Showing papers on "Polygon published in 1968"
TL;DR: Generating functions are obtained, specialized to count those 2-trees embeddable in the plane, thus providing a new approach to the old problem of determining the number of triangulations of a polygon.
Abstract: The higher dimensional concepts corresponding to trees are developed and studied. In order to enumerate these 2-dimensional structures called 2-trees, a dissimilarity characteristic theory is investigated. By an appropriate application of certain combinatorial techniques, generating functions are obtained for the number of 2-trees. These are specialized to count those 2-trees embeddable in the plane, thus providing a new approach to the old problem of determining the number of triangulations of a polygon.
95 citations
TL;DR: This paper considers halfplanes as primitives and their intersections form convex polygons and the unions of the latter are arbitrary polygons, which can be then considered as approximations for arbitrary plane figures.
62 citations
Patent•
03 Sep 1968
TL;DR: In this paper, a planar blank is used for constructing a solid manifold, where each blank has at least one polygonal shaped section and the other ends of the sections are terminated at a cIRCULAR opening.
Abstract: A BLANK FOR CONSTRUCTING SOLID FORMS. THE BLANK COMPRISES A STIFF PLANAR SHEET. THE SHEET HAS AT LEAST ONE POLYGONALLY SHAPED SECTION. EACH OF THE SECTIONS HAS A PLURALITY OF SLITS IN ACCORDANCE WITH THE NUMBER OF CORNERS OF THE POLYGON. EACH OF THE SLITS EMANATES FROM THE CENTER OF A POLYGON SECTION AND EXTENTS OUTWARDLY TOWARDS ONE OF THE CORNERS OF THE SECTIONS. THE OTHER ENDS OF THE SLITS ARE TERMINATED AT A CIRCULAR OPENING. THE BLANK HAS FASTENING ELEMENTS ON EACH OF ITS FREE EDGES WHICH ARE ADAPTED TO FACILIATE CONSTRUCTION OF A SOLID FORM. THE SLITS ENABLE THE SECUREMENT OF THE SOLID FORM TO ANOTHER SOLID FORM HAVING POLYGONALLY SHAPED SECTIONS WITH SIMILARLY FORMED SLITS.
57 citations
13 citations
TL;DR: A technical description of a Danish designed IUD the polygon (Antigon) is given and clinical experiences are related of its 2 years of use in Denmark and a modification of the Antigon has been constructed.
7 citations
Patent•
05 Jul 1968TL;DR: In this paper, an image of a polygon is formed on the imaging medium by recording a number of smaller adjacent polygonal images which integrally form the image of the overall polygon.
Abstract: A process for recording a solid polygonal image on an imaging medium. An artwork tool or apparatus illustrated as an automatic drafting system with an imaging instrument shown as a photoexposure head is employed. The photoexposure head is capable of recording images of different shapes and widths. According to the process, an image of the polygon is formed on the imaging medium by recording a number of smaller adjacent polygonal images which integrally form the image of the overall polygon. The polygon to be recorded is partitioned into a number of integral parts. The time required by the artwork tool to form the image is reduced by selecting the integral parts as rectangles whenever practical so that major portions of the polygonal image can be formed with the widest available images which will fit within a particular rectangle.
3 citations
Patent•
14 Feb 1968
TL;DR: In this paper, a collapsible tube having a metal nozzle, surrounded by a plastics sleeve, is modified in that the nozzle is of a polygonal external shape as viewed in horizontal cross-section.
Abstract: 1,102,863. Collapsible tubes. BETTS & CO. Ltd. 9 June, 1965 [10 June, 1964], No. 24011/64. Addition to 1,045,251. Heading B8D. A collapsible tube having a metal nozzle, surrounded by a plastics sleeve as in Specification 1,045,251, is modified in that the nozzle is of a polygonal external shape as viewed in horizontal cross-section. The polygon may have any number of sides from three to eight.
2 citations
TL;DR: In this paper, the gravitational attraction of a horizontal right-angled polygon is calculated by means of a special line-integral method based on the introduction of a generating function ϑ defined by a derivative.
2 citations
1 citations
01 Jan 1968
TL;DR: In this article, the Fourier series and the isoperimetric problem are discussed, and it is shown that the circle is better than any polygon, even polygonal ones.
Abstract: Publisher Summary This chapter discusses the Fourier series and the isoperimetric problem. The isoperimetric problem is to find the closed curve of a given arc length that encloses the greatest area. The arc length, s, must be measured in a fixed direction from any fixed point on a given curve. Then the coordinates of the general point on the curve can be expressed parametrically in terms of s. Because the curve is closed and has arc length l, both x and y are periodic functions of s with period l. This reasoning shows that the circle is better than any polygon. It is clear, though, that arguments of this type will not solve problem. The restrictions imposed are not really significant, although they may appear so.