimized. Our approach follows a top-down, stepwise refinement of the original problem into simpler subproblems; the combined solution of all of the problems permits the solution of the original problem. We first show that the original problem can be decomposed into two subproblems--one consisting of finding all convex paver polygons and the other of optimal (minimal waste) circumscription of the original figure in the most appropriate paver polygon.

Bounds on the Length of Convex Partitions of Polygons

Abstract: A heuristic for partitioning rectilinear polygons into rectangles, and polygons into convex parts by drawing lines of minimum total length is proposed. For the input polygon with n vertices, k concave vertices and the perimeter of length p, the heuristic draws partitioning lines of total length O(plogk) and runs in time O(nlogn). To demonstrate that the heuristic comes close to optimal in the worst case, a uniform family of rectilinear polygons Qk with k concave vertices, k=1, 2, ... and a uniform family of polygons Pk with k concave vertices, k=1, 2, ... are constructed such that any rectangular partition of Qk has (total line) length Ω(plogk), and any convex partition of Pk has length Ω(plogk/loglogk). Finally, a generalization of the heuristic for minimum length of convex partition of simple polygons to include polygons with polygonal holes is given.

Space sweep solves intersection of convex polyhedra

Abstract: Plane-sweep algorithms form a fairly general approach to two-dimensional problems of computational geometry. No corresponding general space-sweep algorithms for geometric problems in 3- space are known. We derive concepts for such space-sweep algorithms that yield an efficient solution to the problem of solving any set operation (union, intersection, ...) of two convex polyhedra. Our solution matches the best known time bound of O(n log n), where n is the combined number of vertices of the two polyhedra.

An optimal algorithm for computing the minimum vertex distance between two crossing convex polygons

Abstract: LetP={p 1 ,p 2 , ...,p m } andQ={q 1 ,q 2 , ...,q n } be two intersecting convex polygons whose vertices are specified by their cartesian coordinates in order. An optimalO(m+n) algorithm is presented for computing the minimum euclidean distance betweena vertexp i inP and a vertexq j inQ.

Space sweep solves intersection of two convex polyhedra elegantly

Abstract: Plane-sweep algorithms form a fairly general approach to two-dimensional problems of computational geometry. No corresponding three-dimensional space-sweep algorithms for geometric problems in 3-space are known, however. We derive concepts for such space-sweep algorithms that yield an elegant solution to the problem of solving any set operation (union, intersection, ...) of two convex polyhedra. Moreover, our solution matches the best known time bound of O(n log n) where n is the combined number of corners of the two polyhedra.

Covering Polygons with Minimum Number of Rectangles

Abstract: In the fabrication of masks for integrated circuits, it is desirable to replace the polygons comprising the layout of a circuit with as small as possible number of rectangles Let Q be the set of all simple polygons with interior angles ≥ 90 degrees Given a polygon P e Q, let ϑ(P) be the minimum number of (possibly overlapping) rectangles lying within P necessary to cover P, and let r(P) be the ratio between the length of the longest edge of P and the length of the shortest edge of P For every natural n ≥ 5, and k, a uniform polygon P n,k with n corners is constructed such that r(P n,k ) ≥ k and ϑ(P n ) ≥ ω(nloglog(r(P n,k ))) On the other hand, by modifying a known heuristic it is shown that for all convex polygons P in Q with n vertices ϑ(P) ≤ O(nlog(r(P)))

Characteristic impedance of coaxial lines bounded by N-regular polygons

Abstract: A very simple equation based on conformal mapping and domain variational theory is given for the characteristic impedance of a coaxial system consisting of an N-regular polygon and a circle.

Measuring method of working error in precise work

Abstract: PURPOSE:To measure easily and exactly a working error in a precise work by comparing a theoretical Gantt chart of a regular polygon with a product Gantt chart of a regular polygon. CONSTITUTION:Centroid coordinates Gx, Gy, an area, a Fourier spectrum and a circumferential length of a product Gantt chart 2 are derived. In the same way, centroid coordinates, an area, a Fourier spectrum and a circumferential length of a theoretical Gantt chart are derived. Subsequently, a position shift value of a product pattern is derived by comparing two centroid coordinates, and a magnification error value of the product pattern is derived by comparing two areas. Next, a rotational error value of the product pattern is derived by comparing two Fourier spectrums, and a graphic deformation rate of the product pattern is derived by comparing ratios of two circumferential lengths and areas. By processing in this way, a working error in a precise work can be measured easily and exactly.

A shell-fish trap

Abstract: A collapsible trap especially for shell-fish comprises a top plate (11), a bottom plate (12), netting walls (14) and distance members (13) extending between said plates. The top and bottom plates are each designed as a regular polygon with said distance members (13) arranged at the corners of the polygons. The plates (11, 12) are turnable with respect to each other a distance corresponding to the length of a polygon side at which said distance members (13) are brought from a position substantially perpendicular to the plates (11, 12), which in this position are spaced from each other to a position substantially in parallel with said plates, which in this position are brought close to each other. The trap can be provided with as many entrance openings (24) as sides in the polygon.

1/N expansion for Dirichlet boundary problems

Abstract: Dirichlet problems posed by boundaries with a symmetry CN are studied via 1/N expansion. The characteristic impedance of a coaxial system consisting of a regular polygon concentric with a circle is explicitly computed as a typical application of the method.

Generating an integrated graphic display

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. Muftiple, 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.

On domains convex in several directions

Abstract: We discuss a new class of simply connected domains between boundary rotation at most 2(α+1)π and close-to-convex of type α. These domains are convex in a certainset of directions.

Regular Polyhedral Manifolds

Abstract: Publisher Summary A polyhedral 2-manifold is a geometric model of an abstract 2-manifold made up of convex polygons. This chapter presents closed oriented polyhedral 2-manifolds in the ordinary Euclidean E 3 . A polyhedral 2-manifold is regular or equivelar if each of its facets has the same number p of edges and if each of its vertices belongs to the same number q of edges.